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 \(S_n = \xi _1 + \ldots + \xi _n\), where \((\xi _k)_{k \ge 1}\) are independent copies of a nonnegative random variable \(\xi \). 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 \(\xi \), we derive logarithmic asymptotics for the local large deviation probabilities \(P{\hat{N}(t) = \lfloor bE[\hat{N}(t)]}\) as \(t \rightarrow \infty \) 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.