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On a notion of partially conditionally identically distributed sequences

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  • Fortini, Sandra
  • Petrone, Sonia
  • Sporysheva, Polina

Abstract

A notion of conditionally identically distributed (c.i.d.) sequences has been studied as a form of stochastic dependence weaker than exchangeability, but equivalent to it in the presence of stationarity. We extend such notion to families of sequences. Paralleling the extension from exchangeability to partial exchangeability in the sense of de Finetti, we propose a notion of partially c.i.d. dependence, which is shown to be equivalent to partial exchangeability for stationary processes. Partially c.i.d. families of sequences preserve attractive limit properties of partial exchangeability, and are asymptotically partially exchangeable. Moreover, we provide strong laws of large numbers and two central limit theorems. Our focus is on the asymptotic agreement of predictions and empirical means, which lies at the foundations of Bayesian statistics. Natural examples of partially c.i.d. constructions are interacting randomly reinforced processes satisfying certain conditions on the reinforcement.

Suggested Citation

  • Fortini, Sandra & Petrone, Sonia & Sporysheva, Polina, 2018. "On a notion of partially conditionally identically distributed sequences," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 819-846.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:3:p:819-846
    DOI: 10.1016/j.spa.2017.06.008
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    3. Beggs, A.W., 2005. "On the convergence of reinforcement learning," Journal of Economic Theory, Elsevier, vol. 122(1), pages 1-36, May.
    4. Gençay, Ramazan & Dacorogna, Michel & Muller, Ulrich A. & Pictet, Olivier & Olsen, Richard, 2001. "An Introduction to High-Frequency Finance," Elsevier Monographs, Elsevier, edition 1, number 9780122796715.
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    Cited by:

    1. Berti, Patrizia & Dreassi, Emanuela & Pratelli, Luca & Rigo, Pietro, 2021. "Asymptotics of certain conditionally identically distributed sequences," Statistics & Probability Letters, Elsevier, vol. 168(C).

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