PRECISE RATES IN THE LAW OF THE LOGARITHM FOR THE MOMENT CONVERGENCE OF I.I.D. RANDOM VARIABLES

Title & Authors
PRECISE RATES IN THE LAW OF THE LOGARITHM FOR THE MOMENT CONVERGENCE OF I.I.D. RANDOM VARIABLES
Pang, Tian-Xiao; Lin, Zheng-Yan; Jiang, Ye; Hwang, Kyo-Shin;

Abstract
Let {$\small{X,\;X_n;n{\geq}1}$} be a sequence of i.i.d. random variables. Set $\small{S_n=X_1+X_2+{\cdots}+X_n,\;M_n=\max_{k{\leq}n}|S_k|,\;n{\geq}1}$. Then we obtain that for any -1 if and only if EX=0 and $\small{EX^2={\sigma}^2}$<$\small{{\infty}}$.
Keywords
law of the logarithm;moment convergence;rate of convergence;strong approximation;i.i.d. random variables;
Language
English
Cited by
1.
A SUPPLEMENT TO PRECISE ASYMPTOTICS IN THE LAW OF THE ITERATED LOGARITHM FOR SELF-NORMALIZED SUMS,;

대한수학회지, 2008. vol.45. 6, pp.1601-1611
References
1.
P. Billingsley, Convergence of Probability Measures: Second edition, Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999

2.
M. Csorgo and P. Revesz, Strong Approximations in Probability and Statistics, Probability and Mathematical Statistics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981

3.
U. Einmahl, The Darling-Erdos theorem for sums of i.i.d. random variables, Probab. Theory Related Fields 82 (1989), no. 2, 241-257

4.
W. Feller, The law of the iterated logarithm for identically distributed random variables, Ann. of Math. (2) 47 (1946), 631-638

5.
A. Gut and A. Spataru, Precise asymptotics in the law of the iterated logarithm, Ann. Probab. 28 (2000), no. 4, 1870-1883

6.
A. I. Sakhanenko, On estimates of the rate of convergence in the invariance principle, In Advances in Probability Theory: Limit Theorems and Related Problems (A. A. Borovkov, ed.) 124-135, Springer, New York, 1984

7.
A. I. Sakhanenko, Convergence rate in the invariance principle for nonidentically distributed variables with exponential moment, In Advances in Probability Theory: Limit Theorems for Sums of Random Variables (A. A. Borovkov, ed.) 2-73. Springer, New York, 1985. Springer, New York