×

On mixtures of distributions of Markov chains. (English) Zbl 1060.60071

Summary: Let \(X\) be a chain with discrete state space \(I\), and \(V\) be the matrix of entries \(V_{i,n}\), where \(V_{i,n}\) denotes the position of the process immediately after the \(n\)th visit to \(i\). We prove that the law of \(X\) is a mixture of laws of Markov chains if and only if the distribution of \(V\) is invariant under finite permutations within rows (i.e., the \(V_{i,n}\)’s are partially exchangeable in the sense of B. de Finetti). We also prove that an analogous statement holds true for mixtures of laws of Markov chains with a general state space and atomic kernels. Going back to the discrete case, we analyze the relationships between partial exchangeability of \(V\) and Markov exchangeability in the sense of P. Diaconis and D. Freedman. The main statement is that the former is stronger than the latter, but the two are equivalent under the assumption of recurrence. Combination of this equivalence with the aforesaid representation theorem gives the P. Diaconis and D. Freedman basic result of mixtures of Markov chains.

MSC:

60J05 Discrete-time Markov processes on general state spaces
60G05 Foundations of stochastic processes
62A01 Foundations and philosophical topics in statistics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Billingsley, P., Convergence of Probability Measures (1999), Wiley: Wiley New York · Zbl 0172.21201
[2] Diaconis, P.; Freedman, D., De Finetti’s theorem for Markov chains, Ann. Probab., 8, 115-130 (1980) · Zbl 0426.60064
[3] Diaconis, P., Freedman, D., 1980b. De Finetti’s generalizations of exchangeability. In: Jeffrey, R.C. (Ed.), Studies in Inductive Logic and Probability, Vol. II. University of California Press, Berkeley, pp. 233-249.; Diaconis, P., Freedman, D., 1980b. De Finetti’s generalizations of exchangeability. In: Jeffrey, R.C. (Ed.), Studies in Inductive Logic and Probability, Vol. II. University of California Press, Berkeley, pp. 233-249.
[4] de Finetti, B., La prévision: ses lois logiques, ses sources subjectives, Ann. Inst. H. Poincaré, 7, 1-68 (1937) · JFM 63.1070.02
[5] de Finetti, B., 1938. Sur la condition d’ “equivalence partielle”. Actualités Scientifiques et Industrielles, Hermann, Paris, Vol. 739, pp. 5-18.; de Finetti, B., 1938. Sur la condition d’ “equivalence partielle”. Actualités Scientifiques et Industrielles, Hermann, Paris, Vol. 739, pp. 5-18. · JFM 64.0517.08
[6] de Finetti, B., 1959. La probabilità e la statistica nei rapporti con l’induzione secondo i diversi punti di vista. In: Centro Internazionale Matematico Estiro (CIME), Induzione e Statistica. Cremonese, Roma, pp. 1-115. [English translation in B. de Finetti, 1972. Probability, Induction and Statistics. Wiley, New York, pp. 147-227.]; de Finetti, B., 1959. La probabilità e la statistica nei rapporti con l’induzione secondo i diversi punti di vista. In: Centro Internazionale Matematico Estiro (CIME), Induzione e Statistica. Cremonese, Roma, pp. 1-115. [English translation in B. de Finetti, 1972. Probability, Induction and Statistics. Wiley, New York, pp. 147-227.] · Zbl 0124.34801
[7] Fortini, S., Ladelli, L., Petris, G., Regazzini, E., 1999. On mixtures of distributions of Markov chains. Preprint del Dipartimento di Matematica “Francesco Brioschi” del Politecnico di Milano n.369/P.; Fortini, S., Ladelli, L., Petris, G., Regazzini, E., 1999. On mixtures of distributions of Markov chains. Preprint del Dipartimento di Matematica “Francesco Brioschi” del Politecnico di Milano n.369/P. · Zbl 1060.60071
[8] Freedman, D., Mixture of Markov processes, Ann. Math. Statist., 33, 114-118 (1962) · Zbl 0112.09902
[9] Freedman, D., Invariants under mixing which generalize de Finetti’s theorem, Ann. Math. Statist., 33, 916-923 (1962) · Zbl 0201.49501
[10] Freedman, D., 1996. De Finetti’s theorem in continuous time. In: Ferguson, T.S., Sharpley, L.S., MacQueen, J.B. (Eds.), Statistics, Probability and Game Theory. IMS Lecture Notes—Monograph Series, Vol. 30.; Freedman, D., 1996. De Finetti’s theorem in continuous time. In: Ferguson, T.S., Sharpley, L.S., MacQueen, J.B. (Eds.), Statistics, Probability and Game Theory. IMS Lecture Notes—Monograph Series, Vol. 30.
[11] Kallenberg, O., Characterizations and embedding properties in exchangeability, Z. Wahrscheinlichkeitstheorie, 60, 249-281 (1982) · Zbl 0481.60019
[12] Nummelin, E., General Irreducible Markov Chains and Non-Negative Operators (1984), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0551.60066
[13] Zabell, S., Characterizing Markov exchangeable sequences, J. Theoret. Probab., 8, 175-178 (1995) · Zbl 0814.60029
[14] Zaman, A., Urn models for Markov exchangeability, Ann. Probab., 12, 223-229 (1984) · Zbl 0542.60065
[15] Zaman, A., A finite form of de Finetti’s theorem for stationary Markov exchangeability, Ann. Probab., 14, 1418-1427 (1986) · Zbl 0608.60032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.