Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.J. Aldous, Limit theorems for subsequences of arbitrarily-dependent sequences of random variables, Z. Wahrscheinlichkeitstheor. Verw. Geb., 40:59–82, 1977.
T. Aoyama and M. Maejima, Characterizations of subclasses of type G distributions on \( {{\mathbb R}^d} \) by stochastic random integral representation, Bernoulli, 13:148–160, 2007.
O.E. Barndorff-Nielsen, M. Maejima, and K. Sato, Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations, Bernoulli, 12:1–33, 2006.
O.E. Barndorff-Nielsen, J. Rosiński, and S. Thorbjørnsen, General Υ-transformations, ALEA, Lat. Am. J. Probab. Math. Stat., 4:131–165, 2008.
H. Bercovici and D.V. Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J., 42:733–773, 1993.
A. Czyzewska-Jankowska and Z.J. Jurek, Factorization property of s-selfdecomposable measures and class L f distributions, Teor. Veroyatn. Primen., 55(4):812–819, 2010.
B. Grigelionis, Extended Thorin classes and stochastic integrals, Lith. Math. J., 47(4):406–411, 2007.
E. Housworth and Q.-M. Shao, On central limit theorems for shrunken random variables, Proc. Am. Math. Soc., 128(1):261–267, 2000.
A.M. Iksanov, Z.J. Jurek, and B.M. Schreiber, A new factorization property of the selfdecomposable probability measures, Ann. Probab., 32(2):1356–1369, 2004.
Z.J. Jurek, Limit distribution for sums of shrunken random variables, in Second Vilnius Conference on Probability Theory and Mathematical Statistics (Vilnius, 1977), Abstracts of Communications, Vol. 3, Vilnius, 1977, pp. 95–96.
Z.J. Jurek, Limit distributions for sums of shrunken random variables, Diss. Math., 185, 1981.
Z.J. Jurek, Limit distributions and one-parameter groups of linear operators on Banach spaces, J. Multivariate Anal., 13:578–604, 1983.
Z.J. Jurek, The classes L m (Q) of probability measures on Banach spaces, Bull. Acad. Pol. Sci., Sér. Sci. Math., 31:51–62, 1983.
Z.J. Jurek, Polar coordinates in Banach spaces, Bull. Pol. Acad. Sci., Math., 32:61–66, 1984.
Z.J. Jurek, Relations between the s-selfdecomposable and selfdecomposable measures, Ann. Probab., 13(2):592–608, 1985.
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.
Z.J. Jurek, Random integral representation for classes of limit distributions similar to Lévy class L 0, II, Nagoya Math. J., 114:53–64, 1989.
Z.J. Jurek, Series of independent exponential random variables, in S.Watanabe, M. Fukushima, Yu.V. Prokhorov, and A.N. Shiryaev (Eds.), Proceedings of the Seventh Japan–Russia Symposium on Probability Theory and Mathematical Statistics (Tokyo, 1995), World Scientific, Singapore, 1996, pp. 174–182.
Z.J. Jurek, 1D Ising models, compound geometric distributions and selfdecomposability, Rep. Math. Phys., 47(1):21–30, 2001.
Z.J. Jurek, The random integral representation hypothesis revisited: New classes of s-selfdecomposable laws, in Abstract and Applied Analysis, Proceedings of the International Conference (Hanoi, Vietnam, 13–17 August, 2002), World Scientific, Hongkong, 2004, pp. 479–498.
Z.J. Jurek, Random integral representations for free-infinitely divisible and tempered stable distributions, Stat. Probab. Lett., 77:417–425, 2007.
Z.J. Jurek, A calculus on Lévy exponents and selfdecomposability on Banach spaces, Probab. Math. Stat., 28(2):271–280, 2008.
Z.J. Jurek, On relations between Urbanik and Mehler semigroups, Probab. Math. Stat., 29(2):297–308, 2009.
Z.J. Jurek and J.D. Mason, Operator-Limit Distributions in Probability Theory, John Wiley & Sons, New York, 1993.
Z.J. Jurek and B.M. Schreiber, Fourier transforms of measures from classes \( {{\mathcal U}_\beta } - 2 < \beta < - 1 \), J. Multivariate Anal., 41:194–211, 1992.
Z.J. Jurek and W. Vervaat, An integral representation for selfdecomposable Banach space valued random variables, Z. Wahrscheinlichkeitstheor. Verw. Geb., 62:247–262, 1983.
Z.J. Jurek and M. Yor, Selfdecomposable laws associated with hyperbolic functions, Probab. Math. Stat., 24(1):180–190, 2004.
A. Kumar and B.M. Schreiber, Representation of certain infinitely divisible probability measures on Banach spaces, J. Multivariate Anal., 9:288–303, 1979.
M. Maejima and K. Sato, The limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions, Probab. Theory Relat. Fields, 145:119–142, 2009.
M.M. Meerschaert and H.P. Scheffler, Limit Distributions of Sums of Independent Random Vectors, John Wiley & Sons, New York, 2001.
K. Sato, Class L of multivariate distributions and its subclasses, J. Multivariate Anal., 10:201–232, 1980.
K. Sato and M. Yamazato, Operator-selfdecomposable distributions as limit distributions of processes of Ornstein–Uhlenbeck type, Stochastic Processes Appl., 17:73–100, 1984.
N. van Thu, An alternative approach to multiply selfdecomposable probability measures on Banach spaces, Probab. Theory Relat. Fields, 72:35–54, 1986.
K. Urbanik, Lévy’s probability measures on Euclidean spaces, Stud. Math., 44:119–148, 1972.
K. Urbanik, Limit laws for sequences of normed sums satisfying some stability conditions, in P.R. Krishnaiah (Ed.), Multivariate Analysis III, Academic Press, New York, 1973, pp. 225–237.
K. Urbanik, Lévy’s probability measures on Banach spaces, Stud. Math., 63:283–308, 1978.
S.J.Wolfe, On a continuous analogue of the stochastic difference equation X n = ρX n−1 + B n , Stochastic Processes Appl., 12:301–312, 1982.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-011-9132-6