On asymptotic behavior of almost surely extreme values of independent random variables
Kateryna Akbash, Ivan Matsak
TL;DR
This work analyzes the almost-sure asymptotics of the maximum $\bar{\xi}_n$ of a sequence of discrete i.i.d. random variables. It derives precise necessary-and-sufficient conditions for $\bar{\xi}_n$ to equal $a_n + l$ infinitely often, where $a_n$ encodes the tail structure via $\sum_{i\ge k} p_i \ge 1/n$ and $R(n)=-\ln P(\xi\ge n)$ with $r(n)=R(n)-R(n-1)$. The results cover fast Poisson-like tails, intermediate geometric tails, and slow tails, providing a comprehensive discrete analogue to LIL/LTL and extending them via diverging/converging series criteria involving $r(k)$. The paper employs refined Borel–Cantelli tools and moving-threshold lemmas to classify almost-sure occurrences and limits of $\bar{\xi}_n$, including explicit examples for Poisson, quasi-Poisson, and geometric tails. Overall, it supplies a robust framework for the a.s. behavior of extrema in discrete settings and highlights how tail type governs the a.s. ladder of maxima.
Abstract
The article studies the almost surely asymptotics of extreme values $\barξ_n = \max_{1\leq i \leq n} ξ_i$, where $ ξ, ξ_1 , ξ_2 , \ldots$ are discrete identically distributed random variables. One of the main results on this topic is related to the law of the iterated logarithm for the lim sup (LIL) and a law of the triple logarithm for the lim inf (LTL). But, taking into account the specifics of the discrete case, necessary and sufficient conditions are established for $\mathbf{P}(\barξ_n = a_n +l \quad\mbox{infinitely often} ) =1$, where $a_n$ is some given increasing sequence of integers and $l$ is a fixed integer. Note that in the case of discrete random variables whose distribution tails are close to the tails of the Poisson distribution or fall off even faster, Theorems 1-3 of this article are significantly more informative than LIL and LTL. For the geometric distribution (or discrete random variables whose tails fall off even more slowly), the results are given in Theorem 3 and Theorem B, which are an important complement to LIL and LTL.