This paper deals with empirical processes of the type Cn(B)=n−−√{μn(B)−P(Xn+1∈B∣X1,…,Xn)}, where (Xn) is a sequence of random variables and μn=(1/n)∑ni=1δXi the empirical measure. Conditions for supB|Cn(B)| to converge stably (in particular, in distribution) are given, where B ranges over a suitable class of measurable sets. These conditions apply when (Xn) is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029–2052]). By such conditions, in some relevant situations, one obtains that supB|Cn(B)|→P0 or even that n−−√supB|Cn(B)| converges a.s. Results of this type are useful in Bayesian statistics.
Rate of convergence of predictive distributions for dependent data
CRIMALDI, IRENE;P. Rigo
2009
Abstract
This paper deals with empirical processes of the type Cn(B)=n−−√{μn(B)−P(Xn+1∈B∣X1,…,Xn)}, where (Xn) is a sequence of random variables and μn=(1/n)∑ni=1δXi the empirical measure. Conditions for supB|Cn(B)| to converge stably (in particular, in distribution) are given, where B ranges over a suitable class of measurable sets. These conditions apply when (Xn) is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029–2052]). By such conditions, in some relevant situations, one obtains that supB|Cn(B)|→P0 or even that n−−√supB|Cn(B)| converges a.s. Results of this type are useful in Bayesian statistics.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
