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A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables. (English) Zbl 0657.60029
Let \(X_ 1,X_ 2,..\). be a sequence of independent random variables with common non-degenerate distribution function F. For each n let \(X_{1,n}\leq...\leq X_{n,n}\) denote the order statistics based on \(X_ 1,...,X_ n\). For fixed non-negative integers m and k the lightly trimmed sum is defined by \(S_ n(m,k)=\sum^{n-k}_{j=m+1}X_{j,n}\). When \(m=k=0\) this becomes an ordinary untrimmed sum.
The authors present a unified probabilistic approach to the problem of asymptotic distribution of lightly trimmed (as a special case, untrimmed) sums. This approach is based upon the asymptotic behavior of the uniform empirical distribution function in conjunction with an integral representation of these sums.
At first they prove general theorems giving necessary and sufficient conditions for the convergence in law of normalized trimmed (and untrimmed) sums along subsequences of positive integers. Some of these results also show which portions of these sums contribute to ingredients of the limiting laws.
The authors also give a representation of the infinitely divisible random variable in terms of Poisson processes, which is an integral part of their approach. In a sequence of corollaries to these general theorems the authors give necessary and sufficient conditions for these lightly trimmed or untrimmed sums to be in the domain of attraction or partial attraction of a normal or non-normal stable law, in the domain of partial attraction of some infinitely divisible law and a necessary and sufficient condition for the stochastic compactness and subsequential compactness of these sums.
The analytic conditions in these theorems and corollaries are all expressed in terms of quantile function, i.e. left continuous inverse of F.
Reviewer: T.Mori

MSC:
60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
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