# Sequences of expected record values

## TL;DR

This paper characterizes when a sequence of real numbers can be viewed as expected record values from i.i.d. random variables, linking it to the Stieltjes moment problem and providing an inversion formula.

## Contribution

It establishes a necessary and sufficient condition for expected record sequences and introduces a transformation-based method with an explicit inversion formula.

## Key findings

- Provides a characterization linking expected record sequences to the Stieltjes moment problem.
- Develops a transformation and its inverse for analyzing expected record values.
- Offers an explicit inversion formula for the underlying random variable.

## Abstract

We investigate conditions in order to decide whether a given sequence of real numbers represents expected record values arising from an independent, identically distributed, sequence of random variables. The main result provides a necessary and sufficient condition, relating any expected record sequence with the Stieltjes moment problem. The results are proved by means of a useful transformation on random variables. Some properties of this mapping, and its inverse, are discussed in detail, and, under mild conditions, an explicit inversion formula for the random variable that admits a given expected record sequence is obtained. Key words and phrases: characterizations; expected record values; Stieltjes moment problem; transformation of random variables; inversion formula. AMS subject classification: Primary 60E05, 62G30; Secondary 44A60.