text:
\begin{document}

\title{On customer flows in Jackson queuing networks}
\author{Sen Tan\ and\
Aihua Xia\footnote{Corresponding author.
E-mail: xia@ms.unimelb.edu.au}\\
Department of Mathematics and Statistics\\
The University of
Melbourne\\
Parkville, VIC 3052\\
Australia}

\date {{19 January, 2010}}

\maketitle

\begin{abstract} Melamed's theorem states that for a Jackson queuing network, the equilibrium flow along a link
follows Poisson distribution if and only if no customers
can travel along the link more than once. Barbour \& Brown~(1996) considered the Poisson approximate version of Melamed's theorem by
allowing the customers a small probability $p$ of travelling along the link more than once. In this paper, we prove that the customer flow process is a Poisson cluster process and then establish a general approximate version of Melamed's theorem accommodating all possible cases of
$0\le p<1$.

\vskip12pt
\noindent\textit{Key words and phrases.} Jackson queuing network, Palm distribution, Poisson cluster process, over-disperson, Stein's method, negative binomial.

\vskip12pt
\noindent\textit{AMS 2000 subject classifications.} Primary 60G55; secondary 60F05, 60E15.
\end{abstract}

\section{Introduction}
\setcounter{equation}{0}

We consider a Jackson queuing network
with $J$ queues and the following specifications [see Barbour \& Brown~(1996) for more details]. First, we assume that customers can move from one queue to another as well as can enter and leave from any queue. We assume that the exogenous arrival processes are independent Poisson processes with rates $\nu_j$, $1\le j\le J$. Service requirements are assumed to be exponential random variables with parameter 1 and when there are $m$ customers in queue $j$,
the service effort for queue $j$ is $\phi_j(m)$, where $\phi_j(0)=0$, $\phi_j(1)>0$ and  $\phi_j(m)$ is a non-decreasing function of $m$.
Second, we define the switching process as follows.
Let $\lambda_{ij}$ be the probability that an individual moves from queue $i$ to queue $j$,
 $\mu_i$ be the exit probability from queue $i$ and it is natural to assume
 $$\sum_{j=1}^J\lambda_{ij}+\mu_i=1, \ 1\le i\le J.$$
 Without loss of generality, we may assume that the network is irreducible in the sense that all customers can access any queue with a positive probability.
 Set $\alpha_j$  as the total rate of arriving customers (including both exogenous and endogenous arrivals) to queue $j$, then the rates $\{\alpha_j\}$ satisfy
the equations
$$\alpha_j=\nu_j+\sum_{i=1}^J\alpha_i\lambda_{ij},\ 1\le j\le J$$
and they are the unique solution of the equations with $\alpha_j>0$ for all $j$.

For convenience, we define state 0 as the outside of the network, that is, the point of arrival and departure of an individual into and from the system.
We write $\cs:=\{(j,k):\ 0\le j,k\le J\}$ as the set of all possible direct links and use
 $\boldXi^{jk}$ to record the transitions of individuals moving from queue $j$ to queue $k$, then
 $\boldXi=\{\boldXi^{jk},\ 0\le j,k\le J\}$ gives a full account of customer flows in the network, where departures are transitions to 0 and arrivals are transitions from 0. If $\rho_{jk}$ is the rate of equilibrium flow along the link $(j,k)$, then  $\rho_{jk}=\alpha_j\lambda_{jk}$ and the mean measure of $\boldXi$
 is
 $$\bl(ds,(j,k))=\rho_{jk}ds,\ s\in\real,\ (j,k)\in \cs.$$

 Our interest is on the customer flows along the links in $C\subset{\cal S}$ for the time interval $[0,t]$, so we set the carrier space as $\Gamma_{C,t}=[0,t]\times C$ and use $\bXict$ to stand for the transitions along the links in $C$ for the period $[0,t]$. Then the mean measure of $\bXict$
 is
 $$\blct(ds,(j,k))=\rho_{jk}ds,\ 0\le s\le t,\ (j,k)\in C.$$

Melamed's theorem states that $\bXict$ is a Poisson process if and only if no customers travel along the links in $C$ more than once [Melamed~(1979) proved the theorem when $\phi_j(m)=c_j$ for $m\ge 1$ and $1\le j\le J$, and the general case was completed by Walrand \& Varaiya~(1981)]. Barbour \& Brown~(1996) considered the Poisson approximate version of Melamed's theorem by
allowing the customers a small probability of traveling along the links more than once. For convenience, we call the probability of customers traveling along the links in $C$ more than once as the {\it loop probability}. The bounds for the errors of Poisson process approximation are sharpened by Brown, Weinberg \& Xia~(2000) and Brown, Fackrell \& Xia~(2005) and it is concluded in these studies that the accuracy of Poisson approximation depends on how small the loop probability is.

In section~2 of the paper, we use the Palm theory, Barbour-Brown Lemma [Barbour \& Brown~(1996)] and infinite divisibility of point processes to prove that
$\bXict$ is a Poisson cluster process [Daley \& Vere-Jones~(1988), p.~243]. The characterization involves a few quantities which are generally intractable, so in section~3, we prove that $\boldXi$ is over-dispersed [see Brown, Hamza \& Xia~(1998)], i.e., its variance is greater than its mean, and conclude that suitable approximations should be those with the same property, such as the compound Poisson or negative binomial distributions. We then establish a general approximate version of Melamed's theorem for the total number of customers traveling along the links in $C$ for the period $[0,t]$ based on a suitably chosen negative binomial distribution. The approximation error is measured in terms of the total variation distance and the error bound is small when the loop probability is small [cf. the Poisson approximation error bound in Barbour \& Brown~(1996)] and/or $t$ is large with order $\frac{1}{\sqrt{t}}$ [cf. Berry--Esseen bound for normal approximation, Petrov~(1995)].

\section{A characterization of the customer flow process}
\setcounter{equation}{0}

The customer flows are directly linked to the changes of the states of the queue lengths, so we define $N_i(t)$ as the number of customers, including those in service, at queue $i$ and time $t$, $1\le i\le J$. Then
 $\{\bn(t):=(N_1(t),\dots,N_J(t)):\ t\in\real\}$ is a pure Markov jump process on state space $\{0,1,2,\dots\}^J$ with the following transition rates ${\bf n}=(n_1,\dots,n_J)$:
$$q_{{\bf n},{\bf n}+e_j}=\nu_j,\ q_{{\bf n},{\bf n}-e_j+e_k}=\phi_j(n_j)\lambda_{jk},\ q_{{\bf n},{\bf n}-e_j}=\mu_j\phi_j(n_j),\ 1\le j,k\le J,$$
where $e_j$ is the $j$th coordinate vector in $\{0,1,2,\dots\}^J$.
The stationary Markov queue length process has a unique stationary distribution: for each $t>0$, $N_j(t)$, $j=1,\dots,J$ are independent with
$$\prob(N_j(t)=k)=\frac{\alpha_j^k/\prod_{r=1}^k\phi_j(r)}{\sum_{l=0}^\infty
\alpha_j^l/\prod_{r=1}^l\phi_j(r)}.$$

Let $X_i$ be the $i$th queue visited by a given customer, then
 $\{X_i:\ i=1,2,\dots\}$ is called the {\it forward customer chain} and it is a homogeneous finite Markov chain with transition probabilities
\beas
&&p_{00}=0,\ p_{0k}=\frac{\nu_k}{\sum_{j=1}^J\nu_j};\\
&&p_{j0}=\mu_j,\ p_{jk}=\lambda_{jk},\ j,k=1,\dots,J.
\enas
The backward customer chain $X^\ast$ is the forward customer chain for the time-reversed process of $\{\bn(t):=(N_1(t),\dots,N_J(t)):\ t\in\real\}$ [Barbour \& Brown~(1996), p.~475] and it can be viewed as the time-reversal of the forward customer chain $\{X_i:\ i=1,2,\dots\}$ with transition probabilities
\beas
&&p_{00}^\ast=0,\ p_{0j}^\ast=\frac{\mu_j\alpha_j}{\sum_{l=1}^J\mu_l\alpha_l};\\
&&p_{k0}^\ast=\frac{\nu_k}{\alpha_k},\ p_{kj}^\ast=\frac{\alpha_j\lambda_{jk}}{\alpha_k},\ j,k=1,\dots,J.
\enas

We will use the Palm distributions to characterize the distribution of $\bXict$, prove its properties and establish
a general approximate version of Melamed's theorem for $\bXict(\Gamma_{C,t})$.
For the point process $\boldXi$ with locally finite mean measure $\bl(ds,(j,k))=\rho_{jk}ds,\ s\in\real,\ (j,k)\in\cs$, we may consider it as a random measure on the metric space $\real\times \cs$ equipped with the metric
$$d((u_1,(j_1,j_2)),(u_2,(k_1,k_2)))=|u_1-u_2|\bone_{(j_1,j_2)\ne(k_1,k_2)},\ u_1,u_2\in\real\mbox{ and }(j_1,j_2),\ (k_1,k_2)\in\cs,$$ so that we can define the {\it Palm distribution} at $\alpha\in \real\times \cs$ as the distribution of $\boldXi$ conditional on the presence of a point at $\alpha$, that is,
$$P^\alpha(\cdot)=
\frac{\mean \left[1_{[{\scriptsize \boldXi}\in\cdot]}\boldXi(d\alpha)\right]}{\bl(d\alpha)},\
\alpha\in\Gamma_{C,t}\
\bl-\mbox{almost surely},$$
see Kallenberg~(1983), p.~83 for more details.
A process $\boldXi^\alpha$ is called the {\it Palm process} of $\boldXi$ at $\alpha$ if its distribution is $P^\alpha$. In applications, it is often more convenient to work with the {\it reduced Palm process} $\boldXi^\alpha-\delta_\alpha$
[Kallenberg~(1983), p.~84], where $\delta_\alpha$ is the Dirac measure at $\alpha$.

The Palm distributions are closely related to the {\it size-biasing} in
sampling contexts [Cochran~(1977)]. More precisely, if $X$ is a non-negative integer-valued random variable, one may consider it as a point process with the carrier space having only one point so its Palm distribution becomes
$$\prob(X_s=i):=\frac{i\prob(X=i)}{\mean X}.$$
However, this is exactly the definition of the {\it size
biased} distribution of $X$ [see Goldstein \& Xia~(2006)].

\begin{Lemma}\label{BBlemma}
[Barbour \& Brown (1996)]
For the open queuing network, the reduced Palm distribution for the network given a transition at link
$(j,k)$ at time $0$ is the same as that for the original network, save that the network on $(0,\infty)$ behaves as if there were an extra individual at queue $k$ at time 0 and the network on $(-\infty,0)$ behaves as if there were an extra individual in queue $j$ at time 0.
\end{Lemma}

For two random elements $\eta_1$ and $\eta_2$ having the same distribution, we write for brevity $\eta_1\equald\eta_2$.

\begin{Lemma}\label{XiaLemma1} For each $(j,k)\in\cs$, there is a point process $\boldxi^{(0,(j,k))}$ on $\real\times\cs$ independent of $\boldXi$ such that
$$\boldxi^{(0,(j,k))}+\boldXi \equald\boldXi^{(0,(j,k))}.$$
\end{Lemma}

\noindent{\bf Proof.} The proof is adapted from Barbour \& Brown~(1996), p.~480. By Lemma~\ref{BBlemma}, the reduced Palm process $\boldXi^{(0,(j,k))}-\delta_{(0,(j,k))}$ has the same distribution as that of $\boldXi$ except that the network on $(0,\infty)$ behaves as if there were an extra individual at queue $k$ at time 0 and the network on $(-\infty,0)$ behaves as if there were an extra individual in queue $j$ at time 0. Let $\tilde X^{(0)}$ and $X^{(0)}$ be the routes taken by the extra individual on $(-\infty,0)$ and $(0,\infty)$ respectively. Whenever the extra customer is at queue $i$ together with other $m$ customers, we use independently sampled exponential service requirements with instantaneous service rate $\phi_i(m+1)-\phi_i(m)$. Noting that this construction ensures that the extra customer uses the ``spare" service effort and never ``interferes" with the flow of the main traffic, one can see that its transitions are independent of $\boldXi$. The same procedure applies to the construction of the backward route. Let
$\boldxi^{(0,(j,k))}$ be the transitions taken by the extra customer on $(-\infty,0)\cup(0,\infty)$ plus the Dirac measure $\delta_{(0,(j,k))}$, then $\boldxi^{(0,(j,k))}$ is independent of $\boldXi$ and the conclusion of the lemma follows from the construction. \qed

Let $\theta_s, \ s\in\real$, denote the shift operator on
$\real\times\cs$ which translates each point in $\real\times\cs$ by $s$ to the left, i.e. $\theta_s((u,(j,k)))=(u-s,(j,k))$ and we use
$\boldxi^{(s,(j,k))}$ to stand for a copy of $\boldxi^{(0,(j,k))}\circ\theta_s$, $s\in\real$.

From now on, we focus on the point process $\bXict$. With metric $d$, $\Gamma_{C,t}$ is a Polish space and we use $\cb\left(\Gamma_{C,t}\right)$ to stand for the Borel $\sigma$-algebra in $\Gamma_{C,t}$. Let $H_{C,t}$ denote the class of all configurations  (finite nonnegative integer-valued measures) on $\Gamma_{C,t}$
with $\ch_{C,t}$ the $\sigma$-algebra in $H_{C,t}$ generated by the sets
$$\{\xi\in H_{C,t}: \xi(B)=i\},\ i\in\Z_+:=\{0,1,2,\dots\},
\ B\in\cb\left(\Gamma_{C,t}\right),$$
see Kallenberg~(1983), p.~12.

\begin{Theorem}\label{Th1} Let $\{\boldeta_i,\ i\ge 0\}$ be independent and identically distributed random measures on $\Gamma_{C,t}$ having the distribution
\eq\prob\left[\boldeta_0\left(\Gamma_{C,t}\right)\ge 1\right]=1,\ \prob(\boldeta_0\in A)=\mean \sum_{(j,k)\in C}\int_0^t
\frac{\bone_{[{\scriptsize\boldxi}^{(s,(j,k))}\in A]}}{\boldxi^{(s,(j,k))}\left(\Gamma_{C,t}\right)}\cdot\frac{\rho_{jk}}{\theta_{C,t}}ds,\ A\in \ch_{C,t},
\label{Xia2.1}
\en
where
\eq\theta_{C,t}=\mean \sum_{(j,k)\in C}\int_0^t
\frac{1}{\boldxi^{(s,(j,k))}\left(\Gamma_{C,t}\right)}\rho_{jk}ds.\label{Xia2.2}\en
Let $M$ be a Poisson random variable with mean $\theta_{C,t}$ and independent of $\{\boldeta_i,\ i\ge 0\}$, then
$$\bXict\equald \sum_{i=1}^M\boldeta_i.$$
\end{Theorem}

\noindent{\bf Proof.} By Lemma~\ref{XiaLemma1} and Theorem~11.2 of [Kallenberg~(1983)], we can conclude that $\bXict$ is infinitely divisible, hence we obtain from Lemma~6.6 and Theorem~6.1 of [Kallenberg~(1983)] that $\bXict$ is a Poisson cluster process, that is,
$$\bXict\equald \sum_{i=1}^M\boldeta_i,$$
where $\boldeta_i,\ i\ge 0$ are independent and identically distributed random measures on $\Gamma_{C,t}$ such that $\prob\left(\boldeta_0(\Gamma_{C,t})\ge 1\right)=1$, $M$ is a Poisson random variable with mean $\theta_{C,t}$ and independent of  $\{\boldeta_i,\ i\ge 1\}$.
The direct verification ensures that the Palm process of $\sum_{i=1}^M\boldeta_i$ at $\alpha\in\Gamma_{C,t}$ is
$\sum_{i=1}^M\boldeta_i+\boldeta_0^{\alpha}$, where $\boldeta_0^{\alpha}$ is the Palm process of $\boldeta_0$ at $\alpha$ and is independent of $\{M,\boldeta_i,\ i\ge 1\}$. This in turn implies that $\boldxi^{(s,(j,k))}\equald \boldeta_0^{(s,(j,k))}.$

Let $\bm(ds,(j,k))$ denote the mean measure of the point process $\boldeta_0$, then some elementary computation ensures that the mean measure of $\sum_{i=1}^M\boldeta_i$ is $\theta_{C,t}\bm(ds,(j,k))$ for $(j,k)\in C$ and $0\le s\le t$. On the other hand, the mean measure of $\bXict$ is $\blct(ds,(j,k))=\rho_{jk}ds $, $(j,k)\in C$ and $s\in[0,t]$, so we obtain
\eq\bm(ds,(j,k))=\frac{\rho_{jk}}{\theta_{C,t}}ds,\ (j,k)\in C,\ s\in[0,t].\label{Xia2.3}\en
The representation \Ref{Xia2.1} is because of the fact that
$\prob\left(\boldeta_0(\Gamma_{C,t})\ge 1\right)=1$ and
$$\prob\left(\boldeta_0\in A\right)=\mean\int_{\Gamma_{C,t}}\frac{\bone_{[{\scriptsize\boldeta_0}\in A]}}{\boldeta_0\left(\Gamma_{C,t}\right)}\boldeta_0(d\alpha)=\mean \sum_{(j,k)\in C}\int_0^t
\frac{\bone_{[{\scriptsize\boldxi}^{(s,(j,k))}\in A]}}{\boldxi^{(s,(j,k))}\left(\Gamma_{C,t}\right)}\frac{\rho_{jk}}{\theta_{C,t}}ds.$$
In particular, if we take $A=\ch_{C,t}$, then the left hand side becomes 1, so \Ref{Xia2.2} follows.
\qed

Despite the fact that $\theta_{C,t}$ is specified by \Ref{Xia2.2}, since the Palm process $\boldxi^{(s,(j,k))}$ is generally intractable, it is virtually impossible to express $\theta_{C,t}$ explicitly in terms of the specifications of the Jackson queuing network. On the other hand, the relationship \Ref{Xia2.3} yields
$$\mean\boldeta_0(\Gamma_{C,t})=\frac{\rho_C}{\theta_{C,t}}t.$$
The following proposition tells us the range of values that  $\mean\boldeta_0(\Gamma_{C,t})$ and $\theta_{C,t}$ may take. To this end, we define
\eq\epsilon_C(j,k)=\mean\boldxi^{(0,(j,k))}(\real\times C)-1\mbox{ and }\epsilon_C=\sum_{(j,k)\in C}\frac{\rho_{jk}}{\rho_C}\epsilon_C(j,k).\label{Xia2.6}\en
In other words, $\epsilon_C(j,k)$ is the average number of visits in $C$ by the extra customer crossing the link $(j,k)$ and $\epsilon_C$ is the weighted average number of visits by an extra customer crossing links in $C$.

\begin{Proposition} We have
\eq1\le \mean\boldeta_0(\Gamma_{C,t})\le 1+\epsilon_C\label{Xia2.4}\en
and
\eq\frac{\rho_C}{1+\epsilon_C}t\le\theta_{C,t}\le\rho_C t.\label{Xia2.5}\en
\end{Proposition}

\noindent{\bf Proof.} The first inequality of \Ref{Xia2.4} follows immediately from the fact that \\
$\prob(\boldeta_0(\Gamma_{C,t})\ge 1)=1$. For the second inequality of \Ref{Xia2.4}, noting that the mean measure of $\boldeta_0$ is
$$\bm(ds,(j,k))=\frac{\rho_{jk}}{\theta_{C,t}}ds,\ (j,k)\in C,\ s\in[0,t],$$
we have
\beas
\left\{\frac{\rho_C}{\theta_{C,t}}t\right\}^2&=&[\mean\boldeta_0(\Gamma_{C,t})]^2\le\mean\left[\boldeta_0(\Gamma_{C,t})^2\right]\\
&=&\mean\int_{\Gamma_{C,t}}\boldeta_0(\Gamma_{C,t})\boldeta_0(d\alpha)\\
&=&\sum_{(j,k)\in C}\int_0^t\mean \boldeta_0^{(s,(j,k))}(\Gamma_{C,t})\bm(ds,(j,k))\\
&\le&\sum_{(j,k)\in C}(1+\epsilon_C(j,k))\frac{\rho_{jk}}{\theta_{C,t}}t\\
&=&\frac{\rho_C}{\theta_{C,t}}t+\frac{\sum_{(j,k)\in C}\epsilon_C(j,k)\rho_{jk}}{\theta_{C,t}}t\\
&=&\frac{\rho_C}{\theta_{C,t}}t(1+\epsilon_C).
\enas
We divide both sides by $\frac{\rho_C}{\theta_{C,t}}t$ to get
$$\mean\boldeta_0(\Gamma_{C,t})=\frac{\rho_C}{\theta_{C,t}}t\le 1+\epsilon_C.$$
Finally, \Ref{Xia2.5} is an immediate consequence of \Ref{Xia2.4} and the equation $\theta_{C,t}=\frac{\rho_C}{\mean\boldeta_0(\Gamma_{C,t})}t$. \qed

\begin{Remark}{\rm If the loop probability in $C$ is 0, then $\epsilon_C=0$ and $\mean\boldeta_0(\Gamma_{C,t})=1$, so there is only one customer on $\Gamma_{C,t}$. This customer is crossing the link $(j,k)$ with probability $\frac{\rho_{jk}}{\rho_C}$ at a time uniformly distributed on $[0,t].$}
\end{Remark}

\section{A discrete central limit theorem for the customer flow process}
\setcounter{equation}{0}

A random variable is said to be {\it over-dispersed} (resp. {\it under-dispersed}) if its variance to mean ratio is greater (resp. less) than one.
A random measure $\chi$ on a Polish space is said to be {\it over-dispersed} (resp. {\it under-dispersed}) if $\chi(B)$ is over-dispersed (resp. under-dispersed) for all bounded Borel subset $B$ of the Polish space.
It is concluded in Brown, Hamza \& Xia~(1998) that point processes which arise from Markov chains which are
time-reversible, have finitely many states and are irreducible are always
over-dispersed. As our process $\boldXi$ is virtually a multivariate version of point processes studied in Brown, Hamza \& Xia~(1998), the following property can be viewed as a natural extension of the study in Brown, Hamza \& Xia~(1998).

\begin{Proposition} The point process $\boldXi$ is over-dispersed.
\end{Proposition}

\noindent{\bf Proof.} The space $(\real\times\cs,d)$ is a Polish space and for each bounded Borel subset $B$ of $\real\times \cs$, it follows from the definition of the Palm processes [see Kallenberg~(1983), p.~84, equation~(10.4)] that
\beas
&&\mean\left[\boldXi(B)\right]^2=\mean\int_B\boldXi(B)\boldXi(d\alpha)=\mean \int_B\boldXi^\alpha(B)\bl(d\alpha)
\ge\mean \int_B\left(\boldXi(B)+1\right)\bl(d\alpha),
\enas
that is,
\eq\var\left[\boldXi(B)\right]\ge\mean \boldXi(B),\label{xia3.1}
\en
completing the proof. \qed

The inequality in \Ref{xia3.1} is generally strict except that the loop probability is 0,
i.e. $\boldXi$ is a Poisson process.
Hence, suitable approximate models for the distribution of $\Xict:=\bXict(\Gamma_{C,t})$
are necessarily over-dispersed. One potential candidate for approximating the distribution of $\Xict$ is the compound Poisson distribution. However, as it is virtually impossible to extract the distribution of
$\boldxi^{(s,(j,k))}(\Gamma_{C,t})$ for $0\le s\le t$, we face the same difficulty to specify and estimate the approximate distribution if a general compound Poisson is used.  On the other hand, as a special family of the compound Poisson distributions [Johnson, Kemp \& Kotz (2005), pp.~212--213 and p.~346], the negative binomial distribution has been well documented as a natural model for many over-dispersed random phenomena [see Bliss \& Fisher~(1953), Wang \& Xia~(2008) and Xia \& Zhang~(2009)]. The negative binomial distribution
${\rm NB}(r,q)$, $r>0$, $0<q<1$, is defined as
$$\pi_i=\frac{\Gamma(r+i)}{\Gamma(r)i!}q^r(1-q)^i,\ i\in\Z_+.$$
The advantage of using negative binomial approximation is that it suffices to estimate the mean and variance of the approximating distribution, like what we often do in applying the central limit theorem based on the normal approximation.

We will use the total variation distance between the distributions of nonnegative integer-valued random variables $Y_1$ and $Y_2$
$$d_{TV}(Y_1,Y_2):=\sup_{A\subset  \Z_+}|\prob(Y_1\in A)-\prob(Y_2\in A)|$$
to measure the approximation errors in negative binomial approximation.

The discrete central limit theorem is valid under the assumption that the loop probability
is less than 1. More precisely, let $w_C(jk)$ be the probability that a link $(j,k)$ crossing customer crosses the links in $C$ only once, i.e., the only time that the customer crosses the links in $C$ is the one the customer is crossing. Define
$$w_C=\sum_{(j,k)\in C}w_C(jk)\rho_{jk}/\rho_C,$$
the weighted probability of customers crossing links in $C$ only once. Clearly, we have $w_C(jk)\ge\mu_k$, so
$$w_C\ge\sum_{(j,k)\in C}\rho_{jk}\mu_k/\rho_C.$$
The following lemma plays a crucial rule for the estimation of the negative binomial approximation error.

\begin{Lemma}\label{keylemma1}
$d_{TV}\left(\Xict,\Xict+1\right)\le \frac1{\sqrt{2e w_C\rho_Ct}}.$
\end{Lemma}

\noindent{\bf Proof.} We prove the claim by a coupling based on the ``priority principle" [cf. the proof of Lemma~\ref{XiaLemma1}]. We define a customer as a {\it single crossing} ({\it sc} for brevity) customer if the customer crosses links in $C$ only once, otherwise, the customer is labeled as {\it multiple crossing}, or {\it mc} for short. We ``manage" the network by regrouping the customers at each queue into {\it sc} customers and {\it mc} customers. Whenever there are $m_2$ {\it mc} customers together with $m_1$ {\it sc} customers at queue $j$, we use independently sampled exponential service requirements with instantaneous service rate $\phi_j(m_1+m_2)-\phi_j(m_1)$ for all of the {\it mc} customers while the service for the {\it sc} customers is carried out with instantaneous service rate $\phi_j(m_1)$, that is, as if no {\it mc} customers present at the queue. Since the {\it sc} customers take priority over the {\it mc} customers and the {\it mc} customers use the ``spare" service effort and never interrupt the traffic flow of the {\it sc} ones, we can see that its transitions are independent of the transitions of the {\it sc} customers. Let $Z_1^{jk}$ (resp. $Z_2^{jk}$) denote the transitions of {\it sc} (resp. {\it mc}) customers moving from queue $j$ to queue $k$ in the period $[0,t]$, then $\bz_1:=\{Z_1^{jk},\ (j,k)\in C\}$ and
$\bz_2:=\{Z_2^{jk},\ (j,k)\in C\}$ are independent and
$$\bXict\equald \bz_1+\bz_2.$$
By Melamed's theorem, the point process $\bz_1$ is a Poisson process with mean measure
$$\bl_{\bz_1}(ds,(j,k))=w_C(jk)\rho_{jk}ds,\ (j,k)\in C,\ 0\le s\le t,$$
so $\bz_1(\Gamma_{C,t})$ follows Poisson distribution with mean $w_C\rho_Ct$ and
$$d_{TV}\left(\Xict,\Xict+1\right)\le d_{TV}\left(\bz_1(\Gamma_{C,t}),\bz_1(\Gamma_{C,t})+1\right)\le \frac1{\sqrt{2e w_C\rho_Ct}},$$
where the last inequality is because of the fact that the distribution of Poisson is unimodal and Proposition~A.2.7 of [Barbour, Holst \& Janson~(1992), p.~262].
\qed

To state the discrete central limit theorem, we set
$$\sigma_C(j,k)=\mean[\boldxi^{(0,(j,k))}(\real\times C)(\boldxi^{(0,(j,k))}(\real\times C)-1)]\mbox{ and }\sigma_C=\sum_{(j,k)\in C}\frac{\rho_{jk}}{\rho_C}\sigma_C(j,k).$$
That is, $\sigma_C(j,k)$ is the second factorial moment of the number of visits in $C$ by the extra customer crossing the link $(j,k)$ and $\sigma_C$ is the weighted average of the second factorial moments of the number of visits by an extra customer crossing links in $C$ [cf. \Ref{Xia2.6}].

\begin{Theorem}\label{Xia3.2} Let
$$r=\frac{(\rho_Ct)^2}{\var(\Xict)-\rho_Ct},\ q=\frac{\rho_Ct}{\var(\Xict)},$$
then
\bea
d_{TV}\left(\Xict,{\rm NB}(r,q)\right)&\le&\frac1{(\rho_Ct)^2\sqrt{2e w_C\rho_Ct}}
\left\{2(\var(\Xict)-\rho_Ct)^2\right.\nonumber\\
&&\mbox{\hskip0cm}\left.+\rho_Ct(\Xict[3]-\rho_Ct\Xict[2]-2\rho_Ct(\var(\Xict)-\rho_Ct))\right\}\label{Xia3.2.1}\\
&\le&\frac1{\sqrt{2e w_C\rho_Ct}}(2\epsilon_C^2+\sigma_C),\label{Xia3.2.2}
\ena
where $\Xict[n]$ stands for the $n$th factorial moment of $\Xict$ defined as
$$\Xict[n]=\mean[\Xict(\Xict-1)\dots(\Xict-n+1)].$$
\end{Theorem}

\begin{Remark}{\rm The parameters of the approximating negative binomial distribution are chosen so that it matches the mean and variance of $\Xict$.}
\end{Remark}

\begin{Remark}{\rm If
the loop probability in $C$ is 0, then the negative binomial is reduced to Poisson distribution and the upper bound in Theorem~\ref{Xia3.2} becomes 0. This implies half of Melamed's theorem~(1979).}
\end{Remark}

\begin{Remark}{\rm
If the loop probability
is between 0 and 1, then both $\epsilon_C$ and $\sigma_C$ are finite, so the negative binomial approximation error bound is of order $O(1/\sqrt{t})$. Furthermore, if the loop probability is small, then both $\epsilon_C$ and $\sigma_C$ are small, so the negative binomial approximation to the distribution of $\Xict$ is even faster.}
\end{Remark}

\noindent{\bf Proof of Theorem~\ref{Xia3.2}.} The essence of Stein's method is to find a generator which characterizes the approximating distribution, establish a Stein identity to transform the problem of estimating the approximation errors into the study of the structure of the object under investigation.
In the context of negative binomial approximation,
let $a=r(1-q)$, $b=1-q$, then a generator which characterizes ${\rm NB}(r,q)$ is defined as
$$\cb g(i)=(a+bi)g(i+1)-ig(i),\ i\in\Z_+,$$
for all bounded functions $g$ on $\Z_+$ [see Brown \& Phillips~(1999) and Brown \& Xia~(2001)]. The Stein identity is naturally established as
\eq\cb g(i)=f(i)-\pi(f)\label{Steinidentity}\en
for $f\in\cf:=\{f:\ \Z_+\to[0,1]\}$, where $\pi(f)=\sum_{i=0}^\infty f(i)\pi_i$. It was shown in Brown \& Xia~(2001) that, for each $f\in\cf$, the solution $g_f$ to the Stein equation \Ref{Steinidentity} satisfies
\eq\|\Delta g_f\|\le\frac1a,\label{Steinconstant}\en
where $\Delta g_f(\cdot)=g_f(\cdot+1)-g_f(\cdot).$ The Stein identity \Ref{Steinidentity} ensures that
$$\sup_{f\in\cf}\left |\mean f(\Xict)-\pi(f)\right|=\sup_{f\in\cf}\left|\mean \cb g_f(\Xict)\right|,$$
hence, it suffices to estimate $\mean \cb g_f(\Xict)$ for all $f\in\cf$. For convenience, we drop $f$ from the subindex of $g_f$.
By Lemma~\ref{XiaLemma1}, we can take a point process $\boldxi_{C,t}^{(s,(j,k))}$ on $\Gamma_t$ independent of $\bXict$ such that
$$\bXict^{(s,(j,k))}=\bXict+\boldxi_{C,t}^{(s,(j,k))}.$$
Therefore, if we write $\boldxi_{C,t}^{(s,(j,k))}(\Gamma_{C,t})=1+\xi^{(s,(j,k))}$, then
\bea
\mean\cb g(\Xict)&=&\mean[(a+b\Xict)g(\Xict+1)-\Xict g(\Xict)]\nonumber\\
&=&a\mean g(\Xict+1)+b\sum_{(j,k)\in C}\int_0^tg(\Xict+2+\xi^{(s,(j,k))})\rho_{jk}ds\nonumber\\
&&-\sum_{(j,k)\in C}\int_0^tg(\Xict+1+\xi^{(s,(j,k))})\rho_{jk}ds.\label{proof01}
\ena
Let
\eq a+(b-1)\sum_{(j,k)\in C}\rho_{jk}t=0,\label{coefficient1}\en
and $\tXict=\Xict+1$, then it follows from \Ref{proof01} that
\bea
&&\mean\cb g(\Xict)\nonumber\\
&&=\mean \sum_{(j,k)\in C}\int_0^t\left[b\left(g\left(\tXict+1+\xi^{(s,(j,k))}\right)-g\left(\tXict\right)\right)-\left(g\left(\tXict+\xi^{(s,(j,k))}\right)-g\left(\tXict\right)\right)\right]\rho_{jk}ds\nonumber\\
&&=\mean \sum_{(j,k)\in C}\int_0^t\left\{\sum_{r=0}^{\xi^{(s,(j,k))}-1}\left[b\Delta g\left(\tXict+r+1\right)
-\Delta g\left(\tXict+r\right)\right]+b\Delta g\left(\tXict\right)\right\}\rho_{jk}ds.
\label{proof02}\nonumber\\
\ena
Now, set
\eq  b=\frac{\sum_{(j,k)\in C}\int_0^t\mean \xi^{(s,(j,k))}\rho_{jk}ds}{\sum_{(j,k)\in C}\int_0^t\mean \xi^{(s,(j,k))}\rho_{jk}ds+\rho_C t}
=\frac{\var\left(\Xict\right)-\rho_Ct}{\var\left(\Xict\right)}\label{coefficient2},\en
where the last equality is due to the following observation:
\beas
\mean\Xict^2&=&\mean\int_{\Gamma_{C,t}}\Xict\bXict(d\alpha)\\
&=&\sum_{(j,k)\in C}\mean\int_0^t\left(\Xict+1+\xi^{(s,(j,k))}\right)\rho_{jk}ds\\
&=&(\mean\Xict)^2+\rho_Ct+\mean\sum_{(j,k)\in C}\int_0^t\xi^{(s,(j,k))}\rho_{jk}ds,
\enas
and so
\eq\mean\sum_{(j,k)\in C}\int_0^t\xi^{(s,(j,k))}\rho_{jk}ds=\var(\Xict)-\rho_Ct.\label{proof04}\en
We then obtain from \Ref{proof02} that
\bea
&&\mean\cb g(\Xict)\nonumber\\
&&=\mean \sum_{(j,k)\in C}\int_0^t\left\{\sum_{r=0}^{\xi^{(s,(j,k))}-1}\left[b\Delta^2 g\left(\tXict+r\right)-(1-b)\sum_{l=0}^{r-1}\Delta^2 g\left(\tXict+l\right)\right]\right\}\rho_{jk}ds\nonumber\\
&&=\mean \sum_{(j,k)\in C}\int_0^t\left\{\sum_{r=0}^{\xi^{(s,(j,k))}-1}\left[b\mean\Delta^2 g\left(\tXict+r\right)-(1-b)\sum_{l=0}^{r-1}\mean\Delta^2 g\left(\tXict+l\right)\right]\right\}\rho_{jk}ds,
\label{proof03}\nonumber\\
\ena
where the last equation is due to the fact that $\xi^{(s,(j,k))}$ is independent of $\tXict$. On the other hand, using \Ref{Steinconstant}, we have
$$\left|\mean \Delta^2 g(\tXict+l)\right|\le 2\|\Delta g\|d_{TV}(\Xict,\Xict+1)\le \frac{2d_{TV}(\Xict,\Xict+1)}{a},$$
so it follows from \Ref{proof03} that
\bea&&\left|\mean\cb g\left(\Xict\right)\right|\nonumber\\
&&\le\frac{d_{TV}\left(\Xict,\Xict+1\right)}{a}\sum_{(j,k)\in C}\int_0^t\left[2b\mean\xi^{(s,(j,k))}+(1-b)\mean \xi^{(s,(j,k))}\left(\xi^{(s,(j,k))}-1\right)\right]\rho_{jk}ds.
\nonumber\\
\label{proof05}\ena
Using the Palm distributions of $\bXict$ together with \Ref{proof04}, we get
\beas
\Xict[3]&=&\mean\int_{\Gamma_{C,t}}(\Xict-1)(\Xict-2)\bXict(d\alpha)\\
&=&\sum_{(j,k)\in C}\int_0^t\mean\left[\left(\Xict+\xi^{(s,(j,k))}\right)\left(\Xict+\xi^{(s,(j,k))}-1\right)\right]\rho_{jk}ds\\
&=&\rho_Ct\Xict[2]+2\rho_Ct(\var(\Xict)-\rho_Ct)+\sum_{(j,k)\in C}\int_0^t\mean \xi^{(s,(j,k))}\left(\xi^{(s,(j,k))}-1\right)\rho_{jk}ds.
\enas
This in turn ensures
\eq\sum_{(j,k)\in C}\int_0^t\mean \xi^{(s,(j,k))}\left(\xi^{(s,(j,k))}-1\right)\rho_{jk}ds=\Xict[3]-\rho_Ct\Xict[2]-2\rho_Ct(\var(\Xict)-\rho_Ct).\label{proof06}\en
Consequently, combining \Ref{proof04}, \Ref{proof06} with \Ref{proof05} gives \Ref{Xia3.2.1}.

Finally, by the definitions of $\epsilon_C$ and $\sigma_C$, we have
$$\mean\sum_{(j,k)\in C}\int_0^t\xi^{(s,(j,k))}\rho_{jk}ds\le \epsilon_C\rho_Ct$$
and
$$\sum_{(j,k)\in C}\int_0^t\mean \xi^{(s,(j,k))}\left(\xi^{(s,(j,k))}-1\right)\rho_{jk}ds\le
\sigma_C\rho_Ct.$$
Therefore, \Ref{Xia3.2.2} follows from \Ref{Xia3.2.1}, \Ref{proof04} and \Ref{proof06}.
\qed

\def\ac{{Academic Press}~}
\def\aap{{Adv. Appl. Prob.}~}
\def\ap{{Ann. Probab.}~}
\def\anap{{Ann. Appl. Probab.}~}
\def\jap{{J. Appl. Probab.}~}
\def\jws{{John Wiley $\&$ Sons}~}
\def\ny{{New York}~}
\def\ptrf{{Probab. Theory Related Fields}~}
\def\sp{{Springer}~}
\def\spa{{Stochastic Processes Appl.}~}
\def\sv{{Springer-Verlag}~}
\def\tpa{{Theory Probab. Appl.}~}
\def\zw{{Z. Wahrsch. Verw. Gebiete}~}


TITLE: Components of a bounded holomorphic function on annulus
QUESTION [1 upvotes]: I know that a holomorphic function $f$ on an annulus $A=\{z\;:\;r_{1}<\lvert z\rvert<r_{2}\}$ has a unique decomposition $f=f_{1}+f_{2}$, where $f_{1}$ and $f_{2}$ are holomorphic functions outside $\bar{D}(0;r_{1})$ and inside $D(0;r_{2})$, and $\lvert f_{1}(z)\rvert \to 0$ as $\lvert z\rvert\to \infty$. I am asked to show that if $f$ is bounded on $A$, then so are $f_{1}$ and $f_{2}$. I tried using "local" formulae for the functions $f_{1}$ and $f_{2}$: in a smaller annulus $B=\{z\;:\;r_{1}+\epsilon<\lvert z\rvert<r_{2}-\epsilon\}$,
$$ f_{1}(z)=\frac{1}{2\pi i}\int_{\gamma_{1}} \frac{f(\zeta)}{\zeta-z} \:d\zeta, \quad f_{2}(z)=\frac{1}{2\pi i}\int_{\gamma_{2}}\frac{f(\zeta)}{\zeta-z} \:d\zeta $$
where $\gamma_{1}$ and $\gamma_{2}$ are circles about the origin of radii $r_{1}+\epsilon$ and $r_{2}-\epsilon$, oriented clockwise and counterclockwise, respectively. I'm having trouble with this approach because of the denominator in the integrand.

REPLY [1 votes]: Choose any $r \in (r_1, r_2)$. We know that $f_2$ is bounded on $B := \{ z : \lvert z\rvert \leqslant r\}$ by compactness, and that $f_1$ is bounded on $C :=\{ z : \lvert z\rvert \geqslant r\}$, by a similar argument (the same if you know what "analytic at $\infty$" means).
If $f$ is bounded on $A$, then $f_2 = f - f_1$ is bounded on $A\cap C$ as the difference of two bounded functions there, hence $f_2$ is bounded on $(A \cap B) \cup (A \cap C) = A$. Similarly, $f_1 = f - f_2$ is bounded on $A \cap B$, hence on $(A \cap B) \cup (A\cap C) = A$.


TITLE: Convergence of $\int_0^{\infty} s^{2-2\alpha}e^{is^2}ds$
QUESTION [2 upvotes]: I need to evaluate the following integral
$$\int_0^{\infty} s^{2-2\alpha}e^{is^2}ds$$
where $\alpha\in[0,\,1]$. I shall call this integral Gamma-function like integrals since it can be brought to Gamma function as follows:
One first replace $s=\sqrt{it}$, then the integral transforms to
$$\frac{i^{3/2-\alpha}}{2}\int_0^{-i\infty} t^{1/2-\alpha}e^{-t}dt$$
Finally, one can analytically continue the integral from negative $t$ axis to real $t$ axis to obtain $\frac{i^{3/2-\alpha}}{2}\Gamma(3/2-\alpha)$.
When I throw the two integrals into Mathematica, it does gives me the above result. However, it seems to tells me the integrals only converge on $[1/2,1]$ and diverge on $[0, 1/2]$. Could anybody explain why it is the case?

Update: I think now I understand the convergence by myself. Let us start with the second integral. We denote break the second integral into two parts: $I=I_1+I_2$, where
$$I_1=\int_0^{-i\epsilon} t^{\beta}e^{-t}dt$$
$$I_2=\int_{-i\epsilon}^{-i\infty} t^{\beta}e^{-t}dt$$
and $\beta=1/2-\alpha$
To guarantee $\lim_{\epsilon\to0}I_1=0$, we must have
$$\beta>-1$$
Now we would like to analytically continue the integral for $I_2$ to the real positive axis. But the condition of doing this is that the integrand should not have any singularity on the fourth quadrant, apart from the small quarter-circle of radius $\epsilon$ near the origin.
This analyticity of the integrand is guaranteed only if it vanishes at $t=-i\infty$. So we should require
$$\beta<0$$
which leads to
$$I_2=\int_{\epsilon}^{\infty} t^{\beta}e^{-t}dt$$
When $\beta \in(-1,0)$,i.e., $\alpha\in(1/2,3/2)$, we can express $I$ as
$$I=\int_{0}^{\infty} t^{\beta}e^{-t}dt=\Gamma(\beta+1)=\Gamma(3/2-\alpha)$$

REPLY [1 votes]: The following is partially analytical but relies on Mathematica for the closed-forms. We start by deforming the contour to the ray $\arg s=\phi$ where $0<\phi<\pi/2$. This is accomplished by the substitution $s=re^{i\phi}$, obtaining
\begin{align}
I(\phi)
&:=\int_0^{\infty e^{i\phi}} s^{2-2\alpha}\exp(i s^2)\,ds\\
&=e^{(1-2\alpha)i\phi}\int_0^\infty r^{2-2\alpha} \exp(i e^{2i\phi}r^2)\,dr\\
&=e^{(1-2\alpha)i\phi}\int_0^\infty r^{2-2\alpha} \exp(i r^2 \cos(2\phi))\exp(-r^2 \sin(2\phi))\,dr.\\
\end{align}
Since $\sin(2\phi)>0$ due to the assumption on $\phi$, the integrand decays like a Gaussian at infinity and so the integral converges. With this in mind, we can ask Mathematica to carry out the integral under this assumption:
Integrate[Exp[I r^2 Exp[2I \[Phi]]]r^(2-2\[Alpha]),{r,0,\[Infinity]},Assumptions->0<\[Phi]<\[Pi]/4]

That is, the integral converges so long as the real part of $\alpha$ doesn't exceed $3/2$. Including the prefactor and taking the limit $\phi\to 0^+$, we obtain
FullSimplify[Limit[ConditionalExpression[ 1/2 (-I E^(2 I \[Phi]))^(-(3/2) + \[Alpha])Gamma[3/2 - \[Alpha]], \[Alpha] + Conjugate[\[Alpha]] < 3],\[Phi]->0,Direction->"FromAbove"]]

Since Mathematica uses $-i=e^{-i\pi/2}$, we obtain $$I(0^+) =\frac12 e^{i(3/2-\alpha)\pi/2}\tag{Re $\alpha<3/2$}$$
which is seemingly valid on the entirety of $\alpha\in[0,1]$.
A warning, though. On the one hand, for $1/2\leq \alpha<1$ the above formula agrees with the original Mathematica calculation. But the new calculation purports to be valid for $\alpha\leq 1/2$ as well. I'm not sure I understand the issue myself, but note that the exponent $e^{i s^2}$ has modulus 1 along the real line. Hence the rate at which the integrand's modulus grows along the real line is entirely controlled by $s^{2-2\alpha}$: For $\alpha>1/2$ the modulus grows slower than linear, whereas for $\alpha<1/2$ it is faster than linear. Evidently this transition is enough to Mathematica to declare the integral divergent.
By contrast, if we deform the contour just above the real line then the modulus of $e^{i s^2}$ decreases exponentially fast and so there's no issue with convergence. But it does raise the question of what exactly we intend by the original integral: Is it okay to regularize the growth in this way, or is it not? That will probably depend on the original context.


text:
\section{$L^2$ excess decay}\label{sec:decay}

In this section we work towards an excess decay theorem (Proposition \ref{prop:decay}), which gives a decay estimate of the $L^2$ excess $E$ when $M$ looks close to planar, and $\Omega$ looks close to a cone.  At a general point $x$, for most scales $\Omega$ will look like one of only finitely many cones, and so in the next Section we will be able to use Corollary \ref{cor:decay} on each model cone to prove decay on all scales.  As before, in this Section we fix $\OmegaRef$ a polyhedral cone domain of the form $\OmegaRefN^l \times \R^{m}$, where we assume $m \geq 1$.

\begin{prop}\label{prop:decay}
Let $e(\OmegaRefN)$ be the exponent bound of Proposition \ref{prop:expansion}.  For any $\theta \leq 1/10$, there are constants $\delta_0(\OmegaRef, \theta)$, $c_0(\OmegaRef)$, so that the following holds.  Let $\Omega = \Phi(\Psi(\OmegaRef))\cap B_1 \in \cD_{\delta_0}(\OmegaRef)$, and $M \in \cIH_n(\Omega, B_1)$, satisfy
\begin{equation}\label{eqn:decay-hyp}
E_{\delta_0}(\Phi, M, p + V, 0, 1) \leq E \leq \delta_0^2, \quad \theta_M(0, 1) \leq (7/4) \Theta_{T_0\Omega}, \quad d_{\spt M}(0) \leq \delta_0,
\end{equation}
for some $p \in V^\perp$, $V \in \cP_{T_0\Omega}$.

Then there are $V' \in \cP_{T_0\Omega}$, $p' \in {V'}^\perp$, satsifying
\begin{equation}\label{eqn:decay-concl1}
|p - p'| + |\pi_V - \pi_{V'}| \leq c_0 E^{1/2},
\end{equation}
so that
\begin{equation}\label{eqn:decay-concl2}
E_{\delta_0}(\Phi, M, p' + V', 0, \theta) \leq c_0 \theta^{2(e-1)} E .
\end{equation}
\end{prop}

By iterating Proposition \ref{prop:decay}, we obtain the direct Corollary.  As a technical aside, instead of using the monotonicity formula \eqref{eqn:monotonicity} to iterate Proposition \ref{prop:decay}, one could use Lemma \ref{lem:sharp-mass-bound}.
\begin{corollary}\label{cor:decay}
There are constants $\delta_1(\OmegaRef)$, $c_1(\OmegaRef)$, $\beta(\OmegaRef)$ so that the following holds.  Let $\Omega = \Phi(\Psi(\OmegaRef)) \cap B_1 \in \cD_{\delta_1}(\OmegaRef)$, $M \in \cIH_n(\Omega, B_1)$, and suppose that
\begin{equation}\label{eqn:con-decay-hyp}
E(\Phi, M, p + V, 0, 1) \leq E \leq \delta_1^2, \quad \theta_M(0, 1) \leq (3/2) \Theta_{T_0\Omega}, \quad d_{\spt M}(0) \leq \rho
\end{equation}
for some $p \in V^\perp$, $V \in \cP_{T_0\Omega}$, and some $0 \leq \rho \leq 1/2$.

Then there are $V' \in \cP_{T_0\Omega}$,  $p' \in {V'}^\perp \cap B_1$, satisfying
\begin{equation}\label{eqn:con-decay-concl1}
|p - p'| + |\pi_V - \pi_{V'}| \leq (c_1 E)^{1/2},
\end{equation}
such that
\begin{gather}\label{eqn:con-decay-concl2}
E^{tot}(\Phi, M, p' + V', 0, r) \leq  c_1 E r^{2\beta}   \quad \forall \rho \leq r \leq 1/2.
\end{gather}
\end{corollary}

\begin{remark}
The exact form of the upper bound $\Theta_{T_0\Omega}$ is flexible.  One could equally well as use $\Theta_{\OmegaRef}$ or $\omega_{n+1}^{-1}\haus^{n+1}(\Omega)$, provided we take $\delta_1$ sufficiently small.  However in the following section we will find it convenient to use $T_0\Omega$.
\end{remark}

\begin{proof}[Proof of Corollary \ref{cor:decay}]
Choose $\theta(\OmegaRef)$ sufficiently small so that $c_0 \theta^{2(e-1)} \leq 1/2$.  Ensure $\delta_1(\OmegaRef)$ is sufficiently small so that
\[
\delta_1 \leq \min\{ \eps_{mn}, \eps_e \}, \quad (3/2)(1+c_{mn}\delta_1)^{n+1} \leq 7/4 \quad c_e \delta_1 \leq 1/4,
\]
and $\delta_1 \leq \delta_0^2$ with our choice of $\theta$.  Here $\eps_{mn}(n), c_{mn}(n)$ are the constants from Lemma \ref{lem:monotonicity}, $c_e(\OmegaRef, \theta = 1/2, A = 2), \eps_e(\OmegaRef)$ are from Corollary \ref{cor:height-w12-bound}, and $\delta_0(\OmegaRef, \theta), c_0(\OmegaRef)$ are from Proposition \ref{prop:decay}.

First suppose $\rho > \delta_1$.  Then take $p' = p$, $V' = V$, and we have trivially
\[
E(\Phi, M, p' + V', 0, r) \leq \delta_1^{-n-4} r^{2\alpha} E \quad \forall \rho \leq r \leq 1.
\]
Provided we take $c_1 \geq \delta_1^{-n-4}$, this proves the Theorem.

Let us therefore assume $\rho \leq \delta_1$.  Define $r_i = \theta^i$, and let $I$ be the maximal non-negative integer for which $\rho \leq r_I \delta_1$.  If $\rho = 0$, then we set $I = \infty \equiv I+1$.  For $i = 0, 1, \ldots, I+1$, we define inductively a sequence $p_i \in B_1$, $V_i \in \cP_{T_0\Omega}$, so that
\begin{equation}\label{eqn:con-decay-1}
r_i^{-1} |p_{i+1} - p_i| + |\pi_{V_{i+1}} - \pi_{V_i}| \leq c_0 2^{-i/2} \delta_0^{-1} E^{1/2},
\end{equation}
and
\begin{equation}\label{eqn:con-decay-2}
E_{\delta_0}(\Phi, M, p_i + V_i, 0, r_i) \leq 2^{-i} \delta_0^{-2} E.
\end{equation}

We take $p_0 = p$, $V_0 = V$.  Suppose, by inductive hypothesis, we have constructed $p_i$, $V_i$ satisfying \eqref{eqn:con-decay-2} and (if $i \geq 1$) \eqref{eqn:con-decay-1}.  By our choice of $I$ and $\delta_1$, and Remark \ref{rem:scaling-E}, and monotonicity \eqref{eqn:monotonicity} we can apply Proposition \ref{prop:decay} to the rescaled varifold $(\eta_{0, r_i})_\sharp M$ in $\Omega_{0, r_i}$ to obtain a $\tilde p_{i+1}$, $\tilde V_{i+1}$.  Setting $p_{i+1} = r_i \tilde p_{i+1}$ and $V_{i+1} = \tilde V_{i+1}$, the required estimates \eqref{eqn:con-decay-1}, \eqref{eqn:con-decay-2} hold by scaling.  This proves the inductive step, and therefore the existence of the required $p_i$, $V_i$.

If $I < \infty$, then set $p' = p_I$, $V' = V_I$.  Otherwise, if $I = \infty$, then observe that \eqref{eqn:con-decay-1} implies the $p_i$, $V_i$ form Cauchy sequences, and so we can take $p' = \lim_{i\to \infty} p_i$, $V' = \lim_{i \to \infty} V_i$.  For each (finite) $i \leq I$, we have
\begin{equation}\label{eqn:con-decay-3}
r_i^{-1} |p' - p_i| + |\pi_{V'} - \pi_{V_i}| \leq c(\OmegaRef) 2^{-i/2} E^{1/2} .
\end{equation}
In particular taking $i = 0$ gives \eqref{eqn:con-decay-concl1}.

Given $\rho < r \leq 1$, either $r \geq \delta_1$, in which case set $i = 0$, or we have a maximal $i \leq I$ for which $r \leq \delta_1 r_i$.  Using \eqref{eqn:con-decay-3} and \eqref{eqn:upper-reg}, we compute
\begin{align*}
E(\Phi, M, p' + V', 0, r)
&\leq (\theta \delta_1)^{-n-2} E(\Phi, M, p' + V', 0, r_i) \\
&\leq c E(\Phi, M, p_i + V_i, 0, r_i) + c |\pi_{V'} - \pi_{V_i}|^2 + c r_i^{-2} |p' - p_i|^2 \\
&\leq c E 2^{-i} \\
&= c E r_i^{2\beta} \\
&\leq c(\OmegaRef) E r^{2\alpha},
\end{align*}
where $\beta = \log(1/2)/\log(\theta)$.  This proves the required $L^2$ decay of \eqref{eqn:con-decay-concl2}, for $\rho \leq r \leq 1$.  To deduce the decay of $E^{tot}$ we use Remark \ref{rem:scaling-E} to apply Corollary \ref{cor:height-w12-bound} at each scale $\rho \leq r \leq 1/2$.
\end{proof}

\vspace{5mm}

The rest of the section is devoted to proving Proposition \ref{prop:decay}.  We first require a definition and some helper theorems.

\begin{prop}[Fine graphical approximation]\label{prop:graph}
Given any $\tau, \beta$, there is a $\delta(\tau, \beta, \OmegaRef)$ and $c(\OmegaRef)$ so that the following holds.  Let $\Omega = \Phi(\Psi(\OmegaRef)) \cap B_1 \in \cD_\delta(\OmegaRef)$, $M \in \cIH_n(\Omega, B_1)$, satisfy
\begin{equation}\label{eqn:graph-hyp}
E(\Phi, M, p + V, 0, 1) \leq E \leq \delta^2, \quad \theta_M(0, 1) \leq (7/4) \Theta_{T_0\Omega}, \quad d_{\spt M}(0) \leq \delta,
\end{equation}
for some $p \in V^\perp$ and $V \in \cP_{T_0\Omega}$.  Then the following holds:
\begin{enumerate}
\item there is a domain $U \subset p+V$, and $C^{1}$ function $u : U \to V^\perp$, satisfying
\begin{equation}\label{eqn:graph-concl1}
(p+V) \cap\Omega \cap B_{1/4} \setminus B_{2\tau}(\del\Omega) \subset U, \quad r^{-1}|u| + |Du| \leq \beta,
\end{equation}
and
\begin{equation}\label{eqn:graph-concl1.5}
M \llcorner B_{1/2} \setminus B_\tau(\del\Omega) = [\graph(u)] ;
\end{equation}

\item we have the estimates
\begin{equation}
\sup_U |u|^2 + \int_U |Du|^2 \leq c E,
\end{equation}
and
\begin{equation}\label{eqn:graph-concl2.5}
\sup_{z \in \spt \mu_M \cap B_{1/2}} d(z, p + V)^2 \leq c E;
\end{equation}

\item for any $\zeta \in C^1_c(p + V, V^\perp) \cap C^1_c(B_{1/4})$, we have
\[
\left| \int_U Du \cdot D\zeta  \right| \leq c |\zeta|_{C^1} \left( ||H||_{L^\infty(B_1; \mu_M)} + |D^2\Phi|_{C^0(B_2)} + \tau^{1/2} E^{1/2} \right).
\]

\end{enumerate}
\end{prop}

\begin{proof}
Suppose, towards a contradiction, there is a sequence $\delta_i \to 0$, $p_i \in V_i^\perp$, $V_i \in \cP_{T_0\Omega_i}$, $\Omega_i = \Phi_i(\Psi_i(\OmegaRef)) \cap B_1 \in \cD_{\delta_i}(\OmegaRef)$, and $M_i \in \cIH_n(\Omega_i, B_1)$, satisfying \eqref{eqn:graph-hyp}, but failing conclusion 1.  By the height bound \eqref{eqn:sharp-height} and our assumption $\spt M_i \cap B_{\delta_i} \neq \emptyset$, we have $d(p_i + V_i, 0) \to 0$, and by Theorem \ref{thm:first-var} we have that $||\delta M_i||$ is uniformly bounded on compact subsets of $B_1$.  We can therefore find a $V \in \cP_{\OmegaRef}$, and an $M \in \cIV_n(B_1)$, so that after passing to a subsequence, we get $p_i \to 0$, $V_i \to V$, and $M_i \to M$ on compact subsets of $B_1$.  Since $|\Phi_i - \mathrm{Id}|_{C^2(B_2)} \to 0$, $|\Psi_i - \mathrm{Id}| \to 0$, we have $\Theta_{T_0\Omega_i} \to \Theta_{T_0\OmegaRef}$.

Since $E(\Phi_i, M, p_i + V_i, 0, 1) \to 0$, we have $\spt M \subset p + V$ and $||\delta M||(\setint \OmegaRef \cap B_1) = 0$.  Therefore by the constancy theorem $M \llcorner \setint\OmegaRef = k[p + V] \llcorner (\setint\OmegaRef \cap B_1)$ for some constant $k$.  By the non-concentration estimate \eqref{eqn:mass-non-conc} we have $\mu_M(\del\OmegaRef) = 0$, and trivially $\spt \mu_M \subset \OmegaRef$, and so in fact $M = k[p+V] \llcorner (\setint\OmegaRef\cap B_1)$.  From lower-regularity \eqref{eqn:lower-reg}, $\mu_{M_i}(B_{1/2}) \geq 1/c(\OmegaRef)$, and hence $\mu_M(B_1) \geq 1/c(\OmegaRef)$.  This implies $k \geq 1$.  On the other hand,
\[
\mu_M(B_1) \leq \liminf_i \mu_{M_i}(B_1) \leq (3/2) \lim_i \Theta_{T_0\Omega_i} = (3/2) \Theta_{T_0\OmegaRef},
\]
and so $k \leq 1$.  We deduce $k = 1$.  Since for every $W \subset\subset \setint\OmegaRef$ and $i >> 1$ the $M_i \llcorner W$ have bounded mean curvature (tending to zero) and zero boundary, Allard's theorem \cite{All} implies the convergence is $C^1$ on compact subsets of $\setint\OmegaRef \cap B_1$.  This proves conclusion 1.

Let us now fix a $\tau, U$, and prove conclusions 2 and 3.  Ensuring $\delta(\OmegaRef)$ is sufficiently small, There is no loss in assuming $p + V = \R^{n}\times\{0\}$.  Given a function $f : D \subset (p + V) \to V^\perp$, we can extend $f$ to be defined in $D \times V^\perp$ by setting $\tilde f(x) = f(p + \pi_V(x - p))$.  Given $f, g : D \to V^\perp$, then at $\mu_M$-a.e. $x \in \spt M \cap (D \times V^\perp)$ we have the bound
\begin{equation}\label{eqn:graph-1}
|\nabla \tilde f \cdot \nabla \tilde g - D\tilde f \cdot D \tilde g| \leq |\pi_{T_xM} - \pi_V|^2 |D\tilde f ||D\tilde g|.
\end{equation}
In the special case when $f = u$ as defined on $U$, then
\begin{equation}\label{eqn:graph-2}
|\nabla \tilde u|^2 = |\nabla x_{n+1}|^2 \leq |\pi_{T_xM} - \pi_V|^2.
\end{equation}
Combining \eqref{eqn:graph-1}, \eqref{eqn:graph-2} with \eqref{eqn:W12-bound}, and (without loss of generality) ensuring $\beta \leq \beta(n)$, we get
\[
\int_U |Du|^2 \leq (1+c(n) \beta^2) \int_{B_{1/2}} |D \tilde u|^2 d\mu_M \leq c(n) \int_{B_{1/2}} |\pi_{T_zM} - \pi_V|^2 d\mu_M(z) \leq c(\OmegaRef) E.
\]
This completes the $W^{1,2}$ estimate of conclusion 2; the $L^\infty$ estimates both follow from the sharp height bound \eqref{eqn:sharp-height}.

We prove conclusion 3.  Take $\zeta \in C^1_c(p+V, V^\perp) \cap C^1_c(B_{1/4})$.  Let $\phi$ be any function which is $1$ on $B_{1/2}$, and is supported in $B_1$.  By the height bound \eqref{eqn:sharp-height} we can assume that $\spt M \cap \spt \tilde \zeta \subset B_{1/2}$.  Plugging the field $X = \tilde \zeta \phi \tilde e_{n+1}$ into the first variation (where $\tilde e_{n+1}$ as in the proof of Corollary \ref{cor:height-w12-bound}), and noting that $X$ is tangential to $\Omega$, we obtain
\begin{equation}\label{eqn:graph-3}
\left| \int_{B_{1/2}} \nabla \tilde \zeta \cdot \nabla x_{n+1} d\mu_M \right| \leq c(n)|\zeta|_{C^1}( ||H^{tan}_M||_{L^\infty(B_1; \mu_M)} + |D^2 \Phi|_{C^0(B_2)}).

\end{equation}

Let us make some remarks.  First, provided $\beta(n)$ is sufficiently small, the Jacobian $Ju$ of $u$ satisfies the bounds
\begin{equation}\label{eqn:graph-4}
1 \leq Ju \leq 1 + c(n) |Du|^2.
\end{equation}
Second, ensuring $\delta(\tau, \OmegaRef)$ is sufficiently small, by the non-concentration estimate \eqref{eqn:mass-non-conc} we have
\[
\mu_M(B_\tau(\del\Omega) \cap B_{1/2}) \leq c(\OmegaRef) \tau.
\]
Last, using \eqref{eqn:W12-bound}, this implies that
\begin{align}
\left| \int_{B_{1/2} \cap B_{\tau}(\del\Omega)} \nabla \tilde\zeta \cdot e_{n+1} d\mu_M \right|
&\leq c(\OmegaRef) |\zeta|_{C^1} \tau^{1/2} \left( \int_{B_{1/2}} |\pi_{T_zM} - \pi_V|^2 d\mu_M(z) \right)^{1/2} \nonumber \\
&\leq c(\OmegaRef) |\zeta|_{C^1} \tau^{1/2} E^{1/2} \label{eqn:graph-5}
\end{align}

Using conclusions 1 and 2, equations \eqref{eqn:graph-3}, \eqref{eqn:graph-4}, \eqref{eqn:graph-5} we obtain
\begin{align*}
\left| \int_U D \zeta \cdot Du \right|
&\leq c \beta |\zeta|_{C^1} \int_U |Du|^2 + \left| \int_{B_{1/2} \setminus B_\tau(\del\Omega)} D\tilde \zeta \cdot D\tilde u d\mu_M \right| \\
&\leq c |\zeta|_{C^1} \int_U |Du|^2 + c|\zeta|_{C^1} \int_{B_{1/2}} |\pi_{T_zM} - \pi_V|^2 d\mu_M(z) + \left| \int_{B_{1/2}\setminus B_\tau(\del\Omega)} \nabla \tilde \zeta \cdot e_{n+1} d\mu_M \right| \\
&\leq c |\zeta|_{C^1} E+ c |\zeta|_{C^1} \tau^{1/2} E^{1/2} + c |\zeta|_{C^1} (||H||_{L^\infty(B_1; \mu_M)} + |D^2\Phi|_{C^0(B_2)}) \\
&\leq c|\zeta|_{C^1} \tau^{1/2} E^{1/2} + c|\zeta|_{C^1} (||H||_{L^\infty(B_1; \mu_M)} + |D^2\Phi|_{C^0(B_2)}),
\end{align*}
for $c = c(\OmegaRef)$.  The last inequality follows because we can of course assume $\delta^2 \leq \tau$.  This completes conclusion 3, and the proof of Proposition \ref{prop:graph}.
\end{proof}

\begin{definition}
Consider the sequences $\beta_i, \delta_i \in \R$, $\Omega_i = \Phi_i(\Psi_i(\OmegaRef)) \cap B_1 \in \cD_{\delta_i}(\OmegaRef)$, $V_i \in \cP_{T_0\Omega_i}$, $p_i \in V_i^\perp$, $M_i \in \cIH_n(\Omega, B_1)$.  We say $(\OmegaRef, \Omega_i, M_i, p_i + V_i, \beta_i, \delta_i)$ is a blow-up sequence if:
\begin{enumerate}
\item $p_i \to 0$, $V_i \to \R^n \times \{0\}$, $\beta_i \to 0$, $\delta_i \to 0$,
\item $\mu_{M_i}(B_1) \leq (7/4) \Theta_{T_0\Omega_i}$, and $d_{\spt M_i}(0) \to 0$,
\item $\limsup_i \beta_i^{-2} E_{\delta_i}(\Phi_i, M_i, p_i + V_i, 0, 1) < \infty$.
\end{enumerate}
\end{definition}

\begin{prop}\label{prop:blow-up}
Let $(\OmegaRef,  \Omega_i, M_i, p_i + V_i, \beta_i, \delta_i)$ be a blow-up sequence.  Let us write $p_i + V_i = \graph_{\R^n \times \{0\}}(q_i + \phi_i \cdot x)$, for $q_i \in \R$ and $\phi_i \in \{0^l\}\times \R^{m-1}$.  Then there is a $W^{1,2}$ function $v : (\OmegaRefN \times \R^{m-1}) \cap B_{1/4} \to \R$, such that:
\begin{enumerate}
\item $v$ is weakly harmonic with Neumann boundary:
\begin{equation}\label{eqn:blow-up-concl1}
\int_{(\OmegaRefN \times \R^{m-1}) \cap B_{1/4}} Dv \cdot D\zeta = 0 \quad \forall \zeta \in C^1_c(B_{1/4} \cap (\R^n \times \{0\})) ;
\end{equation}

\item $v$ has the $W^{1,2}$ bound
\begin{equation}\label{eqn:blow-up-concl2}
\int_{(\OmegaRefN \times \R^{m-1}) \cap B_{1/4}} v^2 + |Dv|^2 \leq c(\Omega) \left( \limsup_i \beta_i^{-2} E_{\delta_i}(\Phi_i, M_i, p_i + V_i, 0, 1) \right) ;
\end{equation}

\item on any compact set $U \subset\subset B_{1/4} \cap (\OmegaRefN \times \R^{m-1}) \setminus \del\OmegaRef$, we have $L^2$ convergence
\begin{equation}\label{eqn:blow-up-concl3}
\beta_i^{-1} u_i( x + q_i + \phi_i \cdot x) \to v(x);
\end{equation}

\item for any $\rho \leq 1/4$, we have the $L^2$ convergence
\begin{equation}\label{eqn:blow-up-concl4}
\beta_i^{-2} \int_{B_\rho} d_{p_i + V_i}^2 d\mu_M \to \int_{B_\rho \cap (\OmegaRefN \times \R^{m-1})} v^2 .
\end{equation}
\end{enumerate}
\end{prop}

\begin{proof}
Let $\tau_i \to 0$ sufficiently slowly, so that for each $i$ large we can apply Proposition \ref{prop:graph} to to deduce
\[
M_i \cap B_{1/2} \setminus B_{\tau_i}(\del\OmegaRef) = \graph_{p_i + V_i}(u_i),
\]
where $u_i$ is defined on some domain $U_i$ satisfying
\[
(p_i + V_i) \cap B_{1/4} \setminus B_{2\tau_i}(\del\OmegaRef) \subset U_i \subset p_i + V_i.
\]
Write $E_i = E_{\delta_i}(\Phi_i, M_i, p_i + V_i, 0, 1)$.

Fix any $U \subset\subset B_{1/4} \cap (\OmegaRefN \times \R^{m-1}) \setminus \del\OmegaRef$.  Then for sufficiently large $i$, $x\mapsto w_i(x) := u_i(x + q_i + \phi_i \cdot x)$ is well-defined, and parameterizes a subset of $\spt M_i$.  For $x \in U$, we have
\[
Dw_i(x) = (1+o(1)) Du_i (x + q_i + \phi_i \cdot x), \quad 1 \leq J(x) \leq 1 + o(1),
\]
where $J(x)$ is the Jacobian of $x \mapsto x + q_i + \phi_i \cdot x$.  It follows from Proposition \ref{prop:graph} that
\begin{equation}\label{eqn:blow-up-bd}
\int_U |w_i|^2 + |Dw_i|^2 \leq c E_i, \quad \sup_U |w_i|^2 \leq cE_i,
\end{equation}
and for any $\zeta \in C^1_c(B_{1/4})$,
\begin{equation}\label{eqn:blow-up-eqn}
\left| \int_U D\zeta \cdot Dw_i \right| \leq o(1)  |\zeta|_{C^1} E_i^{1/2}.
\end{equation}

If $v_i = \beta_i^{-1} w_i$, then \eqref{eqn:blow-up-bd} (and our definition of blow-up sequence) shows that the $v_i$ are bounded in $W^{1,2}(U)$, with a bound independent of either $i$ or $U$.  A diagonalization argument implies there is a $v \in W^{1,2}(B_{1/4} \cap (\OmegaRefN \times \R^{m-1}))$, satisfying the bound \eqref{eqn:blow-up-concl2}, so that for every $U$ as above $v_i \to v$ strongly in $L^2(U)$ and weakly in $W^{1,2}(U)$.

From \eqref{eqn:blow-up-bd} we have
\begin{equation}\label{eqn:blow-up-height}
\sup |v|^2 \leq c(\OmegaRef),
\end{equation}
and from \eqref{eqn:blow-up-eqn} we get that $v$ satisfies the required \eqref{eqn:blow-up-concl1}.  The strong $L^2$ convergence \eqref{eqn:blow-up-concl4} follows from the $L^\infty$ bounds \eqref{eqn:blow-up-height}, \eqref{eqn:graph-concl2.5}, \eqref{eqn:mass-non-conc} and the fact
\[
\int_{B_{\rho} \setminus B_{\tau_i}(\del\OmegaRef)} d_{p_i + V_i}^2 d\mu_M = (1+o(1))\int_{U_i} |u_i|^2. \qedhere
\]
\end{proof}

\begin{definition}
Let us call any $v$ as obtained in Proposition \ref{prop:blow-up} a \emph{Jacobi field} on $\OmegaRefN \times \R^{m-1}$.
\end{definition}

We are now ready to prove Proposition \ref{prop:decay}.
\begin{proof}[Proof of Proposition \ref{prop:decay}]
Suppose, towards a contradiction, there is a sequence $\delta_i \to 0$, $V_i \in \cP_{T_0\Omega_i}$, $p_i \in V_i^\perp$, $\Omega_i = \Phi_i(\Psi_i(\OmegaRef)) \in \cD_{\delta_i}(\OmegaRef)$, and $M_i \in \cIH_n(\Omega_i, B_1)$, such that
\[
E_i:= E_{\delta_i}(\Phi_i, M_i, p_i + V_i, 0, 1) \leq \delta_i^2, \quad \mu_{M_i}(B_1) \leq (7/4)\Theta_{T_0\Omega_i}, \quad \spt M_i \cap B_{\delta_i} \neq \emptyset,
\]
but for which
\[
E_{\delta_i}(\Phi_i, M_i, p' + V', 0, \theta) \geq c_0 \theta^{2(e-1)} E_i,
\]
for all $p' + V'$ satisfying \eqref{eqn:decay-concl1}.  Here $c_0$ is a constant depending only on $\OmegaRef$ that we will choose later.

Let $\beta_i^2 = E_{\delta_i}(\Phi_i, M_i, p_i + V_i, 0, 1)$, so of course $\beta_i \to 0$ also.  From the height bound \eqref{eqn:sharp-height}, we can assume $p_i \to 0$, and after passing to a subsequence and rotating as necessary, we can assume $V_i \to \R^n \times \{0\}$.  Then $(\OmegaRef, \Omega_i, M_i, p_i + V_i, \beta_i, \delta_i)$ is a blow-up sequence, and we can apply Proposition \ref{prop:blow-up} to obtain a Jacobi field $v$, satisfying
\[
\int_{\OmegaRefN \times \R^{m-1}} Dv \cdot D\zeta = 0 \quad \forall \zeta \in C^1_c(B_{1/4} \cap (\R^n \times \{0\})),
\]
and the $W^{1,2}$ estimate
\[
\int_{(\OmegaRefN \times \R^{m-1}) \cap B_{1/4}} |v|^2 + |Dv|^2 \leq c(\OmegaRef).
\]
From Proposition \ref{prop:expansion}, we can expand in $W^{1,2}$
\begin{equation}\label{eqn:decay-jacobi-exp}
v(x = r\omega) = b + A \cdot x + \sum_i a_i r^{\alpha_i} \phi_i(\omega),
\end{equation}
for some $A \in \{0^l\}\times \R^{m-1}$, and $\alpha_i \geq e > 1$, for $e(\OmegaRefN)$ as in \ref{prop:expansion}.  Using the $L^2( (\OmegaRefN \times \R^{m-1}) \cap S^{n-1})$-orthogonality of this expansion for every fixed $r$, we get
\begin{equation}\label{eqn:decay-jacobi-bd}
|b|^2 + |A|^2 + \sum_i \frac{a_i^2 (1/4)^{2\alpha_i + n}}{2\alpha_i + n} \leq c(\OmegaRef).
\end{equation}

Let us write
\[
p_i + V_i = \graph_{\R^n\times \{0\}} ( q_i + \phi_i \cdot x),
\]
for $q_i \in \R$, and $\phi_i \in \{0^l\}\times \R^{m-1}$.  Now define the new affine planes
\[
p_i' + V_i' = \graph_{\R^n \times \{0\}} (q_i + \beta_i b + (\phi_i + \beta_i A) \cdot x).
\]
Since $A \in \{0^l\}\times \R^{m-1}$, we have $V_i' \in \cP_\OmegaRef$, and we can take $p_i' \in {V_i'}^\perp$.  By definition of the $p_i'$, $V_i'$, and by considering the analytic maps
\[
\phi \mapsto \pi_\phi := \pi_{\graph_{\R^n\times\{0\}}(\phi \cdot x)}, \quad (q, \phi) \mapsto \pi^\perp_\phi(q e_{n+1})
\]
we have
\begin{equation}
|p_i - p_i'| + |\pi_{V_i} - \pi_{V_i'}| \leq c(n) \beta_i (|b| + |A|) \leq c_{d1}(\OmegaRef) \beta_i.
\end{equation}
Therefore, $(\OmegaRef, \Omega_i, M_i, p_i' + V_i', \beta_i, \delta_i)$ is a blow-up sequence also, and we can again use Proposition \ref{prop:blow-up} to obtain a new Jacobi field $v'$ (with, a priori, a slightly worse $W^{1,2}$ bound than $v$).

There is a sequence $\tau_i \to 0$, so that we can write
\begin{equation}\label{eqn:decay-M-graph}
M_i \cap B_{1/2} \setminus B_{\tau_i}(\del\OmegaRef) = \graph_{p_i + V_i}(u_i) = \graph_{p_i' + V_i'}(u_i'),
\end{equation}
where, for any compact subset $U  \subset\subset B_{1/4} \cap (\mathrm{int} \OmegaRefN \times \R^{m-1})$, we have
\begin{equation}\label{eqn:decay-M-C1}
|u_i(x + q_i + \phi_i \cdot x)|_{C^1(U)} + |u_i'(x + q_i + \beta_i b + (\phi_i + \beta_i A)\cdot x)|_{C^1(U)} \to 0,
\end{equation}
and (from Proposition \ref{prop:blow-up}.3)
\begin{equation}\label{eqn:decay-u-lim}
\beta_i^{-1} u_i(x + q_i + \phi_i \cdot x) \to v, \quad \beta_i^{-1} u_i'(x + q_i + \beta_i p + (\phi_i + \beta_i A) \cdot x) \to v'
\end{equation}
in $L^2(U)$.

From \eqref{eqn:decay-M-graph}, \eqref{eqn:decay-M-C1} and since $|p_i| + |\phi_i| \to 0$, we have that for every $x \in U$,
\begin{equation}\label{eqn:decay-u-u'}
|u_i'( x + q_i + \beta_i p + (\phi_i + \beta_i A) \cdot x) - u_i(x + q_i + \phi_i \cdot x) - \beta_i p - \beta_i  A\cdot x| \leq o(1) \beta_i,
\end{equation}
where $o(1) \to 0$ as $i \to \infty$.  By \eqref{eqn:decay-u-lim} and \eqref{eqn:decay-u-u'}, we deduce that
\[
v'(x) = v(x) - b - A \cdot x = \sum_i a_i r^{\alpha_i} \phi_i(x),
\]
where $a_i$ as in equation \eqref{eqn:decay-jacobi-exp}.

Therefore, using \eqref{eqn:decay-jacobi-bd}, $4 \theta \leq 1$ and $|\alpha_i| \geq 1+ \alpha$, we have:
\[
\int_{B_\theta \cap (\OmegaRefN \times \R^{m-1})} |v'|^2 = \sum_i \frac{a_i^2 \theta^{2\alpha_i + n}}{2\alpha_i + n} \leq c_{d2}(\OmegaRef)(4\theta)^{n+2e}.
\]
Provided we take $c_0$ larger than $c_{d1}$ and $8^{4+n} c_{d2}$, then by the strong $L^2$ convergence
\[
\beta_i^{-2} \int_{B_\theta} d_{p_i' + V_i'}^2 d\mu_M \to \int_{B_\theta \cap (\OmegaRefN \times \R^{m-1})} |v'|^2,
\]
we obtain a contradiction (we remind the reader that the $H$ and $\Phi$ terms of $E$ decay gratuitously).  This completes the proof of Proposition \ref{prop:decay}.
\end{proof}

config: arxiv


TITLE: Can I infer that if I'm told $f(x)=\ln x$ then $f(y)=\ln y$
QUESTION [1 upvotes]: I have a lengthy economics question, I am told that $f(x)=\ln x$ (the natural log of $x$) but I am also given the functions $f(y)$ to work with. Since $f(.)$ is a function which applies a transformation, am I able to imply that $f(y)=\ln y$ from the transformation $f(x)$ even thought I am not told this?

REPLY [3 votes]: Yes, because when we write 'let $f(y)=\ln y$', this is simply an abbreviation of the statement 'let $f:\mathbb{R^+}\mapsto\mathbb{R}$ be the function defined by $f(y)=\ln y$ for all $y$'. The qualifying statement 'for all $y'$ tells us that $y$ is a dummy variable—the only purpose it is serving is to tell us that each number is mapped to its natural logarithm. Hence, it makes no difference to write 'let $f(x)=\ln x$'.


text:
\section{Other non-commutative systems}

In this section, we consider further examples of non-commutative recursions which
have the positive Laurent property. In the following,
all the variables we consider belong to some skew field of rational fractions $\cA$.

\subsection{Affine rank $2$ non-commutative systems}

The non-commutative $A_1$ $Q$-system \eqref{aoneNCQsystem} corresponds to one of the so-called
rank two affine cluster algebras, namely with initial exchange matrix $B_0$
related to affine Cartan matrices
of rank 2. These read $B_0=\begin{pmatrix} 0 & -c \\ b & 0 \end{pmatrix}$ with $bc=4$, $b,c\in\Z_{>0}$.
While the $A_1$ case corresponds to $b=c=2$, the two other cases $(b,c)=(1,4)$ or $(4,1)$
were shown in Ref. \cite{DFK09b}
to be related to the following non-commutative system for $\bR_n\in\cA$, $n\in\Z$:
\begin{equation}\label{onefour}
\bR_{n+1}(\bR_n)^{-1}\bR_{n-1} \bR_n= \left\{ \begin{matrix} (\bR_n)^b +1 \hfill & {\rm if} & n\, {\rm odd}\\
(\bR_n)^c +1 \hfill & {\rm if} & n\, {\rm even}
\end{matrix}\right.
\end{equation}
Let us fix $b=1,c=4$.

\begin{thm}\label{onefourthm}
The solution $\bR_n$ of the system \eqref{onefour} is a positive Laurent polynomial of any initial
data $(\bR_i,\bR_{i+1})$, for any $i,n\in\Z$.
\end{thm}

We sketch the proof below.
As in the $A_1$ case, one finds two conserved quantities and a linear recursion relation for $\bR_n$
with constant coefficients.
Finally, the generating function for $\bR_{2n}\bR_0^{-1}$ reads:
$$\bF_0(t)=\sum_{n\geq 0} t^n \, \bR_{2n} \bR_0^{-1}
=\left(1-t \bz_1 -t^2(1-t \bz_3)^{-1} \bz_2\right)^{-1}$$
where
\begin{eqnarray*}
\bz_1&=&\bu_1 \bu_0^{-1}=(\bR_1^{-1}+1)\bR_0^{-1}\bR_1\bR_0^{-1}\\
\bz_2&=&(\bK-\bz_1)\bz_1 -\bC= \big( \bR_1^{-1}\bR_0^2\bR_1^{-1}+
(1+\bR_1^{-1})\bR_0^{-2}(1+\bR_1^{-1})
\big)\bR_0^{-1}\bR_1\bR_0^{-1}\\
\bz_3&=&\bK-\bz_1= \bR_1^{-1}\bR_0^2+(1+\bR_1^{-1})\bR_0^{-2}
\end{eqnarray*}
and the Laurent positivity follows for $(\bR_0,\bR_1)$.
The mutated expressions in terms of $(\bR_i,\bR_{i+1})$
are again obtained by manifestly positive non-commutative rearrangements of $\bF_0(t)$,
and the general Laurent positivity follows from symmetries of the system.

\subsection{Some rank $2k+1$ cases}

For any fixed $k\in\Z_+$,
we consider the ``rank $2k+1$" recursion relation for $n\in\Z$, for variables $\bu_n\in\cA$:
\begin{eqnarray}
\bu_{2n+2k+1} \bu_{2n}&=& \bu_{2n+1}\bu_{2n+2k}+{\bf 1}\label{rk3one}\\
\bu_{2n+1}\bu_{2n+2k+2}&=& \bu_{2n+2k+1}\bu_{2n+2} +{\bf 1}\label{rk3two}
\end{eqnarray}

Admissible initial data for this system are $\bx_i=(\bu_i,\bu_{i+1},\ldots, \bu_{i+2k})$ for all $i\in\Z$.
We have the following positivity theorem.

\begin{thm}\label{positrkthree}
The general solution $\bu_n$ to the system \eqref{rk3one}-\eqref{rk3two} is a positive Laurent
polynomial of any initial data $\bx_i$, for all $n,i\in\Z$.
\end{thm}

This is proved in the Appendix \ref{appendixb}, by  using the integrability of the system
and constructing non-commutative weighted path models for $u_n$. The commutative version of
the system \eqref{rk3one}-\eqref{rk3two} corresponds to the affine $A_{2k}$ cluster algebra \cite{AssemReit}.
Note that the exchange matrix of the fundamental seed is singular, so there is no quantum
cluster algebra version of the system \eqref{rk3one}-\eqref{rk3two}.

\subsection{A non-coprime rank 2 case related to rank 3}

The following non-commutative recursion relation
also satisfies the positive Laurent property:
\begin{equation}\label{firla}\bR_{n+1}(\bR_n)^{-1} \bR_{n-1}=\bR_n+1 \end{equation}
Note that we still have the quasi-commutation relation $\bR_n\bR_{n+1}=\bR_{n+1}\bC\bR_n$
from the conserved quantity $\bC$, and that we may equivalently write
$\bR_{n+1}\bC\bR_{n-1}=\bR_n(\bR_n+1)$,
or
\begin{equation}\label{lastla}\bR_{n-1}(\bR_n)^{-1}\bR_{n+1}=\bR_n+\bC^{-1}\end{equation}
In the commuting case, it would correspond to the recursion relation $R_{n+1}R_{n-1}=R_n(R_n+1)$
which does not correspond to a cluster algebra mutation, as the two terms on the r.h.s. are not coprime.
However, this relation may be connected to a non-commutative rank 3 system, with the
positive Laurent property,  as follows.
Introduce the change of variables:
$$ \bR_{2n+1}=\bu_{2n+1}\bu_{2n}, \qquad \bR_{2n}=\bu_{2n-1}\bu_{2n}$$
for some variables $\bu_n$, $n\in \Z$. We have:
\begin{eqnarray*}
\bu_{2n+1}\bu_{2n-2}&=&\bu_{2n-1}\bu_{2n}+\bone\\
\bu_{2n-1}\bu_{2n+2}&=&\bu_{2n+1}\bu_{2n}+\bC^{-1}
\end{eqnarray*}
This is nothing but the $k=1$ system \eqref{rk3one}-\eqref{rk3two} with an additional
fixed coefficient $\bC^{-1}$. Choosing the initial data $\bu_0=\bR_1$, $\bu_1=\bone$ and $\bu_2=\bR_2$,
with $\bC^{-1}=\bu_0\bu_2^{-1}\bu_0^{-1}\bu_2$, we may repeat the proof of Appendix
\ref{appendixb} so as to include the fixed coefficient $\bC^{-1}$, and deduce the
Laurent positivity property in terms of $(\bR_1,\bR_2)$.

\section{Towards a non-commutative Lindstr\"om-Gessel-Viennot Theorem}

All the cases we have been investigating so far are expressible in
terms of non-commutative weighted path models. More precisely, we have
been able to represent the generating function $\sum_{n\geq 0} t^n\bR_n\bR_0^{-1}$ as the partition
function for paths with non-commutative weights. In the application of the $A_r$
case to the quantum $A_r$ $Q$-system of Section \ref{section5}, we also have variables
$\bR_{\al,n}$, $n>1$, expressed as ``quantum Wronskians" (see Remark \ref{quantumwronsk}).
This is actually a non-commutative generalization of the ``Wronskian" expressions
for the $Q$-system solutions in the commuting case \eqref{discwronskco}, which take the form:
$${R_{\al,n+m_1}\over (R_{1,m_1})^\al}
=\det_{1\leq i,j\leq \al}\left( {R_{1,i+j-\al-1+n+m_1}\over R_{1,m_1} }\right) $$
In the cases $\bm=\bm_0$ or $\bm_1$ for instance
(fundamental initial data and Stieltjes point, both with $m_1=0$)
the Lindstr\"om-Gessel-Viennot theorem \cite{LGV1} \cite{LGV2} allows to interpret directly the l.h.s. as the partition
function for a family of $\al$ non-intersecting lattice paths,
namely paths traveled by walkers
starting at the root of the graph $\Gamma_\bm$ at times $0,2,...,2\al-4,2\al-2$ and
returning at times $2n+2\al-2,2n+2\al-4,...,2n+2,2n$,
and such that no two walkers meet at any vertex of
$\Gamma_\bm$. In the case of general $\bm$, we have a more subtle application of the theorem,
in which the paths must be strongly non-intersecting (see Ref. \cite{DFK3} for details).

It would be extremely interesting to extend the classical
Lindstr\"om-Gessel-Viennot theorem
to the case of paths with non-commutative weights.
Such a generalization was introduced in \cite{FG} in the context
of non-commutative Schur functions,
but does not seem to apply directly to the present framework.

So far, we have found such an interpretation only in the fully non-commutative
$A_1$ $Q$-system case. We wish to interpret the $Q$-system relation
$\bR_{n+1}\bC\bR_{n-1}-\bR_n^2=1$
in terms of pairs of non-intersecting paths on the graph $\Gamma_{(0)}$, i.e. the integer segment $[1,4]$.
Let $\bP_n=\bR_n\bR_0^{-1}$ denote the partition function of paths from vertex $0$ to itself
on $\Gamma_{(0)}$ with the non-commutative weights $\by_1,\by_2,\by_3$ respectively for the steps
$2\to 1$, $3\to 2$ and $4\to 3$, while the up steps $1\to2$, $2\to 3$ and $3\to 4$
receive the trivial weight $1$. Multiplying the $Q$-system relation by $\bC$, we have:
\begin{equation}\label{cnilp}
\bP_{n+1}\bR_0\bC\bP_{n-1}\bR_0\bC-\bP_n\bR_0\bP_n\bR_0\bC=\bC
\end{equation}
To proceed, we note the following important relations between the weights $\by_1,\by_2,\by_3$
and $\bR_0$, $\bC$:
\begin{equation}\label{commutaone}
(\by_3)^n\by_2\by_1\bR_0\bC(\by_1)^n=\by_2\by_1\bR_0\bC=\bR_0^{-1} \qquad
(\by_1)^n \bR_0 (\by_3)^n =\bR_0 \quad (n\in \Z)
\end{equation}

\begin{figure}
\centering
\includegraphics[width=11.cm]{lgv}
\caption{\small A typical contribution to $\bP_{n+1}\bR_0\bC\bP_{n-1}\bR_0\bC$ for $n=8$ is made of
a pair of paths $A\to B$ (blue, left to right) and $C\to D$ (red, right to left),
with two intermediate stations at vertex $1$, $B\to C$ and $D\to A$.
When paths intersect (like here), this may also be
interpreted as a contribution to $\bP_n\bR_0\bP_n\bR_0\bC$, upon reversing the direction of travel of the
portions of paths to the right of their rightmost intersection $X$,
with a weight $\bR_0$ for the step $C\to B$.}
\label{fig:lgv}
\end{figure}

We now interpret the quantity $\bP_{n+1}\bR_0\bC\bP_{n-1}\bR_0\bC$ as the partition function for
a path loop $A\to B\to C\to D\to A$ (see Fig. \ref{fig:lgv}), where the path $A\to B$
(partition function $\bP_{n+1}$) is traveled from left to right
and assigned the usual descending step weights $\by_i$ for a step $i+1\to i$, while the
portions $B\to C$, $D\to A$ receive the weight $\bR_0\bC$, and the path $C\to D$
(partition function $\bP_{n-1}$)
is traveled from right to left, and receives the usual descending step weights
$\by_i$ for a step $i+1\to i$.
The contribution for the configuration in the top of
Fig.\ref{fig:lgv} is $\by_3^2\by_2^2\by_1\by_3^2\by_2\by_1\bR_0\bC \by_1^2\by_3\by_2\by_1^3\bR_0\bC$.

\begin{thm} The quantity
\begin{equation}\label{cancelaone}
\bP_{n+1}\bR_0\bC\bP_{n-1}\bR_0\bC-\bP_n\bR_0\bP_n\bR_0\bC
\end{equation}
is the partition function of loops $A\to B\to C\to D\to A$
formed by pairs of {\emph non-intersecting} paths on $[1,4]$ respectively from
$A=(0,1)$ to $B=(2n+2,1)$, then from $C=(2n,1)$ to $D=(2,1)$, with the two
intermediate stations $B\to C$ and $D\to A$ at the root receiving the weight $\bR_0\bC$.
\end{thm}
\begin{proof}

Recall first that the portions of loop from $B\to C$, and $D\to A$ receive the weight $\bR_0\bC$.
We define the weight for the same portions, but traveled in the other direction ($C\to B$ and $A\to D$)
to be $\bR_0$.
Let us assume the two paths $B\to C$, and $D\to A$ intersect, namely that they have at least a common vertex.
Let us denote by $X$ the rightmost such common vertex.

We may now decompose
the above loop into three pieces: that from $A\to X$, $X\to B\to C\to X$ and that from $X\to A$.
The central portion $X\to B\to C\to X$, with weight $W_{XBCX}$, may be ``flipped" into
$X\to C\to B\to X$,
with new weight $W_{XCBX}$, by
reversing the direction of travel, so that
the new configuration now corresponds to pairs of paths $A\to C$ and $B\to D$
of equal length $2n$, now separated by the two intermediate stations $C\to B$ and $D\to A$
with respective weights $\bR_0$ and $\bR_0\bC$, thus contributing
to $\bP_n\bR_0\bP_n\bR_0\bC$.

We now note that $W_{XCBX}=W_{XBCX}$.
Indeed, depending on the height of $X$, we have the two following possibilities
(the first case is illustrated in Fig.\ref{fig:lgv}):
\begin{itemize}
\item (i) $X=(2t+1,2)$ for some $t$, then the blue path $X\to B$
is $3\to 3\to 4 \to 3\to 4\to \cdots \to 4\to 3\to 2$ (weight $\by_3^m \by_2\by_1$, $m=n-t$)
and the red path $C\to X$ is $1\to 2\to 1\to 2\to \cdots \to 1\to 2$ (weight $\by_1^m$).
The weight of the central loop $X\to B\to C\to X$ is:
$$W_{XBCX}= \by_3^m (\by_2\by_1 \bR_0\bC) \by_1^m=\bR_0^{-1}
=\by_1\bR_0\by_2= \by_1^{m+1}\bR_0  \by_3^m\by_2=W_{XCBX}$$
by use of \eqref{commutaone}.
\item (ii) $X=(2t+2,3)$ for some $t$, then the blue path $X\to B$
is $3\to 4 \to 3\to 4\to \cdots \to 4\to 3\to 2$ (weight $\by_3^m \by_2\by_1$, $m=n-t$)
and the red path $C\to X$ is $1\to 2\to 1\to 2\to \cdots \to 1\to 2\to 3$ (weight $\by_1^{m-1}$).
The weight of the central loop $X\to B\to C\to X$ is:
$$ W_{XBCX}= \by_3^m (\by_2\by_1 \bR_0\bC) \by_1^{m-1} =\by_3\bR_0^{-1}
=\bR_1^{-1}=\by_2\bR_0=\by_2 \by_1^{m}\bR_0 \by_3^m =W_{XCBX}$$
by use of \eqref{commutaone}.
\end{itemize}

The above flipping procedure is a bijection between pairs of paths $A\to B$ and $C\to D$ that
intersect and pairs of paths $A\to C$ and $B\to D$. They cancel each other in the formula
\eqref{cancelaone}, leaving us only with non-intersecting paths, and the theorem follows.
\end{proof}

Note that there is a unique configuration of non-intersecting paths, where $A\to B$ is
$1\to 2\to 3\to 4\to 3\to \cdots \to 4\to 3\to 2\to 1$ and $C\to D$ is $1\to 2\to 1\to 2\to \cdots \to 2\to 1$.
The corresponding weight is $\by_3^{n-1} (\by_2\by_1 \bR_0\bC) \by_1^{n-1}\bR_0\bC=\bC$ by
\eqref{commutaone}, and we recover \eqref{cnilp}.

\section{Conclusion: Towards non-commutative cluster algebras}

The rank 2 non-commutative cluster algebras are defined as follows.
Let us consider a bipartite
chain ${\mathbb T}_2$ (Cayley tree of degree 2), with alternating black and white vertices
indexed by $v\in\Z$ (black if $v$ is even, white otherwise), and
edges with alternating labels $1,2$, and a fixed element $\bC\in\cA$.
To each vertex $v$ of ${\mathbb T}_2$, we attach a cluster, i.e. a vector of the form $\bx_v=(\ba,\bb)$ in $\cA^2$,
where the cluster variables $\ba,\bb$ obey the quasi-commutation relations
$$\begin{matrix}\ba\, \bb=\bb\bC \ba \hfill & {\rm if} &v\, {\rm is}\, {\rm black}\\
\bb\, \ba=\ba\bC \bb \hfill & {\rm if} &v\, {\rm is}\, {\rm white} \end{matrix}$$

To each (doubly oriented) edge labeled $i$ connecting a black and a white vertex,
we associate the two following mutations: (i) the forward mutation $\mu_i^{+}$ corresponding to the oriented
edge from the black vertex to the white (ii) the backward mutation  $\mu_i^{-}=(\mu_i^{+})^{-1}$
corresponding to the oriented edge from the white vertex to the black.
Depending on the color of the vertex $v$,
the forward and backward mutations of $x_v=(\ba,\bb)$ are
defined as follows:
$$\begin{matrix} v\, {\rm black}: &\mu_1^{+}:(\ba,\bb)\mapsto (\ba',\bb)\hfill &{\rm with} & \ba'\bC \ba =(\bb)^b+\bone \hfill\\
&\mu_2^{-}:(\ba,\bb)\mapsto (\ba,\bb')\hfill &{\rm with} & \bb\bC \bb' =(\ba)^c+\bone \hfill\\
v\, {\rm white}: &\mu_2^{+}:(\ba,\bb)\mapsto (\ba,\bb')\hfill &{\rm with} & \bb'\bC \bb =(\ba)^c+\bone \hfill\\
&\mu_1^{-}:(\ba,\bb)\mapsto (\ba',\bb)\hfill &{\rm with} & \ba\bC \ba' =(\bb)^b+\bone \hfill
\end{matrix}$$
These definitions are compatible with an exchange matrix equal to $B_\bullet=\begin{pmatrix} 0 & -c\\ b &0\end{pmatrix}$
for black vertices and $B_\circ=-B_\bullet$ for white ones, evolving according to the rules of the usual rank 2 cluster
algebras.
The general conjecture of \cite{Kon} may be rephrased as follows: {\it for any positive integers $b,c$,
the cluster variables at any node of ${\mathbb T}_2$ may be expressed as positive Laurent polynomials of
the cluster variables at any other node of ${\mathbb T}_2$}.
This is easily proved in the ``finite cases" for $bc<4$ (leading to only finitely many distinct cluster
variables in the commuting case), where one can show explicitly that there exists a positive integer $m_{b,c}$
such that $\bx_{v+m_{b,c}}=\bC\bx_v\bC^{-1}$, with $m_{1,1}=5$, $m_{1,2}=6$, and $m_{1,3}=8$,
leaving us with only finitely many cluster variables to inspect. In the affine case $bc=4$, this was proved in
\cite{DFK09b} (see the $A_1$ case $b=c=2$ in the introduction,
and the case $b=1,c=4$ Theorem \ref{onefourthm} above),
while the case $bc>4$ is still open.

To generalize this construction, we may think of using the results of the present paper in the following way.
Assuming that the $A_r$ $Q$-system cluster algebra introduced in \cite{Ke07} may be generalized to a
non-commutative setting, we would expect the relevant cluster variables to still be expressible
in terms of non-commutatively weighted paths on target graphs, in terms of given initial data. In this paper,
we have shown how to introduce mutations of non-commutative weights
via non-commutative continued fractions.
Concentrating first on the Stieltjes point,  we may trace
the expressions for $\Delta_{1,n}=\bR_{1,n}$ back to the initial variables
$\Delta_{\al,\al}$ and $\Delta_{\al,\al-1}$, $\al=1,2,...,r+1$ (see Theorem \ref{stilnc}).
This seems to indicate, in this case at least, that
the quasi-Wronskians are good non-commutative cluster variables.
However, the general expressions of Theorem \ref{ncweightcalc} for $\bR_{1,n}$
in terms of quasi-Wronskians or the more compact $C,D$ variables
(see \eqref{defCD} and \eqref{evalC}) do not clearly
display which initial data they are related to. We clearly need another definition
of the non-commutative cluster variables.

We hope to report on this issue in a later publication.


TITLE: Does the relation between absolute continuity of functions and absolute continuity of measures hold in $\mathbb{R}^2$?
QUESTION [3 upvotes]: If $\mu$ is a Borel measure on $\mathbb{R}$, then $\mu$ is absolutely continuous with respect to Lebesgue measure if and only if $\mu\left((-\infty,x]\right)$ is an absolutely continuous function from $\mathbb{R}$ to $\mathbb{R}$.
I'm wondering if this can be generalized to two dimensions.  Let $\mu$ be a Borel measure on $\mathbb{R}^2$.  My question is, is there some notion of absolute continuity for multivariable functions such that we can say that $\mu$ is absolutely continuous with respect to Lebesgue measure if and only if $\mu\left((-\infty,x]\times(-\infty,y]\right)$ is an absolutely continuous function from $\mathbb{R}^2$ to $\mathbb{R}$?
This paper, this paper, and this paper discuss generalizations of absolute continuity to functions of multiple variables.  Do any of these generalizations yield the notion I want?
This is all an attempt to better understand jointly continuous random variables.

REPLY [1 votes]: Let $\mu$ be absolutely continuous w.r.t. Lebesgue measure. For a rectangle $R=(a,b]\times (c,d]$ define $F(R)=F(b,d)-F(a,d)-F(b,c)+F(a,c)$. Then, given $\epsilon >0$ three exists $\delta >0$ such that for any finite disjoint collection of $(R_i)$ of rectangles with total area less than $\delta$ we have $\sum F(R_i) <\epsilon$. The converse is also true.


TITLE: How to prove that the limit belongs to $(0, 1)$?
QUESTION [2 upvotes]: In a sequence of positive numbers $$a_{1}, a_{2},\dots,a_{n},\dots,$$
each of the terms $a_n$ is equal to either
$\dfrac{ a_{n-1}  }{ 2}$ or $\sqrt{a_{n - 1}}$.
Can this sequence have a limit belonging to the interval $(0, 1)$?

REPLY [2 votes]: Assume that $\lim_{n \to \infty} a_n = a$ exists, with $0 < a < 1$. Then there exists an $N$ such that for all $n \ge N$
$$
 \frac 34 a < a_n < \frac 54 a \, .
$$
It follows that for $n> N$
$$
 \color{red}{a_n} > \frac 34 a > \frac 12 \cdot \frac 54 a \color{red}{> \frac 12 a_{n-1}}
$$
and consequently
$$
 a_n = \sqrt{a_{n-1}} \, .
$$
The limit must therefore satisfy
$$
 a = \sqrt a
$$
which is not possible for $0 < a < 1$.
Therefore such a sequence does not exist.