On the asymptotic behavior of randomly weighted averages

In this paper we assume that X1, X2, . . . is a sequence of independent continuous centered random variables with finite variances σ2 1 , σ2 2 , . . .. Then we present a central limit theorem for the randomly weighted averages Sn = R1X1 + · · · + RnXn, where the random weights R1, . . . , Rn are the cuts of (0, 1) by the order statistics of a random sample of size n−1 from a uniform distribution on (0, 1). Indeed we prove that under certain assumptions on the variances, √ n + 1Sn converges in distribution to the normal distribution with mean zero and variance 2c, c = limn→∞(1/n) ni =1 σ2 i .

Central limit theorem, limiting distribution, randomly weighted average