On tail behavior of infinite sums of independent indicators
Alexander Iksanov, Valeriya Kotelnikova
TL;DR
This work provides a precise first-order description of tail and point probabilities for infinite sums of independent indicators, revealing a classification based on the tilted-variance ψ''(s_n). By introducing an exponential change of measure, the authors reduce the problem to analyzing Y under a tilted law and derive sharp asymptotics in several decay regimes for r_k, including polynomial and stretched-exponential forms. The results unify and extend existing asymptotics, connect to Hayman-admissible analytic frameworks and total positivity, and apply to a variety of models such as ranges of Poissonized samples and decoupled renewal processes. The analysis yields explicit leading terms, subleading corrections, and even extension results for asymptotically equivalent probabilities, with potential relevance to combinatorics, random measures, and stochastic process ranges.
Abstract
Let $Y=\sum_{k\ge 1} 1_{A_k}$ be an infinite sum of the indicators of independent events. We investigate a precise (as opposed to logarithmic) first-order asymptotic behavior of the tail probabilities $\mathbb{P}\{Y\ge n\}$ and the point probabilities $\mathbb{P}\{Y=n\}$ as $n\to\infty$. Our analysis provides a reasonably complete classification of the asymptotic behaviors covering most cases of practical interest. These general results are then applied to specific examples where the success probabilities $r_k:=\mathbb{P}(A_k)$ decay polynomially $r_k\sim ck^{-β}$ or (sub-, super-) exponentially $r_k\sim ce^{-k^β}$, yielding the asymptotic tail and point probabilities in explicit forms. As briefly discussed in the paper, infinite sums of independent indicators arise naturally in numerous settings as diverse as the range of Poissonized samples, the infinite Ginibre point processes and decoupled renewal processes, and records in the $F^α$ scheme. We also explore the connection of our research to the theory of Hayman-admissible functions and the notion of total positivity.