ABSTRACT
Thc proof for the case of simple r.v.s X , ; i -- 1,2, ..., I L is cornplctetl by observirlg that the r.v. C;=, a j W j follows a corripolind Poisson distribution. More specifically, the characteristic function of aj M/, is
We shall now trcat the general case, X , > 0. Define
fbr S = 1, 2, .... For each i - 1, 2, ..., n, the sequence of simple nonnegative r.v.s @,(X,) converges (as s + m) to X , almost surely (as . ) [e.g. see Billingsley (1986, pp. 185, 259)]. From the first part of the proof it follows that, for each s = 1 ,2 , ...
