Central Limit Theorem and Almost Sure Central Limit Theorem for the Product of Some Partial Sums

Let (Xn)n≥1 be a sequence of independent identically distributed (i.i.d.) positive random variables (r.v.). Recently there have been several studies to the products of partial sums. It is well known that the products of i.i.d. positive, square integrable random variables are asymptotically log-normal. This fact is an immediate consequence of the classical central limit theorem (CLT). This point, up to the knowledge of the author, was first argued by Arnold and Villaseñr [1], who considered the limiting properties of the sums of records. In their paper Arnold and Villaseñr obtained the following version of the CLT for a sequence of i.i.d. exponential r.v.’s (Xn)n≥1 with the mean equal to one: