On the rate of convergence in the central limit theorem for random sums of strongly mixing random variables

Abstract

We present upper bounds for supx ∈ ℝ|P{Z N < x} − Φ(x)|, where Φ(x) is the standard normal distribution function, for random sums $$ {Z}_N={S}_N/\sqrt{\mathbf{V}{S}_N} $$ with variances VS N > 0 (S N = X1 + ⋯ + X N ) of centered strongly mixing or uniformly strongly mixing random variables X1, X2, . . . . Here the number of summands N is a nonnegative integer-valued random variable independent of X1,X2, . . . .