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The random integral representation conjecture: a quarter of a century later

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In [Z.J. Jurek, Relations between the s-selfdecomposable and selfdecomposable measures, Ann. Probab., 13(2):592–608, 1985] and [Z.J. Jurek, Random integral representation for classes of limit distributions similar to Lévy class L 0, Probab. Theory Relat. Fields, 78:473–490, 1988] the random integral representation conjecture was stated. It claims that (some) limit laws can be written as the probability distributions of random integrals of the form \( \int {_{\left( {a,b} \right]}h(t){\text{d}}{Y_v}\left( {r(t)} \right)} \) for some deterministic functions h, r, and a Lévy process \( {Y_v}(t),\;t \geqslant 0 \). Here we review situations where such a claim holds. Each theorem is followed by a remark that gives references to other related papers, results, and historical comments. Moreover, some open questions are stated.

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  1. Institute of Mathematics, University of Wroclaw, Pl. Grunwaldzki 2/4, 50-384, Wroclaw, Poland

    Zbigniew J. Jurek

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Jurek, Z.J. The random integral representation conjecture: a quarter of a century later. Lith Math J 51, 362–369 (2011). https://doi.org/10.1007/s10986-011-9132-6

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