On the asymptotic behavior of randomly weighted averages

In this paper we assume that X1,X2,…X1,X2,… is a sequence of independent continuous centered random variables with finite variances σ12,σ22,…. Then we present a central limit theorem for the randomly weighted averages Sn=R1X1+⋯+RnXnSn=R1X1+⋯+RnXn, where the random weights R1,…,RnR1,…,Rn are the cuts of (0,1)(0,1) by the order statistics of a random sample of size n−1n−1 from a uniform distribution on (0,1)(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 2c2c, c=limn→∞(1/n)∑i=1nσi2.