The Random Integral Representation Conjecture: a quarter of a century later

TL;DR

This paper reviews the history, current status, and open questions of the random integral representations conjecture, which suggests certain limit laws can be expressed as distributions of specific stochastic integrals.

Contribution

It provides a comprehensive review of cases where the conjecture holds, summarizes related results, and discusses open problems in the field.

Findings

Identification of conditions where the conjecture holds

Compilation of related results and references

Presentation of open questions in the area

Abstract

In Jurek 1985 and 1988 the random integral representations conjecture was stated. It claims that (some) limit laws can be written as probability distributions of random integrals of the form $\int_{(a,b]}h(t)dY_{\nu}(r(t))$, for some deterministic functions $h$, $r$ and a L\'evy process $Y_{\nu}(t),t\ge 0$. Here we review situations where a such claim holds true. Each theorem is followed by a remark which gives references to other related papers, results as well as some historical comments. Moreover, some open questions are stated.