The central limit theorem and the law of iterated logarithm for empirical processes under local conditions

The Central Limit Theorem for Empirical Processes Under Local Conditions: The Case of Radon Infinitely Divisible Limits without Gaussian Component

“…Actually our main interest is in the case Xnj = 6xjbn with X3 i.i.d. and bn | oo, that is, the p-stable limit case, in fact only for p E [1,2): the case p = 2 is considered in Andersen et al (1988) and the case p < 1 is trivial in the sense that condition (ii) of Proposition 2.2 becomes superfluous (as bn/n -► oo) and condition (i) is also necessary for the CLT, at least in the measurable case. Another reason for not considering the case p < 1 is that it has no relevance regarding the law of large numbers.…”

“…In §4 we prove the main theorem, which is a CLT for the randomized empirical processes (5Z"=1 £njf(Xnj)/bn: / E^} where Xnj, j = l,...,n, are independent (S, J^)-valued random variables and &~ is a class of measurable functions on S. The method of proof is similar to that of Theorem 3.1 in Andersen et al (1988). Of course the majorizing measure and the local modulus conditions are quite different in the present situation.…”

“…§3 contains a generalization of an inequality for sums of positive random variables that allowed replacing entropy by majorizing measure conditions in Andersen et al (1988), and is used here for the same purpose.…”

“…In §6 we apply a recent result of Ledoux and Talagrand that essentially reduces the law of the iterated logarithm in separable Banach spaces to a law of large numbers, to obtain a LIL under bracketing that complements the one obtained in Andersen et al (1988) and Ledoux and Talagrand (1986). So, the unifying theme in this article is the use of majorizing measures combined with local conditions to prove convergence in probability to zero of randomized and suitably normalized empirical processes.…”

“…Notation and definitions are as in Andersen et al (1988). When we refer to triangular arrays Xnj of S-valued independent random variables and to independent Rademacher arrays {enj}, we mean that these variables are defined on a product probability space, the Xnj's being coordinate functions and the {enj} being defined License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use in another factor of the product, that is, (n,s,Pr) = (nn)J5,nn>J^,nnipni) x ([o,i],^,a), the Xnj are the coordinate functions UnjS -► S and the {£nj} are independent Bernoulli symmetric random variables defined on [0,1].…”