On sequential maxima of exponential sample means, with an application to ruin probability

Dimitris Cheliotis,Nickos Papadatos

doi:10.1214/19-ecp273

Abstract

We obtain the distribution of the maximal average in a sequence of independent identically distributed exponential random variables. Surprisingly enough, it turns out that the inverse distribution admits a simple closed form. An application to ruin probability in a risk-theoretic model is also given.

Highlights

Consider a sequence (Xi)i≥1 of independent identically distributed (i.i.d.) random variables, each having exponential distribution with mean 1
We obtain the distribution of the maximal average in a sequence of independent identically distributed exponential random variables
For each i ∈ N+ define the sample mean of the first i variables as Xi := (X1 + X2 + · · · + Xi)/i

Summary

Introduction

Consider a sequence (Xi)i≥1 of independent identically distributed (i.i.d.) random variables, each having exponential distribution with mean 1. For each i ∈ N+ define the sample mean of the first i variables as Xi := (X1 + X2 + · · · + Xi)/i. The supremum of this sequence, Z∞ := sup{Xi : i ∈ N+}, is finite because the sequence converges to 1 with probability 1. In this note we compute the distribution function, F∞, of Z∞. What has nice form is the inverse of this distribution function. (b) The restriction of F∞ on (1, ∞) is one to one and onto (0, 1) with inverse.

Maxima of exponential sample means

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