[Paper Review] Exponential rate of L_p-convergence of intrinsic martingales in supercritical branching random walks

Gerold Alsmeyer, Alexander Iksanov+2 more|ArXiv.org|Mar 23, 2009

Stochastic processes and statistical mechanics5 citations

TL;DR

This paper establishes necessary and sufficient conditions for the $L_p$-convergence of the series $\sum_{n \geq 0} e^{an}(W - W_n)$, where $W_n$ is the intrinsic martingale in a supercritical branching random walk and $W$ its almost sure limit. The key result identifies the exponential rate of $L_p$-convergence via moment conditions on the offspring distribution and the spectral radius of the generating function $m(r)$, with sharp criteria depending on $p$ and the parameter $a > 0$. The analysis relies on martingale inequalities, moment estimates, and the convexity of $L_p$-norms via Burkholder-type arguments.

ABSTRACT

Let $W_n, n\in\mn_{0}$ be an intrinsic martingale with almost sure limit $W$ in a supercritical branching random walk. We provide criteria for the $L_p$-convergence of the series $\sum_{n\ge 0} e^{an}(W-W_n)$ for $p>1$ and $a>0$. The result may be viewed as a statement about the exponential rate of convergence of $\me |W-W_n|^p$ to zero.

Motivation & Objective

• To determine the exponential rate of $L_p$-convergence of the intrinsic martingale $W_n$ to its limit $W$ in a supercritical branching random walk.
• To derive necessary and sufficient conditions for the almost sure and $L_p$-convergence of the series $\sum_{n \geq 0} e^{an}(W - W_n)$ for $p > 1$ and $a > 0$.
• To characterize the decay rate of $\mathbb{E}|W - W_n|^p$ in terms of the moment generating function $m(r)$ of the offspring distribution.
• To establish sharp criteria involving the spectral radius $m^{1/r}(r)$ and the $p$-th moment of the initial weight distribution.

Proposed method

• Use of the intrinsic martingale $W_n = Z_n / m^n(1)$, where $Z_n$ is the total weight of the $n$-th generation in a weighted branching process.
• Application of Burkholder's inequality and moment estimates to control $\mathbb{E}|W_{n+1} - W_n|^p$ via conditional variances and $L_2$-norms.
• Decomposition of the increment $W_{n+1} - W_n$ as a sum over individuals in generation $n$, weighted by $L_v^2(W_1(v) - 1)^2$.
• Use of convexity and superadditivity of $x \mapsto x^{p/2}$ for $p \geq 2$ to derive lower bounds on $\mathbb{E}|W_{n+1} - W_n|^p$.
• Application of Proposition 1.1 and Lemma 3.5 to bound $\mathbb{E}(Z_n^{(2)})^{p/2}$ in terms of $m^n(p)$ and $m^n(2)$, depending on the relative size of $m(p)$ and $m^{p/2}(2)$.
• Use of Minkowski’s inequality in $L_{p/2}$ to control the $L_p$-norm of the series $\widehat{A}$, reducing the problem to summability of $e^{pan} \mathbb{E}|W_{n+1} - W_n|^p$.

Experimental results

Research questions

• RQ1 Under what conditions does the series $\sum_{n \geq 0} e^{an}(W - W_n)$ converge in $L_p$ for $p > 1$ and $a > 0$?
• RQ2 What is the precise exponential rate at which $\mathbb{E}|W - W_n|^p$ decays to zero in a supercritical branching random walk?
• RQ3 How do the $p$-th moment of the initial weight distribution and the function $m(r)$ influence the convergence behavior of the martingale increments?
• RQ4 Is the condition $e^a m^{1/p}(p) < 1$ necessary and sufficient for $L_p$-convergence when $p > 2$?
• RQ5 What role does the relative size of $m(p)$ and $m^{p/2}(2)$ play in determining the convergence criteria?

Key findings

• The $L_p$-convergence of $\sum_{n \geq 0} e^{an}(W - W_n)$ holds if and only if $\mathbb{E}W_1^p < \infty$ and $e^a m^{1/p}(p) < 1$ for $p > 2$, with the condition $e^a m^{1/2}(2) < 1$ being necessary when $m(p) \geq m^{p/2}(2)$.
• For $p = 2$, the $L_2$-convergence holds if and only if $e^a m^{1/2}(2) < 1$, which is equivalent to $e^{2a} m(2) < 1$.
• When $p > \vartheta$, the necessity of $e^a m^{1/p}(p) < 1$ is confirmed by comparing $s_n(2)$ with $m^n(p)$, where $s_n(2)$ is the second-order moment of the increment.
• The condition $e^{ap} m(p) < 1$ is necessary for $L_p$-convergence when $p \geq 2$, derived via Jensen’s inequality and moment bounds on $\mathbb{E}|W_{n+1} - W_n|^p$.
• The sufficiency of the conditions is proven by showing $e^{pan} \mathbb{E}|W_{n+1} - W_n|^p = O(q^n)$ for some $q < 1$, using bounds on $\mathbb{E}(Z_n^{(2)})^{p/2}$ and the convexity of $x \mapsto x^{p/2}$.
• In Case 1 ($m(p) < m^{p/2}(2)$), the bound $\mathbb{E}(Z_n^{(2)})^{p/2} = O(m^n(p))$ ensures convergence with $q = e^{ap} m^{p/2}(2)$, while in Case 2 ($m(p) \geq m^{p/2}(2)$), the bound $\mathbb{E}(Z_n^{(2)})^{p/2} = O(n^c m^n(p))$ allows convergence with $q = \delta e^{pa} m(p)$ for $\delta > 1$ close to 1.