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Proceedings of the American Mathematical Society

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On a conjecture of Révész
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by Qi Man Shao PDF
Proc. Amer. Math. Soc. 123 (1995), 575-582 Request permission

Abstract:

Let $\{ {X_n},n \geq 1\}$ be i.i.d. random variables with $P({X_i} = \pm 1) = \frac {1}{2}$. Révész (1990) proved \[ \begin {array}{*{20}{c}} {1 \le \lim \inf \limits _{n \to \infty } \max \limits _{0 \le j < n} \max \limits _{1 \le k \le n - j} {{(2k\log n)}^{ - 1/2}}({S_{j + k}} - {S_j})} \hfill \\ { \le \lim \sup \limits _{n \to \infty } \max \limits _{0 \le j < n} \max \limits _{1 \le k \le n - j} {{(2k\log n)}^{ - 1/2}}({S_{j + k}} - {S_j}) \le K\quad {\rm {a}}.{\rm {s}}.} \hfill \\ \end {array}\] and conjectured $K = 1$, where ${S_n} = \sum \nolimits _{i = 1}^n {{X_i}}$. In this we show that Révész’s conjecture is true but the conclusion is not valid for general i.i.d. random variables with finite moment generating function.
References
  • Herman Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Statistics 23 (1952), 493–507. MR 57518, DOI 10.1214/aoms/1177729330
  • M. Csörgő and P. Révész, Strong approximations in probability and statistics, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 666546
  • D. A. Darling and P. Erdös, A limit theorem for the maximum of normalized sums of independent random variables, Duke Math. J. 23 (1956), 143–155. MR 74712
  • Paul Erdős and Alfréd Rényi, On a new law of large numbers, J. Analyse Math. 23 (1970), 103–111. MR 272026, DOI 10.1007/BF02795493
  • D. L. Hanson and Ralph P. Russo, Some results on increments of the Wiener process with applications to lag sums of i.i.d. random variables, Ann. Probab. 11 (1983), no. 3, 609–623. MR 704547
  • D. L. Hanson and Ralph P. Russo, Some limit results for lag sums of independent, non-i.i.d., random variables, Z. Wahrsch. Verw. Gebiete 68 (1985), no. 4, 425–445. MR 772191, DOI 10.1007/BF00535337
  • M. Csörgő and P. Révész, A new method to prove Strassen type laws of invariance principle. I, II, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75), 255–259; ibid. 31 (1974/75), 261–269. MR 375411, DOI 10.1007/BF00532865
  • P. Révész, Random walk in random and non-random environments, World Scientific, Singapore, 1990. Q. M. Shao, Limit theorems for sums of dependent and independent random variables, Ph.D. Dissertation, Univ. of Science and Technology of China, Hefei, People’s Republic of China, 1989.
  • Qi Man Shao, Random increments of a Wiener process and their applications, Studia Sci. Math. Hungar. 29 (1994), no. 3-4, 443–480. MR 1304897
  • Qi Man Shao, Strong approximation theorems for independent random variables and their applications, J. Multivariate Anal. 52 (1995), no. 1, 107–130. MR 1325373, DOI 10.1006/jmva.1995.1006
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 575-582
  • MSC: Primary 60F15; Secondary 60G17
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1231304-9
  • MathSciNet review: 1231304