Operator-self-similar processes in a finite-dimensional space

Hudson, William N.; Mason, J. David

doi:10.1090/S0002-9947-1982-0664042-7

Operator-self-similar processes in a finite-dimensional space
HTML articles powered by AMS MathViewer

by William N. Hudson and J. David Mason

Trans. Amer. Math. Soc. 273 (1982), 281-297

DOI: https://doi.org/10.1090/S0002-9947-1982-0664042-7

PDF | Request permission

Abstract:

A general representation for an operator-self-similar process is obtained and its class of exponents is characterized. It is shown that such a process is the limit in a certain sense of an operator-normed process and any limit of an operator-normed process is operator-self-similar.

References

Patrick Billingsley, Convergence of types in $k$-space, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5 (1966), 175–179. MR 198510, DOI 10.1007/BF00536653
P. M. Cohn, Lie groups, Cambridge Tracts in Mathematics and Mathematical Physics, No. 46, Cambridge University Press, New York, N.Y., 1957. MR 0103940
Charles W. Curtis, Linear algebra: an introductory approach, 3rd ed., Allyn and Bacon, Inc., Boston, Mass., 1974. MR 0345978

On convergence to semi-stable Gaussian processes

22

Kenneth Hoffman and Ray Kunze, Linear algebra, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0276251
H. Kesten and F. Spitzer, A limit theorem related to a new class of self-similar processes, Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 5–25. MR 550121, DOI 10.1007/BF00535672
R. G. Laha and V. K. Rohatgi, Operator self-similar stochastic processes in $\textbf {R}_{d}$, Stochastic Process. Appl. 12 (1982), no. 1, 73–84. MR 632393, DOI 10.1016/0304-4149(81)90012-0
John Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104 (1962), 62–78. MR 138128, DOI 10.1090/S0002-9947-1962-0138128-7
John Lamperti, Semi-stable Markov processes. I, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 205–225. MR 307358, DOI 10.1007/BF00536091

Analysis

Michael Sharpe, Operator-stable probability distributions on vector groups, Trans. Amer. Math. Soc. 136 (1969), 51–65. MR 238365, DOI 10.1090/S0002-9947-1969-0238365-3
Ja. G. Sinaĭ, Self-similar probability distributions, Teor. Verojatnost. i Primenen. 21 (1976), no. 1, 63–80 (Russian, with English summary). MR 0407959
Murad S. Taqqu, A representation for self-similar processes, Stochastic Process. Appl. 7 (1978), no. 1, 55–64. MR 492691, DOI 10.1016/0304-4149(78)90037-6

Self-similar processes and related ultraviolet and infrared catastrophes

Ishay Weissman, On convergence of types and processes in Euclidean space, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37 (1976/77), no. 1, 35–41. MR 423456, DOI 10.1007/BF00536296