A NEW KIND OF THE LAW OF THE ITERATED LOGARITHM FOR PRODUCT OF A CERTAIN PARTIAL SUMS

Title & Authors
A NEW KIND OF THE LAW OF THE ITERATED LOGARITHM FOR PRODUCT OF A CERTAIN PARTIAL SUMS
Zang, Qing-Pei;

Abstract
Let {X, $\small{X_{i};\;i{\geq}1}$} be a sequence of independent and identically distributed positive random variables. Denote $\small{S_n= \sum\array\\_{i=1}^nX_i}$ and $\small{S\array\\_n^{(k)}=S_n-X_k}$ for n $\small{{\geq}}$1, $\small{1{\leq}k{\leq}n}$. Under the assumption of the finiteness of the second moments, we derive a type of the law of the iterated logarithm for $\small{S\array\\_n^{(k)}}$ and the limit point set for its certain normalization.
Keywords
law of the iterated logarithm;product of partial sums;strong law of large numbers;
Language
English
Cited by
References
1.
B. C. Arnold and J. A. Villasenor, The asymptotic distributions of sums of records, Extremes 1 (1999), no. 3, 351-363.

2.
P. Y. Chen, On the law of iterated logarithm for products of sums, Acta Math. Sci. Ser. A Chin. Ed. 28 (2008), no. 1, 66-72.

3.
K. A. Fu and W. Huang, A weak invariance principle for self-normalized products of sums of mixing sequences, Appl. Math. J. Chinese Univ. Ser. B 23 (2008), no. 2, 183- 189.

4.
K. Gonchigdanzan, An almost sure limit theorem for the product of partial sums with stable distribution, Statist. Probab. Lett. 78 (2008), no. 18, 3170-3175.

5.
K. Gonchigdanzan and K. M. Kosinski, On the functional limits for partial sums under stable law, Statist. Probab. Lett. 79 (2009), no. 17, 1818-1822.

6.
K. Gonchigdanzan and G. Rempala, A note on the almost sure limit theorem for the product of partial sums, Appl. Math. Lett. 19 (2006), no. 2, 191-196.

7.
K. M. Kosinski, On the functional limits for sums of a function of partial sums, Statist. Probab. Lett. 79 (2009), no. 13, 1522-1527.

8.
Y. X. Li and J. F.Wang, Asymptotic distribution for products of sums under dependence, Metrika 66 (2007), no. 1, 75-87.

9.
W. D. Liu and Z. Y. Lin, Asymptotics for self-normalized random products of sums for mixing sequences, Stoch. Anal. Appl. 25 (2007), no. 4, 739-762.

10.
X. W. Lu and Y. C. Qi, A note on asymptotic distribution of products of sums, Statist. Probab. Lett. 68 (2004), no. 4, 407-413.

11.
P. Matula and I. Stepien, Weak convergence of products of sums of independent and non-identically distributed random variables, J. Math. Anal. Appl. 353 (2009), no. 1, 49-54.

12.
Y. C. Qi, Limit distributions for products of sums, Statist. Probab. Lett. 62 (2003), no. 1, 93-100.

13.
G. Rempala and J. Wesolowski, Asymptotics for products of sums and v-statistics, Electron. Comm. Probab. 7 (2002), 47-54.

14.
L. X. Zhang and W. Huang, A note on the invariance principle of the product of sums of random variables, Electron. Comm. Probab. 12 (2007), 51-56.