A Sequence of Nested Exponential Random Variables with Connections to Two Constants of Euler

TL;DR

This paper explores a recursively defined sequence of exponential random variables and uncovers a surprising connection to Euler's constants, providing new insights into these mathematical quantities.

Contribution

It introduces a novel recursive sequence of exponential variables and reveals their unexpected link to Euler-Gompertz and Euler-Mascheroni constants, enhancing theoretical understanding.

Findings

Sequence of distributions linked to Euler constants

New analytical perspective on Euler-Gompertz and Euler-Mascheroni constants

Theoretical insights into parameter uncertainty models

Abstract

We investigate a recursively generated sequence of random variables that begins with an Exponential random variable with parameter (i.e., inverse-mean) 1, and continues with additional Exponentials, each of whose random parameter possesses the distribution of the prior term in the sequence. Although such sequences enjoy some (limited) applicability as models of parameter uncertainty, our present interest is primarily theoretical. Specifically, we observe that the implied sequence of distribution functions manifests a surprising connection to two well-known mathematical quantities first studied by Leonhard Euler: the Euler-Gompertz and Euler-Mascheroni constants. Through a close analysis of one member of this distribution-function sequence, we are able to shed new light on these constants.