Limiting Distributions for Sums of Independent Random Products

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 ${\lim }_{n\to \mathrm{\infty }}{e}^{-cn}{N}_n=1$ for some $c>0$. Our main object of interest is the sum of independent random products $$Z_n=\sum _{i=1}^{N_n}\prod _{j=1}^n{e}^{X_{i,j}}\mathrm{.}$$ It is shown that the limiting properties of $Z_n$, as $n\to \mathrm{\infty }$, 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, $Z_n\mathrm{/}\mathbb{E}{Z}_n$ 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].