Summary
A stochastic process {X t ∶t ≧0} onR d is called wide-sense self-similar if, for eachc>0, there are a positive numbera and a functionb(t) such that {X ct } and {aX t +b(t)} have common finite-dimensional distributions. If {X t } is widesense self-similar with independent increments, stochastically continuous, andX 0=const, then, for everyt, the distribution ofX t is of classL. Conversely, if μ is a distribution of classL, then, for everyH>0, there is a unique process {X t (H) } selfsimilar with exponentH with independent increments such thatX 1 has distribution μ. Consequences of this characterization are discussed. The properties (finitedimensional distributions, behaviors for small time, etc.) of the process {X t (H) } (called the process of classL with exponentH induced by μ) are compared with those of the Lévy process {Y t } such thatY 1 has distribution μ. Results are generalized to operator-self-similar processes and distributions of classOL. A process {X t } onR d is called wide-sense operator-self-similar if, for eachc>0, there are a linear operatorA c and a functionb c (t) such that {X ct } and {A c X t +b c (t)} have common finite-dimensional distributions. It is proved that, if {X t } is wide-sense operator-self-similar and stochastically continuous, then theA c can be chosen asA c =c Q with a linear operatorQ with some special spectral properties. This is an extension of a theorem of Hudson and Mason [4].
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References
Fristedt, B.E.: Sample function behavior of increasing processes with stationary, independent increments. Pac. J. Math.21, 21–33 (1967)
Gihman, I.I., Skorohod, A.V.: The theory of stochastic processes, vol. 1. Berlin Heidelberg New York: Springer 1975
Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. 2nd ed. Reading, Mass: Addison-Wesley 1968
Hudson, W.N., Mason, J.D.: Operator-self-similar processes in a finite-dimensional space. Trans. Am. Math. Soc.273, 281–297 (1982)
Kawazu, K.: Some topics on random walk in random environment. Seminar talk at Nagoya University on March 13, 1989
Kawazu, K.: Diffusion process in random environment and related topics. In: Fifth internat. Vilnius conference on probab. theory and math. stat. Abstracts of communications, vol. 1, pp. 244–245 Vilnius: 1989
Khintchine, A.Ya.: Limit laws for sums of independent random variables. Moscow Leningrad: ONTI 1938 (Russian)
Khintchine, A.Ya.: Sur la croissance locale des processus stochastiques homogènes à accroissements indépendants. Izv. Akad. Nauk SSSR, Ser. Mat.3, 487–508 (1939) [Russian]
Laha, R.G., Rohatgi, V.K.: Operator self similar stochastic processes inR d . Stochastic Processes Appl.12, 73–84 (1982)
Lamperti, J.W.: Semi-stable stochastic processes. Trans. Am. Math. Soc.104, 62–78 (1962)
Lévy, P.: Théorie de l'addition des variables aléatoires. 2ème éd. Paris: Gauthier-Villars 1954 (1ère éd. 1937)
Loève, M.: Probability theory, vol. 1 and 2, 4th ed. New York Heidelberg Berlin: Springer 1977/78
Sato, K.: ClassL of multivariate distributions and its subclasses. J. Multivariate Anal.10, 207–232 (1980)
Sato, K.: Absolute continuity of multivariate distributions of classL. J. Multivariate Anal.12, 89–94 (1982)
Sato, K.: Lectures on multivariate infinitely divisible distributions and operator-stable processes. Technical Report Series, Lab. Res. Statist. Probab. Carleton Univ. and Univ. Ottawa, No. 54, 1985
Sato, K.: Strictly operator-stable distributions. J. Multivariate Anal.22, 278–295 (1987)
Sato, K., Yamazato, M.: On distribution functions of classL. Z. Wahrscheinlichkeitstheor. Verw. Geb.43, 273–308 (1978)
Sato, K., Yamazato, M.: On higher derivatives of distribution functions of classL. J. Math. Kyoto Univ.21, 575–591 (1981)
Sato, K., Yamazato, M.: Operator-self-decomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type. Stochastic Processes Appl.17, 73–100 (1984)
Sato, K., Yamazato, M.: Completely operator-self-decomposable distributions and operatorstable distributions. Nagoya Math. J.97, 71–94 (1985)
Sharpe, M.: Operator-stable probability distributions on vector groups. Trans. Am. Math. Soc.136, 51–65 (1969)
Shtatland, E.S.: On local properties of processes with independent increments. Theory Probab. Appl.10, 317–322 (1965)
Taqqu, M.S.: A bibliographical guide to self-similar processes and long-range dependence. In: Eberlein, E., Taqqu, M.S. (eds.) Dependence in probability and statistics, pp. 137–162 Boston: Birkhäuser 1986
Urbanik, K.: Lévy's probability measures on Euclidean spaces. Stud. Math.44, 119–148 (1972)
Wolfe, S.J.: On the unimodality ofL functions. Ann. Math. Stat.42, 912–918 (1971)
Wolfe, S.J.: On the continuity properties ofL functions. Ann. Math. Stat.42, 2064–2073 (1971)
Wolfe, S.J.: On the unimodality of multivariate symmetric distribution functions of classL. J. Multivariate Anal.8, 141–145 (1978)
Yamazato, M.: Unimodality of infinitely divisible distributions of classL. Ann. Probab.6, 523–531 (1978)
Yamazato, M.: Absolute continuity of operator-self-decomposable distributions onR d. J. Multivariate Anal.13, 550–560 (1983)
Yamazato, M.:OL distributions on Euclidean spaces. Theory Probab. Appl.29, 1–17 (1984)
Zolotarev, V.M.: The analytic structure of infinitely divisible laws of classL. Litov. Mat. Sb.3, 123–40 (1963) [Russian]
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Sato, Ki. Self-similar processes with independent increments. Probab. Th. Rel. Fields 89, 285–300 (1991). https://doi.org/10.1007/BF01198788
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DOI: https://doi.org/10.1007/BF01198788