A Journal "Theory of Probability and Mathematical Statistics"
2022
2021
2020
2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970


Archive

About   Editorial Board   Contacts   Template   Publication Ethics   Peer Review Process   Special Issues   History  

Theory of Probability and Mathematical Statistics



Study of the limiting behavior of delayed random sums under non-identical distributions setup and a chover type LIL

M. Sreehari, P. Chen

Download PDF

Abstract: We consider delayed sums of the type Sn+an − Sn where an is possibly a positive integer valued random variable satisfying certain conditions and S n is the sum of independent random variables Xn with distribution functions Fn∈{G1, G2}. We study the limiting behavior of the delayed sums and prove laws of the iterated logarithm of Chover type. These results extend the results in Vasudeva and Divanji (1992) and Chen (2008).

Keywords: Stable distribution, domain of normal attraction, Chover type law of the iterated logarithm, delayed random sum.

Bibliography:
1. P. Chen, Limiting behavior of weighted sums with stable distributions, Stat. Probab. Lett., 60 (2002), 367–375.
2. P. Chen, Limiting behavior of delayed sums under a non-identically distribution setup, Ann. Braz. Acad. Sci., 80 (2008), 617–625.
3. J. Chover, A law of the iterated logarithm for stable summands, Proc. Amer. Math. Soc., 17 (1966), 441–443.
4. Y. S. Chow, T. L. Lai, Limiting behavior of weighted sums of independent random variables, Ann. Probab., 1 (1973), 810–824.
5. G. Divanji, A law of iterated logarithm for delayed random sums, Research and Reviews: J. Statist., 6 (2017), 24–32.
6. G. Divanji, K. N. Raviprakash, A log log law for subsequences of delayed random sums, J. Ind. Soc. Probab. Statist., 18 (2017), 159–175.
7. W. Feller, An introduction to probability theory and its applications, vol. II, Wiley, New York, 1971.
8. A. Gut, Stopped Random walks: Limit theorems and applications Springer, New York, 2009.
9. C. C. Heyde, A note concerning the behaviour of iterated logarithm type, Proc. Amer. Math. Soc., 23 (1969), 85–90.
10. T.-C. Hu, M. O. Cabrera, A. Volodin, Almost sure lim sup behavior of dependent bootstrap means, Stoc. Anal. Appl., 24 (2006), 934–952.
11. T. L. Lai, Limit theorems for delayed sums, Ann. Probab., 2 (1974), 432–440.
12. D. Li, P. Chen, A characterization of Chover-type law of iterated logarithm, SpringerPlus, 3:386 (2014).
13. M. Sreehari, On a class of limit distributions for normalized sums of independent random variables, Theory Probab. Appl., 15 (1970), 258–281.
14. W. F. Stout, Almost sure convergence, Academic Press, New York, 1974.
15. R. Vasudeva, G. Divanji, LIL for delayed sums under non-identically distribution setup, Theory Probab. Appl., 37 (1993), 534–542.
16. N. M. Zinchenko, A modified law of iterated logarithm for stable random variables, Theory of Probab. Math. Statist., 49 (1994), 69–76.