A Central Limit Theorem for Predictive Distributions

Patrizia Berti, Luca Pratelli and Pietro Rigo
Additional contact information
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

Mathematics, 2021, vol. 9, issue 24, 1-11

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 ) .

Keywords: bayesian predictive inference; central limit theorem; conditional identity in distribution; exchangeability; predictive distribution; stable convergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
https://www.mdpi.com/2227-7390/9/24/3211/pdf (application/pdf)
https://www.mdpi.com/2227-7390/9/24/3211/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:9:y:2021:i:24:p:3211-:d:700631

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().