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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On a necessary conditon for the Erdős-Rényi law of large numbers

Author: Josef Steinebach
Journal: Proc. Amer. Math. Soc. 68 (1978), 97-100
MSC: Primary 60F15
MathSciNet review: 0461637
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Abstract: For a sequence $ {\{ {X_i}\} _{i = 1,2, \ldots }}$ of independent, identically distributed random variables with existing moment-generating function $ \varphi (t) = E\;\exp (t{X_i})$ in some nondegenerate interval, Erdös and Rényi (1970) studied the maximum $ D(N,K)$ of the $ N - K + 1$ sample means $ {K^{ - 1}}({S_{n + K}} - {S_n}),\;0 \leqslant n \leqslant N - K$, where $ {S_0} = 0,\;{S_n} = {X_1} + \cdots + {X_n}$. They showed that for a certain range of numbers a there exist positive constants $ C({\mathbf{a}})$ such that $ {\lim _{N \to \infty }}D(N,[C({\mathbf{a}})\log N]) = {\mathbf{a}}$ with probability one. In the present paper it is shown that the existence of the moment-generating function is also a necessary condition, i.e. that $ \lim {\sup _{N \to \infty }}D(N,[C\log N]) = \infty $ for every positive constant C, if the moment-generating function does not exist for any positive number t.

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Keywords: Erdös-Rényi law of large numbers, large deviations, moment-generating functions
Article copyright: © Copyright 1978 American Mathematical Society