A Central Limit Theorem for Predictive Distributions

Patrizia Berti,Pietro Rigo,Luca Pratelli

doi:10.3390/math9243211

Abstract

Let S be a Borel subset of a Polish space and F the set of bounded Borel functions f:S→R. Let an(·)=P(Xn+1∈·∣X1,…,Xn) be the n-th predictive distribution corresponding to a sequence (Xn) of S-valued random variables. If (Xn) is conditionally identically distributed, there is a random probability measure μ on S such that ∫fdan⟶a.s.∫fdμ for all f∈F. Define Dn(f)=dn∫fdan−∫fdμ for all f∈F, where dn>0 is a constant. In this note, it is shown that, under some conditions on (Xn) and with a suitable choice of dn, the finite dimensional distributions of the process Dn=Dn(f):f∈F stably converge to a Gaussian kernel with a known covariance structure. In addition, Eφ(Dn(f))∣X1,…,Xn converges in probability for all f∈F and φ∈Cb(R).

Highlights

Let S be a Borel subset of a Polish space and F the set of bounded Borel functions f : S → R
All random elements appearing in the sequel are defined on a common probability space, say (Ω, A, P)
We denote by S a Borel subset of a Polish space and by B the Borel σ-field on S

Summary

Introduction

All random elements appearing in the sequel are defined on a common probability space, say (Ω, A, P). (v) From a Bayesian point of view, μ can be seen as a random parameter of the data sequence X. This is quite clear if X is exchangeable, for, in this case, X is conditionally i.i.d. given μ. In a Bayesian framework, conditions (4)–(5) may be useful to make (asymptotic) inference about μ To this end, an alternative could be proving a limit theorem for Wn = wn (μn − μ), where wn is a suitable constant and μn = (1/n) ∑nj=1 δX j the empirical measure. The predictive distributions investigated in this note have been introduced in connection with Bayesian prediction problems; see [3] Another example is the asymptotic behavior of certain urn schemes. Without any claim of being exhaustive, a list of references is: [3,5,9,10,11,12,13,14,15,16,17,18,19,20,21]

Preliminaries

Main Result