Probabilistic proofs of large deviation results for sums of semiexponential random variables and explicit rate function at the transition

TL;DR

This paper develops probabilistic methods to analyze large deviation probabilities for sums of semiexponential variables, providing explicit rate functions at critical transition points with simplified proofs.

Contribution

It introduces a probabilistic approach to derive large deviation asymptotics for semiexponential sums and explicitly characterizes the rate function at transition thresholds.

Findings

Derived rough asymptotics at logarithmic scale

Provided explicit rate function at the transition point

Simplified proofs using classical large deviation tools

Abstract

Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$ $\times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated, in particular when X 1 is not exponentially integrable. For instance, A.V. Nagaev formulated exact asymptotics results for P(S n > x n) when x n > n 1/2 (see, [13, 14]). In this paper, we derive rough asymptotics results (at logarithmic scale) with shorter proofs relying on classical tools of large deviation theory and expliciting the rate function at the transition.