Tail Asymptotic of Sum and Product of Random Variables with Applications in the Theory of Extremes of Conditionally Gaussian Processes

TL;DR

This paper derives the tail asymptotics for sums and products of independent random variables with known tail behaviors, and extends results to Gaussian processes with random drifts, using Laplace asymptotic methods.

Contribution

It provides new asymptotic formulas for sums and products of independent variables and extends Gaussian process tail results to cases with random multiplicative drifts.

Findings

Tail asymptotics for sums and products of independent variables derived

Asymptotic behavior of Gaussian processes with random drifts characterized

Laplace method employed for precise asymptotic computations

Abstract

We consider two independent random variables with the given tail asymptotic (e.g. power or exponential). We find tail asymptotic for their sum and product. This is done by some cumbersome but purely technical computations and requires the use of the Laplace method for asymptotic of integrals. We also recall the results for asymptotic of a self-similar locally stationary centered Gaussian process plus a deterministic drift; and we find the asymptotic for the same probability after multiplying the drift by a random variable, which is independent of this process. Keywords: tail asymptotic, Laplace method, self-similar processes, Gaussian processes, locally stationary processes.