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A Central Limit Theorem for Predictive Distributions

Author

Listed:
  • Patrizia Berti

    (Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio-Emilia, Via Campi 213/B, 41100 Modena, Italy)

  • Luca Pratelli

    (Accademia Navale di Livorno, 57127 Livorno, Italy)

  • Pietro Rigo

    (Dipartimento di Scienze Statistiche “P. Fortunati”, Università di Bologna, Via delle Belle Arti 41, 40126 Bologna, Italy)

Abstract

Let S be a Borel subset of a Polish space and F the set of bounded Borel functions f : S → R . Let a n ( · ) = P ( X n + 1 ∈ · ∣ X 1 , … , X n ) be the n -th predictive distribution corresponding to a sequence ( X n ) of S -valued random variables. If ( X n ) is conditionally identically distributed, there is a random probability measure μ on S such that ∫ f d a n ⟶ a . s . ∫ f d μ for all f ∈ F . Define D n ( f ) = d n ∫ f d a n − ∫ f d μ for all f ∈ F , where d n > 0 is a constant. In this note, it is shown that, under some conditions on ( X n ) and with a suitable choice of d n , the finite dimensional distributions of the process D n = D n ( f ) : f ∈ F stably converge to a Gaussian kernel with a known covariance structure. In addition, E φ ( D n ( f ) ) ∣ X 1 , … , X n converges in probability for all f ∈ F and φ ∈ C b ( R ) .

Suggested Citation

  • Patrizia Berti & Luca Pratelli & Pietro Rigo, 2021. "A Central Limit Theorem for Predictive Distributions," Mathematics, MDPI, vol. 9(24), pages 1-11, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:24:p:3211-:d:700631
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Crimaldi, Irene & Pratelli, Luca, 2005. "Two inequalities for conditional expectations and convergence results for filters," Statistics & Probability Letters, Elsevier, vol. 74(2), pages 151-162, September.
    4. Sandra Fortini & Sonia Petrone, 2020. "Quasi‐Bayes properties of a procedure for sequential learning in mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(4), pages 1087-1114, September.
    5. 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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    Cited by:

    1. Emanuele Dolera, 2022. "Preface to the Special Issue on “Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini’s 75th Birthday”," Mathematics, MDPI, vol. 10(15), pages 1-4, July.

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