The law of the iterated logarithm on subsequences-characterizations

Abstract

Let §> be any increasing sequence of integers and M > 1 we connect to them in a very simply way, an increasing unbounded function φ: 3 -> R + . Let also X u X 2 , be a sequence of i.i.d. random vectors with value in euclidian space R m . We prove that the cluster set of the sequence {(XΊ + + X n )l<\/~nφ(ri), n e 3} almost surely coincides with the unit ball of R m , if, and only if, the covariance matrix of X x is the identity matrix of R m and EX λ is the zero vector of R m . We define a functional A on the set of increasing sequences of integers as follows:We prove that P{limsup β9Λ _ M (-Xi + • + X n )N%n loglog n > 0} > 0, for at least one sequence X u X, of i.i.d. real r.v.'s with EX X = 0 and E(X x f < oo, if, and only if Λ{%) > 0; further the definition of A( ) does not depend of the value of M. Different characterizations are also established. Further, the law of the iterated logarithm in the sense of Strassen is considered. We finally show a functional law of the iterated logarithm on subsequences for lipschitzian random functions. § 1. Introduction This work is a natural continuation of our previous results obtained in [16]. Let X u X 2 , be a sequence of independent, identically distributed (i.i.d), real random variables (r.v.'s). Set Vn, > 1, S n = Xi + + X n .We investigate the asymptotic behavior of the sequence {S n } when n runs on arbitrary subsequences of integers. This is a quite natural

Critical probabilistic characteristics of the Cramér model for primes and arithmetical properties

“…almost surely, follows from the proof of Theorem 3.3 in [31]. This is rather easy to observe from estimates (3.22), (3.23), (3.24) in [31].…”

“…sums) has slower amplitude than the one given by the classical normalizing factor √ 2n log log n, when n is restricted to subsequences. For instance, if n runs along the subsequence N = {2 2 k , k ≥ 1}, then the LIL restricted to N holds with normalizing factor √ 2n log log log n. See [31] for a characterization of the LIL for subsequences. The same phenomenon holds in fact -with no additional requirementfor the Cramér model.…”

The law of the iterated logarithm for subsequences: A simple proof

“…There are various studies [2,3] on the asymptotic behavior of subsequences of sums of i.i.d. The following result given by Weber [4] determined the speed of divergence of every subsequences. Theorem 1.…”

Some results on increments of the partially observed empirical process

“…Towards this goal, a methodology exposed by Weber [26], Weber [27], Lifshits and Weber [17] is used. We refer to these articles for the study of limit laws for the partially observed Wiener process.…”