An extension of almost sure central limit theorem for self-normalized products of sums for mixing sequences

ABSTRACT

Let X, X₁, X₂, … be a sequence of strictly stationary φ-mixing random variables with EX = μ > 0. In this paper, we show that a self-normalized version of almost sure central limit theorem (ASCLT) holds under the assumptions that the mixing coefficients satisfy ∑^∞_{n = 1}φ^1/2(2ⁿ) < ∞ and the weight sequence {d_k} satisfies a mild growth condition similar to Kolmogorov’s condition for the LIL. This shows that logarithmic averages, used traditionally in ASCLT for products of sums, can be replaced by other averages, leading to considerably sharper results.