On a conjecture of Révész

Author:
Qi Man Shao

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1007/BF00532865
P. Révész, - 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 https://doi.org/10.1006/jmva.1995.1006

*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.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
60F15,
60G17

Retrieve articles in all journals with MSC: 60F15, 60G17

Additional Information

Keywords:
Increments,
partial sums,
a.s. convergence,
random walk

Article copyright:
© Copyright 1995
American Mathematical Society