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

Sreehari, M.; Chen, P.

doi:10.1090/tpms/1103

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

Authors: M. Sreehari and P. Chen
Journal: Theor. Probability and Math. Statist. 100 (2020), 153-168
MSC (2010): Primary 60F15
DOI: https://doi.org/10.1090/tpms/1103
Published electronically: August 5, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider delayed sums of the type $S_{n+a_n}-S_n$ where $a_n$ is possibly a positive integer valued random variable satisfying certain conditions and $S_n$ is the sum of independent random variables $X_n$ with distribution functions $F_n \in \{G_1, G_2\}$. We study the limiting behavior of the delayed sums and prove Chover’s type laws of the iterated logarithm. These results extend the results in Vasudeva and Divanji (1992) and Chen (2008).

References

Chen Pingyan, Limiting behavior of weighted sums with stable distributions, Statist. Probab. Lett. 60 (2002), no. 4, 367–375. MR 1947176, DOI https://doi.org/10.1016/S0167-7152%2802%2900286-9
Pingyan Chen, Limiting behavior of delayed sums under a non-identically distribution setup, An. Acad. Brasil. Ciênc. 80 (2008), no. 4, 617–625. MR 2478597, DOI https://doi.org/10.1590/S0001-37652008000400003
Joshua Chover, A law of the iterated logarithm for stable summands, Proc. Amer. Math. Soc. 17 (1966), 441–443. MR 189096, DOI https://doi.org/10.1090/S0002-9939-1966-0189096-2
Y. S. Chow and T. L. Lai, Limiting behavior of weighted sums of independent random variables, Ann. Probability 1 (1973), 810–824. MR 353426, DOI https://doi.org/10.1214/aop/1176996847

G. Divanji, A law of iterated logarithm for delayed random sums, Research and Reviews: J. Statist. 6 (2017), 24–32.

G. Divanji and K. N. Raviprakash, A log log law for subsequences of delayed random sums, J. Ind. Soc. Probab. Statist. 18 (2017), 159–175.

William Feller, An introduction to probability theory and its applications. Vol. II., 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403

Allan Gut, Stopped random walks, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, New York, 2009. Limit theorems and applications. MR 2489436

C. C. Heyde, A note concerning behaviour of iterated logarithm type, Proc. Amer. Math. Soc. 23 (1969), 85–90. MR 251772, DOI https://doi.org/10.1090/S0002-9939-1969-0251772-3

Tien-Chung Hu, Manuel Ordóñez Cabrera, and Andrei Volodin, Almost sure lim sup behavior of dependent bootstrap means, Stoch. Anal. Appl. 24 (2006), no. 5, 939–952. MR 2258910, DOI https://doi.org/10.1080/07362990600869969

Tze Leung Lai, Limit theorems for delayed sums, Ann. Probability 2 (1974), 432–440. MR 356193, DOI https://doi.org/10.1214/aop/1176996658

D. Li and P. Chen, A Characterization of Chover-Type Law of Iterated Logarithm, SpringerPlus, 3:386 (2014).

M. Sreehari, On a class of limit distributions for normalized sums of independent random variables, Teor. Verojatnost. i Primenen. 15 (1970), 269–290 (English, with Russian summary). MR 0270423

William F. Stout, Almost sure convergence, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Probability and Mathematical Statistics, Vol. 24. MR 0455094

R. Vasudeva and G. Divanji, LIL for delayed sums under a non-identically distributed setup, Teor. Veroyatnost. i Primenen. 37 (1992), no. 3, 534–542; English transl., Theory Probab. Appl. 37 (1992), no. 3, 497–506. MR 1214358, DOI https://doi.org/10.1137/1137099

N. M. Zinchenko, A modified law of iterated logarithm for stable random variables, Teor. Ĭmovīr. Mat. Stat. 49 (1993), 99–109 (Ukrainian, with Ukrainian summary); English, with English Ukrainian summary transl., Theory Probab. Math. Statist. 49 (1994), 69–76 (1995). MR 1445249