On the central limit theorem along subsequences of sums of i.i.d. random variables

摘要

Let N = {1, 2, 3,...}. Let {X, X_n; n ∈ N} be a sequence of i.i.d. random variables, and let S_n = ∑_(i=1)~n X_i, n ∈ N. Then S_n/√n ? N(0, σ~2) for some σ~2 < ∞ whenever, for a subsequence {n_k; k ∈ N} of N, S_(nk) /√n_k ? N(0, σ~2). Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when {√n; n ∈ N} is replaced by {√na_n; n ∈ N} with lim_(n→∞) a_n = ∞. We show that, for given positive nondecreasing sequence {a_n; n ∈ N} with lim_(n→∞) a_n =∞and lim_(n→∞) a_(n+1)/a_n = 1 and given nondecreasing function h(·): (0,∞) → (0,∞) with lim_(x→∞) h(x)=∞, there exists a sequence {X, Xn; n ∈ N} of symmetric i.i.d. random variables such that Eh(|X|) = ∞and, for some subsequence {n_k; k ∈ N} of N, S_(nk) /√n_ka_n_(k) ? N(0, 1). In particular, for given 0 < p < 2 and given nondecreasing function h(·): (0,∞) → (0,∞) with lim_(x→∞) h(x) = ∞, there exists a sequence {X, X_n; n ∈ N} of symmetric i.i.d. random variables such that Eh(|X|)= ∞and, for some subsequence {n_k; k ∈ N} of N, S_n_(k) _k~(1/p) ? N(0, 1).