On local large deviations for decoupled random walks

2025/08/07 by Buraczewski, Dariusz, Iksanov, Alexander, Marynych, Alexander · 1 citation
#60F05 #FOS: Mathematics #Primary: 60F10 #Probability (math.PR) #Secondary: 60G55

paper · doi:10.48550/arxiv.2508.05178

Abstract

A decoupled standard random walk is a sequence of independent random variables (S_n)_{n ≥ 1} such that, for each n ≥ 1, the distribution of S_n is the same as that of S_n = ξ₁ + … + ξ_n, where (ξ_k)_{k ≥ 1} are independent copies of a nonnegative random variable ξ. We consider the counting process (N(t))_{t≥ 0} defined as the number of terms S_n in the sequence (S_n)_{n ≥ 1} that lie within the interval [0, t]. Under various assumptions on the tail distribution of ξ, we derive logarithmic asymptotics for the local large deviation probabilities ℙ\N(t) = \lfloor b 𝔼[N(t)] \rfloor\ as t → ∞ 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.

Citations

• Convolution powers of unbounded measures on the positive half-line
• On decoupled standard random walks
• Large Deviation Probabilities for Sums of Random Variables with Heavy or Subexponential Tails
• Large gap asymptotics on annuli in the random normal matrix model
• Precise deviations for disk counting statistics of invariant\n determinantal processes
• Renewal shot noise processes in the case of slowly varying tails
• Some statistical properties of the eigenvalues of complex random matrices

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• Limit theorems for decoupled renewal processes

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