Summary
Stochastic processes with certain underlying strong symmetries are characterized as mixtures of well-known “standard”-processes as for example Brownian motions and bridges, Poisson processes, and homogenous (recurrent) Markov chains. The proofs are based on an abstract integral representation for certain functions on semigroups.
References
Aldous, D.: Exchangeability and related topics. In: Hennequin, P.L. (ed) Ecole d' été St. Flour. (Lect. Notes Math. vol. 1117, pp. 1–198. Berlin Heidelberg New York: Springer 1985
Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic analysis on semigroups. Berlin Heidelberg New York: Springer 1984
Diaconis, P., Freedman, D.: De Finetti's theorem for Markov chains. Ann. Probab. 8, 115–130 (1980)
Feller, W.: An introduction to probability theory and its applications, vol. II. New York: Wiley 1971
Freedman, D.: Invariants under mixing which generalize de Finetti's theorem: continuous time parameter. Ann. Math. Stat. 34, 1194–1216 (1963)
Kallenberg, O.: Canonical Representations and Convergence Criteria for Processes with Interchangeable Increments. Z. Wahrscheinlichkeitstheor. Verw. Geb. 27, 23–36 (1973)
Kallenberg, O.: On symmetrically distributed random measures. Trans. Amer. Math. Soc. 202, 105–121 (1975)
Kallenberg, O.: Random measures, 3rd ed. Berlin: Akademie-Verlag
Lauritzen, S.L.: Extreme point Models in Statistics. Scand. J. Statistics 11, 65–91 (1984)
Lindahl, R.J., Maserick, P.H.: Positive-definite functions on involution semigroups. Duke Math. J. 38, 771–782 (1971)
Ressel, P.: De Finetti-type theorems: an analytical approach. Ann. Probab. 13, 898–922 (1985)
Ross, K.A.: A note on extending semicharacters on semigroups. Proc. Amer. Math. Soc. 10, 579–583 (1959)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ressel, P. Integral representations for distributions of symmetric stochastic processes. Probab. Th. Rel. Fields 79, 451–467 (1988). https://doi.org/10.1007/BF00342235
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00342235
Keywords
- Markov Chain
- Stochastic Process
- Brownian Motion
- Probability Theory
- Statistical Theory