Large deviation results for triangular arrays of semiexponential random variables

TL;DR

This paper extends large deviation asymptotics for sums of i.i.d. semiexponential variables to triangular arrays without assuming absolute continuity, providing a broader understanding of deviation probabilities.

Contribution

It generalizes existing results on semiexponential large deviations from i.i.d. variables to triangular arrays without the absolute continuity assumption.

Findings

Established asymptotic deviation probabilities for triangular arrays of semiexponential variables.

Extended large deviation results to non-absolutely continuous cases.

Provided a unified approach to deviation asymptotics in this setting.

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 1 has a semiexponential distribution (see, [16, 17]). In the same setting, the authors of [4] derived deviation results at logarithmic scale with shorter proofs relying on classical tools of large deviation theory and expliciting the rate function at the transition. In this paper, we exhibit the same asymptotic behaviour for triangular arrays of semiexponentially distributed random variables, no more supposed absolutely continuous.