On local large deviations for decoupled random walks

Abstract

A decoupled standard random walk is a sequence of independent random variables $({\hat{S}}_n{)}_{n\ge 1}$ such that, for each $n\ge 1$, the distribution of ${\hat{S}}_n$ is the same as that of $S_n=\hat{I}{¾}_1+\dots +\hat{I}{¾}_n$, where $(\hat{I}{¾}_k{)}_{k\ge 1}$ are independent copies of a nonnegative random variable $\hat{I}¾$. We consider the counting process $(\hat{N}(t){)}_{t\ge 0}$ defined as the number of terms ${\hat{S}}_n$ in the sequence $({\hat{S}}_n{)}_{n\ge 1}$ that lie within the interval $[0,t]$. Under various assumptions on the tail distribution of $\hat{I}¾$, we derive logarithmic asymptotics for the local large deviation probabilities $\mathbb{P}\{\hat{N}(t)=\lfloor b\mathbb{E}[\hat{N}(t)]\rfloor \}$ as $t\to \infty$ for a fixed constant $b>0$. These results are then applied to obtain a logarithmic local large deviations asymptotic for the counting process associated with the infinite Ginibre ensemble and, more generally, for determinantal point processes with the Mittag-Leffler kernel.