Large deviation probabilities for sums of censored random variables with regularly varying distribution tails

TL;DR

This paper derives the asymptotic probabilities of large deviations for sums of censored i.i.d. random variables with heavy tails, revealing a transition in behavior near multiples of the censoring threshold.

Contribution

It establishes the asymptotics of large deviation probabilities for censored sums with regularly varying tails, highlighting the multiple large jumps principle and transition phenomena.

Findings

Asymptotic formulas obeying the multiple large jumps principle.

Different asymptotic behaviors near and away from multiples of the threshold.

Smooth transition in large deviation representations as deviations cross thresholds.

Abstract

Let $\xi_1, \xi_2,\ldots$ be a sequence of independent and identically distributed random variables with zero mean, finite second moment and regularly varying right distribution tail. Motivated by a stop-loss insurance model, we consider a threshold sequence $M_n\gg (n\ln n)^{1/2},$ $n\to \infty,$ and establish the asymptotics of the probabilities of the large deviations of the form $ \sum_{j=1}^n(\xi_j \wedge M_n)>x$ in the whole spectrum of $x$-values in the region $O(M_n).$ The asymptotic representations for these probabilities obey the "multiple large jumps principle" and have different forms in the vicinities of the multiples $kM_n$ of the censoring threshold values, on the one hand, and inside intervals of the form $((k-1)M_n, kM_n),$ on the other. We show that there is a "smooth transition" of these representations from one to the other when the deviation $x$ increases to a multiple of $M_n$, "crosses" it and then moves away from it.