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An extension of the Erdős-Rényi new law of large numbers


Author: Stephen A. Book
Journal: Proc. Amer. Math. Soc. 48 (1975), 438-446
MSC: Primary 60F15
DOI: https://doi.org/10.1090/S0002-9939-1975-0380950-0
MathSciNet review: 0380950
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Abstract: If $ {S_n}$ is the $ n$th partial sum of a sequence of independent, identically distributed random variables $ {X_1},{X_2} \cdots $ such that $ E({X_1}) = 0$ and $ E(\exp (t{X_1})) < \infty $ for some nonempty interval of $ t$'s, then, for a wide range of positive numbers $ \lambda $, Erdös and Rényi (1970) showed that $ \Sigma (N,[C(\lambda )\log N])$ converges with probability one to $ \lambda $ as $ N \to \infty $, where $ \Sigma (N,K)$ is the maximum of the $ N - K + 1$ averages of the form $ {K^{ - 1}}({S_{n + K}} - {S_n})$ for $ 0 \leq n \leq N - K$, and $ C(\lambda )$ is a known constant depending on $ \lambda $ and the distribution of $ {X_1}$. The objective of the present article is to state and prove the Erdös-Rényi theorem for the $ N - K + 1$ ``averages'' of the form $ {K^{ - 1/r}}({S_{n + K}} - {S_n})$, where $ 1 < r < 2$. This form of the Erdös-Rényi theorem arises from the extended form of the strong law of large numbers which asserts that, if $ E(\vert{X_1}{\vert^r}) < \infty $ for some $ r,1 \leq r < 2$, and $ E({X_1}) = 0$, then $ {n^{ - 1/r}}{S_n}$ converges with probability one to 0 as $ n \to \infty $.


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  • [1] R. R. Bahadur and R. Ranga Rao, On deviations of the sample mean, Ann. Math. Statist. 31 (1960), 1015-1027. MR 22 #8549. MR 0117775 (22:8549)
  • [2] S. A. Book, The Erdös-Rényi new law of large numbers for weighted sums, Proc. Amer. Math. Soc. 38 (1973), 165-171. MR 46 #10044. MR 0310946 (46:10044)
  • [3] H. Cramér, Sur un nouveau théoreme-limite de la théorie des probabilités, Actualités Sci. Indust., no. 736, Hermann, Paris, 1938, pp. 5-23.
  • [4] P. Erdös and A. Rényi, On a new law of large numbers, J. Analyse Math. 23 (1970), 103-111. MR 42 #6907. MR 0272026 (42:6907)
  • [5] W. Feller, An introduction to probability theory and its applications. I, 3rd ed., Wiley, New York, 1968. MR 37 #3604. MR 0228020 (37:3604)
  • [6] I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff, Groningen, 1971. MR 0322926 (48:1287)
  • [7] M. Loève, Probability theory, 3rd ed., Van Nostrand, Princeton, N. J., 1963. MR 34 #3596. MR 0203748 (34:3596)
  • [8] V. V. Petrov, Generalization of Cramér's limit theorem, Uspehi Mat. Nauk 9 (1954), no. 4 (62), 195-202. (Russian) MR 16, 378. MR 0065058 (16:378e)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0380950-0
Keywords: Strong limit theorems, laws of large numbers, large deviations, moment-generating functions
Article copyright: © Copyright 1975 American Mathematical Society

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