Limiting Distributions for Sums of Independent Random Products

TL;DR

This paper investigates the asymptotic distributional behavior of sums of independent random products, revealing phase transitions at two critical points and connecting to known results in Gaussian cases.

Contribution

It introduces a detailed analysis of the limiting distributions of sums of independent random products, identifying phase transitions and extending previous Gaussian results to more general distributions.

Findings

For c > c_2, Z_n satisfies a central limit theorem.

For c < c_2, Z_n converges to a totally skewed alpha-stable law.

Z_n / E[Z_n] converges in probability to 1 if and only if c > c_1.

Abstract

Let $\{X_{i,j}:(i,j)\in\mathbb N^2\}$ be a two-dimensional array of independent copies of a random variable $X$, and let $\{N_n\}_{n\in\mathbb N}$ be a sequence of natural numbers such that $\lim_{n\to\infty}e^{-cn}N_n=1$ for some $c>0$. Our main object of interest is the sum of independent random products $$Z_n=\sum_{i=1}^{N_n} \prod_{j=1}^{n}e^{X_{i,j}}.$$ It is shown that the limiting properties of $Z_n$, as $n\to\infty$, undergo phase transitions at two critical points $c=c_1$ and $c=c_2$. Namely, if $c>c_2$, then $Z_n$ satisfies the central limit theorem with the usual normalization, whereas for $c<c_2$, a totally skewed $\alpha$-stable law appears in the limit. Further, $Z_n/\mathbb E Z_n$ converges in probability to 1 if and only if $c>c_1$. If the random variable $X$ is Gaussian, we recover the results of Bovier, Kurkova, and L\"owe [Fluctuations of the free energy in the REM and the $p$-spin SK models. Ann. Probab. 30(2002), 605-651].