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On mixtures of distributions of Markov chains

Author

Listed:
  • Fortini, Sandra
  • Ladelli, Lucia
  • Petris, Giovanni
  • Regazzini, Eugenio

Abstract

Let X be a chain with discrete state space I, and V be the matrix of entries Vi,n, where Vi,n denotes the position of the process immediately after the nth 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 Vi,n's are partially exchangeable in the sense of 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 Diaconis and 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 Diaconis and Freedman basic result for mixtures of Markov chains.

Suggested Citation

  • Fortini, Sandra & Ladelli, Lucia & Petris, Giovanni & Regazzini, Eugenio, 0. "On mixtures of distributions of Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 147-165, July.
    • Handle: RePEc:eee:spapps:v:100:y::i:1-2:p:147-165
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    Cited by:

    1. Fortini, S. & Petrone, S., 2012. "Hierarchical reinforced urn processes," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1521-1529.
    2. Sigeo Aki, 2012. "Statistical modeling for discrete patterns in a sequence of exchangeable trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(3), pages 633-655, June.
    3. Epifani, I. & Fortini, S. & Ladelli, L., 2002. "A characterization for mixtures of semi-Markov processes," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 445-457, December.
    4. Peluso, Stefano & Mira, Antonietta & Muliere, Pietro, 2015. "Reinforced urn processes for credit risk models," Journal of Econometrics, Elsevier, vol. 184(1), pages 1-12.

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