Convergence in law of random sums with non-random centering
Let \(\{X_ n, n \geq 1\}\) be a sequence of independent random variables such that for some norming constants \(\{B_ n, n \geq 1\}\), \(B_ n \to \infty\), as \(n \to \infty\), and some centering constants \(\{A_ n, n \geq 1\}\) the distribution function of \({1 \over B_ n} \sum^ n_{k = 1} X_ k + A_ n\) converges weakly, as \(n \to \infty\), to some distribution function \(F\). Now suppose \(\{N_ n, n \geq 1\}\) is a sequence of positive integer-valued random variables independent of \(\{X_ n, n \geq 1\}\). One problem of nonrandom centering is this: under what conditions on \(\{N_ n, n \geq 1\}\) does the distribution function of \((X_ 1 + \cdots + X_{N_ n})/B_ n + A_ n\) converge weakly, as \(n \to \infty\), to the same \(F\)? Further, one could also ask under what conditions this sequence converges weakly to any distribution function. Let us observe that the centering by \(A_ n\) is nonrandom centering; random centering would involve \(A_{N_ n}\). Also norming by \(B_ n\) is nonrandom norming; random one would involve \(B_{N_ n}\). The authors present necessary and sufficient conditions for weak convergence of these random variables under nonrandom centering and nonrandom norming, and in some cases, under random norming. The distribution functions for the limit laws are determined, too. Some results concerning necessary and sufficient conditions for weak convergence of random sums of independent random variables have also been obtained by \textit{A. Krajka} and the reviewer [Commentat. Math. Univ. Carol. 34, No. 3, 465-482 (1993; Zbl 0785.60016)].
necessary and sufficient conditions for weak convergence
scientific article; zbMATH DE number 637649