On local large deviations for decoupled random walks

TL;DR

This paper investigates the probabilities of large deviations in the count of a decoupled random walk within a fixed interval, deriving asymptotic formulas under various tail assumptions, with applications to determinantal point processes.

Contribution

It provides new logarithmic asymptotics for local large deviation probabilities of decoupled random walks, extending to processes like the Ginibre ensemble and Mittag-Leffler kernel determinantal processes.

Findings

Derived asymptotics for large deviation probabilities as time grows

Applied results to the Ginibre ensemble counting process

Extended analysis to determinantal point processes with Mittag-Leffler kernel

Abstract

A decoupled standard random walk is a sequence of independent random variables $(\hat{S}_n)_{n \geq 1}$ such that, for each $n \geq 1$, the distribution of $\hat{S}_n$ is the same as that of $S_n = \xi_1 + \ldots + \xi_n$, where $(\xi_k)_{k \geq 1}$ are independent copies of a nonnegative random variable $\xi$. We consider the counting process $(\hat{N}(t))_{t\geq 0}$ defined as the number of terms $\hat{S}_n$ in the sequence $(\hat{S}_n)_{n \geq 1}$ 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 $\mathbb{P}\{\hat{N}(t) = \lfloor b \, \mathbb{E}[\hat{N}(t)] \rfloor\}$ 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.