Central Limit Theorems for Dependent Random Variables
نویسنده
چکیده
1. Limiting distributions of sums of 'small' independent random variables have been extensively studied and there is a satisfactory general theory of the subject (see e.g. the monograph of B.V. Gnedenko and A.N. Kolmogorov [2]). These results are conveniently formulated for double arrays Xn k (k = 1, . . . , kn ; n = 1, 2, . . . ) of random variables where the Xn k (k = 1, . . . , kn), the random variables in the n-th row, are assumed mutually independent for every n. The smallness assumption is that, as n -* <*>, the random variables Xn k converge in probability to 0, uniformly in k. This assumption implies that the limit distributions of Xn ! + . . . + Xn k are infinitely divisible ones, and necessary and sufficient conditions on the distributions of the summands are known for convergence to any specific infinitely divisible distribution.
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