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Asymptotics of certain conditionally identically distributed sequences

Author

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  • Berti, Patrizia
  • Dreassi, Emanuela
  • Pratelli, Luca
  • Rigo, Pietro

Abstract

The probability distribution of a sequence X=(X1,X2,…) of random variables is determined by its predictive distributions P(X1∈⋅) and P(Xn+1∈⋅∣X1,…,Xn), n≥1. Motivated by applications in Bayesian predictive inference, in Berti et al. (2020), a class C of sequences is introduced by specifying such predictive distributions. Each X∈C is conditionally identically distributed. The asymptotics of X∈C is investigated in this paper. Both strong and weak limit theorems are provided. Conditions for X to converge a.s., and for X not to converge in probability, are given in terms of the predictive distributions. A stable CLT is provided as well. Such a CLT is used to obtain approximate credible intervals.

Suggested Citation

  • Berti, Patrizia & Dreassi, Emanuela & Pratelli, Luca & Rigo, Pietro, 2021. "Asymptotics of certain conditionally identically distributed sequences," Statistics & Probability Letters, Elsevier, vol. 168(C).
    • Handle: RePEc:eee:stapro:v:168:y:2021:i:c:s0167715220302261
    DOI: 10.1016/j.spl.2020.108923
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    References listed on IDEAS

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    1. 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.
    2. Edoardo M. Airoldi & Thiago Costa & Federico Bassetti & Fabrizio Leisen & Michele Guindani, 2014. "Generalized Species Sampling Priors With Latent Beta Reinforcements," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(508), pages 1466-1480, December.
    3. P. Richard Hahn & Ryan Martin & Stephen G. Walker, 2018. "On Recursive Bayesian Predictive Distributions," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(523), pages 1085-1093, July.
    4. Patrizia Berti & Irene Crimaldi & Luca Pratelli & Pietro Rigo, 2009. "Rate of Convergence of Predictive Distributions for Dependent Data," Quaderni di Dipartimento 091, University of Pavia, Department of Economics and Quantitative Methods.
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    Cited by:

    1. Patrizia Berti & Luca Pratelli & Pietro Rigo, 2021. "A Central Limit Theorem for Predictive Distributions," Mathematics, MDPI, vol. 9(24), pages 1-11, December.

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