A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X-n)n greater than or equal to 1 is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (g(n))(n greater than or equal to 0), if it is adapted to (g(n))(ngreater than or equal to 0) and, for each n greater than or equal to 0, (X-k)(k>n) is identically distributed given the past g(n). In case g(0) is trivial and g(n) = sigma(X-1,...,X-n), a result of Kallenberg implies that (X-n)(n greater than or equal to 1) is exchangeable if and only if it is stationary and c.i.d. After giving some natural examples of nonexchangeable c.i.d. sequences, it is shown that (X-n)(n greater than or equal to 1) is exchangeable if and only if (X-tau(n))(n greater than or equal to 1) is c.i.d. for any finite permutation tau of {1, 2,...}, and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-sigma-field. Moreover, (1/n) Sigma(k=1)(n) X-k converges a. s. and in L-1 whenever (X-n)(n greater than or equal to 1) is (real-valued) c.i.d. and E[|X-1|]<infinity. As to the CLT, three types of random centering are considered. One such centering, significant in Bayesian prediction and discrete time filtering, is E[Xn+1\g(n)]. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.

Limit theorems for a class of identically distributed random variables / Berti, Patrizia; L., Pratelli; P., Rigo. - In: ANNALS OF PROBABILITY. - ISSN 0091-1798. - STAMPA. - 32:(2004), pp. 2029-2052.

Limit theorems for a class of identically distributed random variables

BERTI, Patrizia;
2004-01-01

Abstract

A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X-n)n greater than or equal to 1 is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (g(n))(n greater than or equal to 0), if it is adapted to (g(n))(ngreater than or equal to 0) and, for each n greater than or equal to 0, (X-k)(k>n) is identically distributed given the past g(n). In case g(0) is trivial and g(n) = sigma(X-1,...,X-n), a result of Kallenberg implies that (X-n)(n greater than or equal to 1) is exchangeable if and only if it is stationary and c.i.d. After giving some natural examples of nonexchangeable c.i.d. sequences, it is shown that (X-n)(n greater than or equal to 1) is exchangeable if and only if (X-tau(n))(n greater than or equal to 1) is c.i.d. for any finite permutation tau of {1, 2,...}, and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-sigma-field. Moreover, (1/n) Sigma(k=1)(n) X-k converges a. s. and in L-1 whenever (X-n)(n greater than or equal to 1) is (real-valued) c.i.d. and E[|X-1|]
32
2029
2052
Limit theorems for a class of identically distributed random variables / Berti, Patrizia; L., Pratelli; P., Rigo. - In: ANNALS OF PROBABILITY. - ISSN 0091-1798. - STAMPA. - 32:(2004), pp. 2029-2052.
Berti, Patrizia; L., Pratelli; P., Rigo
File in questo prodotto:
File Dimensione Formato  
1089808418.pdf

non disponibili

Tipologia: Versione dell'autore revisionata e accettata per la pubblicazione
Dimensione 213.8 kB
Formato Adobe PDF
213.8 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
Pubblicazioni consigliate

Caricamento pubblicazioni consigliate

Licenza Creative Commons
I metadati presenti in IRIS UNIMORE sono rilasciati con licenza Creative Commons CC0 1.0 Universal, mentre i file delle pubblicazioni sono rilasciati con licenza Attribuzione 4.0 Internazionale (CC BY 4.0), salvo diversa indicazione.
In caso di violazione di copyright, contattare Supporto Iris

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/305605
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 42
  • ???jsp.display-item.citation.isi??? 36
social impact