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We present an almost sure limit theorem for the product of the partial sums of i.i.d. positive random variables. We also prove a corresponding almost sure limit theorem for a triangular array.

Abstract.We prove an almost sure functional limit theorem for the product of partial sums of i.i.d.positive random variables with finite second moment.Mathematics Subject Classification. 60F05, 60F15.

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We prove an almost sure central limit theorem (ASCLT) for strongly mixing sequence of random variables with a slightly slow mixing rate α(n) = O((log log n) −1−δ ). We also show that ASCLT holds for an associated sequence of random variables without a stationarity assumption.2000 Mathematics Subject Classification: 60F05, 60F15.