# Limiting Distributions for Sums of Independent Random Products - Mathematics > Probability

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Abstract: Let $\{X {i,j}:i,j\in\mathbb N^2\}$ be a two-dimensional array ofindependent copies of a random variable $X$, and let $\{N n\} {n\in\mathbb N}$be a sequence of natural numbers such that $\lim {n\to\infty}e^{-cn}N n=1$ forsome $c>0$. Our main object of interest is the sum of independent randomproducts $$Z n=\sum {i=1}^{N n} \prod {j=1}^{n}e^{X {i,j}}.$$ It is shown thatthe limiting properties of $Z n$, as $n\to\infty$, undergo phase transitions attwo 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$cc 1$. Ifthe 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 SKmodels. Ann. Probab. 302002, 605-651.

Autor: Zakhar Kabluchko

Fuente: https://arxiv.org/