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.
From: Alexander Marynych [view email]
[v1]
Thu, 7 Aug 2025 09:15:08 UTC (36 KB)
[v2]
Wed, 26 Aug 2026 19:43:39 UTC (38 KB)
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