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Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type. (English) Zbl 0533.60021
A random variable X is said to have an operator-selfdecomposable or Lévy distribution if there are sequences \(\{X_ n\}\), \(\{A_ n\}\) and \(\{a_ n\}\) of respectively independent random variables, linear operators, and vectors such that the random variables \(A_ n(X_ 1+...+X_ n)-a_ n\) converge completely to X and in addition an infinitesimal condition is satisfied. Processes of Ornstein-Uhlenbeck type on Euclidean spaces are analogues of the Ornstein-Uhlenbeck process with the Brownian motion part replaced by general processes with homogeneous independent increments. The class of operator- selfdecomposable distributions is characterized as the class of limit distributions of processes of Ornstein-Uhlenbeck type. Integro- differential equations for operator-selfdecomposable distributions are established.
Reviewer: St.Wolfe

60E07 Infinitely divisible distributions; stable distributions
60J25 Continuous-time Markov processes on general state spaces
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[1] Brockwell, P.J., Stationary distributions for dams with additive input and content dependent release rate, Adv. appl. prob., 9, 645-663, (1977) · Zbl 0374.60102
[2] Cinlar, E.; Pinsky, M., A stochastic integral in storage theory, Z. wahrsch. verw. gebiete, 17, 227-240, (1971) · Zbl 0195.47503
[3] Cuppens, R., Decomposition of multivariate probabilities, (1975), Academic Press New York · Zbl 0363.60012
[4] Dynkin, E.B., Markov processes, Vol. 1, (1965), Springer Berlin · Zbl 0132.37901
[5] Hengartner, W.; Theodorescu, R., Concentration functions, (1973), Academic Press New York · Zbl 0323.60015
[6] Hudson, W.N.; Mason, J.D., Operator-stable laws, J. multivar. anal., 11, 434-447, (1981) · Zbl 0466.60016
[7] Hudson, W.N.; Mason, J.D., Operator-stable distributions on \(R\)^{2} with multiple exponents, Ann. probability, 9, 482-489, (1981) · Zbl 0465.60020
[8] Jurek, Z.J., Structure of a class of operator-selfdecomposable probability measures, Ann. probability, 10, 849-856, (1982) · Zbl 0489.60007
[9] Jurek, Z.J., An integral representation of operator-selfdecomposable random variables, Bull. acad. polon. Sér. sci. math., 30, 385-393, (1982) · Zbl 0503.60063
[10] Z.J. Jurek, Limit distributions and one-parameter groups of linear operators on Banach spaces, J. Multivar. Anal., to appear. · Zbl 0533.60004
[11] Jurek, Z.J.; Vervaat, W., An integral representation for selfdecomposable Banach space valued random variables, Z. wahrsch. verw. gebiete, 62, 247-262, (1983) · Zbl 0488.60028
[12] Loève, M., Probability theory, (1963), Van Nostrand New York · Zbl 0108.14202
[13] Parthasarathy, K.R., Probability measures on metric spaces, (1967), Academic Press New York · Zbl 0153.19101
[14] Sato, K., A note on infinitely divisible distributions and their Lévy measures, Sci. rep. Tokyo kyoiku daigaku, 12, 101-109, (1973), Sect. A · Zbl 0279.60010
[15] Sato, K., Class L of multivariate distributions and its subclasses, J. multivar. anal., 10, 207-232, (1980) · Zbl 0425.60013
[16] Sato, K., Absolute continuity of multivariate distributions of class L, J. multivar. anal., 12, 89-94, (1982) · Zbl 0485.60010
[17] K. Sato and M. Yamazato, On distribution functions of class L, Z. Wahrsch. Verw. Gebiete 43 273-308. · Zbl 0395.60019
[18] K. Sato and M. Yamazato, Stationary processes of Ornstein-Uhlenbeck type, Probability Theory and Mathematical Statistics, IV, USSR-Japan Symposium, Lecture Notes in Math. (Springer, Berlin, to appear). · Zbl 0532.60065
[19] Sharpe, M., Operator-stable probability distributions on vector groups, Trans. amer. math. soc., 136, 51-65, (1969) · Zbl 0192.53603
[20] -Nagy, B.Sz., Séries et intégrales de Fourier des fonctions monotones non bornées, Acta sci. math., 13, 118-135, (1949), (Szeged) · Zbl 0039.29501
[21] Urbanik, K., Self-decomposable probability distributions on \(R\)^{m}, Zastos. mat., 10, 91-97, (1969) · Zbl 0233.62003
[22] Urbanik, K., Lévy’s probability measures on Euclidean spaces, Studia math., 44, 119-148, (1972) · Zbl 0251.60022
[23] Watanabe, S., A limit theorem of branching processes and continuous state branching processes, J. math. Kyoto univ., 8, 141-167, (1968) · Zbl 0159.46201
[24] Wolfe, S.J., On the continuity properties of L functions, Ann. math statist., 42, 2064-2073, (1971) · Zbl 0227.60014
[25] Wolfe, S.J., On the unimodality of multivariate symmetric distribution functions of class L, J. multivar. anal., 8, 141-145, (1978) · Zbl 0379.60017
[26] Wolfe, S.J., A characterization of Lévy probability distribution functions on Euclidean spaces, J. multivar. anal., 10, 379-384, (1980) · Zbl 0438.60020
[27] Wolfe, S.J., On a continuous analogue of the stochastic difference equation Xn = pxn-1 + bn, Stochastic process. appl., 12, 301-312, (1982)
[28] Wolfe, S.J., A characterization of certain stochastic integrals, Stochastic process. appl., 12, 136, (1982), (Tenth Conference on Stochastic Processes and Their Applications, Contributed Papers)
[29] Yamazato, M., Unimodality of infinitely divisible distributions of class L, Ann. probability, 6, 523-531, (1978) · Zbl 0394.60017
[30] M. Yamazato, Absolute continuity of operator-selfdecomposable distributions on \(R\)^{d}, J. Multivar. Anal. to appear. · Zbl 0531.60022
[31] M. Yamazato, OL distributions on Euclidean spaces, Teor. Verojatnost. i Primenen, to appear.
[32] Zolotarev, V.M., The analytic structure of infinitely divisible laws of class L, Litovsk. mat. sb., Selected transl. in math. statist. and probability, 15, No. 1, 15-31, (1981), English translation: · Zbl 0477.60018
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