A Note on the Invariance Principle of the Product of Sums of Random Variables

where N(0, 1) is a standard normal random variable. Later Rempala and Wesolowski (2002) extended such a central limit theorem to general i.i.d. positive random variables. Recently, the central limit theorem for product of sums has also been studied for dependent random variables (c.f., Gonchigdanzan and Rempala (2006)). In this note, we will show that this kind of result follows from the invariance principle. Let {Sn;n ≥ 1} be a sequence of positive random variables. To present our main idea, we assume that (possibly in an enlarged probability space in which the sequence {Sn;n ≥ 1} is redefined without changing its distribution) there exists a standard Wiener process {W (t) : t ≥ 0} and two positive constants μ and σ such that Sn − nμ− σW (n) = o( √ n) a.s. (2)