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.
P. Berti, I. Crimaldi, L. Pratelli, P. Rigo (2009). Rate of convergence of predictive distributions for dependent data. BERNOULLI, 15(4), 1351-1367 [10.3150/09-BEJ191].
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.
