Abstract
Let X1, X2, ... be a sequence of independent, identically distributed random variables, and let . The rate of convergence of probabilities of the form and is studied for any ɛ > ɛ0 and some r and ɛ0. Moreover, necessary and sufficient conditions are given that the relations be satisfied uniformly with respect to x in the region 0 ≤ x ≤ c√log n, where σ and c are some positive constants, and . Local limit theorems are also presented.
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