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

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 inte...

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Bibliographic Details
Published in:Lithuanian mathematical journal 2011-07, Vol.51 (3), p.362-369
Main Author: Jurek, Zbigniew J.
Format: Article
Language:English
Subjects:
Actuarial Sciences
Mathematics
Mathematics and Statistics
Number Theory
Ordinary Differential Equations
Probability Theory and Stochastic Processes
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Summary: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 for some deterministic functions h , r , and a Lévy process . 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.
ISSN:0363-1672
1573-8825
DOI:10.1007/s10986-011-9132-6