On the other LIL for variables without finite variance
Authors:
R. P. Pakshirajan and M. Sreehari
Journal:
Theor. Probability and Math. Statist. 107 (2022), 159-171
MSC (2020):
Primary 60F15; Secondary 60G50, 60J65
DOI:
https://doi.org/10.1090/tpms/1179
Published electronically:
November 8, 2022
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: In this paper we give a simpler proof of Jain’s [Z. Wahrsch. Verw. Gebiete 59 (1982), no. 1, 117–138] result concerning the Other Law of the Iterated Logarithm for partial sums of a class of independent and identically distributed random variables with infinite variance but in the domain of attraction of a normal law. Jain’s result is less restrictive than ours but depends heavily on the techniques of Donsker and Varadhan in the theory of Large deviations. Our proof involves elementary properties of slowly varying functions. We assume that the distribution of random variables $X_n$ satisfies the condition that $\lim _{ x\rightarrow \infty } \frac {\log H(x)}{(\log x)^\delta } = 0$ for some $0 <\delta < 1/2$, where $H(x)=\mathsf E\left (X_1^2 I(|X_1|\le x)\right )$ is a slowly varying function. The condition above is not very restrictive.
References
- Leo Breiman, On the tail behavior of sums of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 (1967), 20–25. MR 226707, DOI 10.1007/BF00535464
- Boris Buchmann and Ross Maller, The small-time Chung-Wichura law for Lévy processes with non-vanishing Brownian component, Probab. Theory Related Fields 149 (2011), no. 1-2, 303–330. MR 2773034, DOI 10.1007/s00440-009-0255-1
- Guang-Hui Cai, On the other law of the iterated logarithm for self-normalized sums, An. Acad. Brasil. Ciênc. 80 (2008), no. 3, 411–418 (English, with English and Portuguese summaries). MR 2444529, DOI 10.1590/S0001-37652008000300002
- S. Cho, P.Kim,and J. Lee, General law of the iterated logarithm for Markov processes, arXiv 2102.01917v1 [math.PR] 3 Feb 2021.
- Kai Lai Chung, On the maximum partial sums of sequences of independent random variables, Trans. Amer. Math. Soc. 64 (1948), 205–233. MR 26274, DOI 10.1090/S0002-9947-1948-0026274-0
- E. Csáki, On the lower limits of maxima and minima of Wiener process and partial sums, Z. Wahrsch. Verw. Gebiete 43 (1978), no. 3, 205–221. MR 0494527, DOI 10.1007/BF00536203
- Miklós Csörgő and ZhiShui Hu, A strong approximation of self-normalized sums, Sci. China Math. 56 (2013), no. 1, 149–160. MR 3016589, DOI 10.1007/s11425-012-4434-7
- Alejandro de Acosta, Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm, Ann. Probab. 11 (1983), no. 1, 78–101. MR 682802
- Uwe Einmahl, On the other law of the iterated logarithm, Probab. Theory Related Fields 96 (1993), no. 1, 97–106. MR 1222366, DOI 10.1007/BF01195884
- William Feller, An extension of the law of the iterated logarithm to variables without variance, J. Math. Mech. 18 (1968/1969), 343–355. MR 0233399, DOI 10.1512/iumj.1969.18.18027
- 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
- David Freedman, Brownian motion and diffusion, 2nd ed., Springer-Verlag, New York-Berlin, 1983. MR 686607, DOI 10.1007/978-1-4615-6574-1
- Naresh C. Jain, A Donsker-Varadhan type of invariance principle, Z. Wahrsch. Verw. Gebiete 59 (1982), no. 1, 117–138. MR 643792, DOI 10.1007/BF00575529
- Naresh C. Jain and William E. Pruitt, Maxima of partial sums of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 27 (1973), 141–151. MR 358929, DOI 10.1007/BF00536625
- Naresh C. Jain and William E. Pruitt, The other law of the iterated logarithm, Ann. Probability 3 (1975), no. 6, 1046–1049. MR 397845, DOI 10.1214/aop/1176996232
- R. P. Pakshirajan, On the maximum partial sums of sequences of independent random variables, Teor. Veroyatnost. i Primenen. 4 (1959), 398–404 (English, with Russian summary). MR 0115205
- R. P. Pakshirajan, Probability theory, Texts and Readings in Mathematics, vol. 63, Hindustan Book Agency, New Delhi, 2013. A foundational course. MR 3051706, DOI 10.1007/978-93-86279-54-5
- Tian-xiao Pang, Li-xin Zhang, and Jian-feng Wang, Precise asymptotics in the self-normalized law of the iterated logarithm, J. Math. Anal. Appl. 340 (2008), no. 2, 1249–1262. MR 2390926, DOI 10.1016/j.jmaa.2007.09.054
- B. A. Rogozin, Distribution of the first laddar moment and height, and fluctuations of a random walk, Teor. Verojatnost. i Primenen. 16 (1971), 539–613 (Russian, with English summary). MR 0290473
- Qi Man Shao, A Chung type law of the iterated logarithm for subsequences of a Wiener process, Stochastic Process. Appl. 59 (1995), no. 1, 125–142. MR 1350259, DOI 10.1016/0304-4149(95)00025-3
- A. V. Skorokhod, Studies in the theory of random processes, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. Translated from the Russian by Scripta Technica, Inc. MR 0185620
- Hao Yu, A strong invariance principle for associated sequences, Ann. Probab. 24 (1996), no. 4, 2079–2097. MR 1415242, DOI 10.1214/aop/1041903219
- Qingpei Zang, Precise asymptotics in the law of the iterated logarithm under dependence, Int. J. Nonlinear Sci. 6 (2008), no. 2, 154–159. MR 2453356
References
- L. Breiman, On the tail behavior of sums of independent random variables, Z. Wahrsch. Verw. Gebiete 9 (1967), 20–25. MR 226707
- B. Buchmann, and R. Maller, The small-time Chung-Wichura law for Lévy processes with non-vanishing Brownian component, Probab. Theory and Related Fields149 (2011), no. 1-2, 303–330. MR 2773034
- G-H. Cai, On the other law of the iterated logarithm for self-normalized sums, An. Acad. Brasil. Ciênc. 80 (2008), no. 3, 411–418. MR 2444529
- S. Cho, P.Kim,and J. Lee, General law of the iterated logarithm for Markov processes, arXiv 2102.01917v1 [math.PR] 3 Feb 2021.
- K. L. Chung, On the maximum partial sums of sequences of independent random variables, Trans. Amer. Math. Soc. 64 (1948), 205–233. MR 26274
- E. Csáki, On the lower limits of maxima and minima of Wiener process and partial sums, Z. Wahrsch. Verw. Gebiete 43 (1978), no. 3, 205–221. MR 0494527
- M. Csörgö, and Z. Hu, A strong approximation of self-normalized sums, Sci. China Math. 56 (2013), no. 1, 149–160. MR 3016589
- A. De Acosta, Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm, Ann. Probab. 11 (1983), no. 1, 78–101. MR 682802
- U. Einmahl, On the other law of the iterated logarithm, Probab. Theory Related Fields 96 (1993), no. 1, 97–106. MR 1222366
- W. Feller, An extension of the law of the iterated logarithm to variables without variances, J. Math. Mech. 18 (1968/1969), 343–355. MR 0233399
- W. Feller, An introduction to Probability Theory and its Applications, Vol. II, second edition, John Wiley & Sons, Inc., New York, 1971. MR 0270403
- D. Freedman, Brownian motion and Diffusion, second edition, Springer-Verlag, New York, 1983. MR 686607
- N. C. Jain, A Donsker-Varadhan type of invariance principle, Z. Wahrsch. Verw. Gebiete 59 (1982), no. 1, 117–138. MR 643792
- N. C. Jain, and W. E. Pruitt, Maxima of partial sums of indenendent random variables, Z. Wahrsch. Verw. Gebiete 27 (1973), 141–151. MR 358929
- N. C. Jain, and W. E. Pruitt, The other law of the iterated logarithm, Ann. Probab. 3 (1975), no. 6, 1046–1049. MR0397845 MR 397845
- R. P. Pakshirajan, On the maximum partial sums of sequences of independent random variables, Teor. Verojatnost. i. Premenen., 4 (1959), 398–404. MR 0115205
- R. P. Pakshirajan, Probability theory-A foundational course, Texts and Readings in Mathematics, 63, Hindustan Book Agency, New Delhi, 2013. MR 3051706
- T-X. Pang, L-X. Zhang, and J-F. Wang, Precise asymptotics in the self-normalized law of the iterated logarithm, J. Math. Anal. Appl. 340 (2008), no. 2, 1249–1262. MR 2390926
- B. A. Rogozin, The distribution of the first ladder moment and height and fluctuations of a random variable, Teor. Verojatnost. i Primenen. 16 (1971), 539–613. MR 0290473
- Q-M. Shao, A Chung type law of the iterated logarithm for subsequences of a Wiener process, Stochastic Process. Appl. 59 (1995), no. 1, 125–142. MR 1350259
- A. V. Skorohod, Studies in the Theory of Random Processes, translated from the Russian by Scripta Technica, Inc, Addison-Wesley Publishing Co., Inc., Reading, MA, 1965. MR 0185620
- H. Yu, A strong invariance principle for associated sequences, Ann. Probab. 24 (1996), no. 4, 2079–2097. MR 1415242
- L-X. Zhang. Precise rates in the law of iterated logarithm, arXiv:math/0610519v1 [math.PR] 17 Oct 2006. MR 2453356
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2020):
60F15,
60G50,
60J65
Retrieve articles in all journals
with MSC (2020):
60F15,
60G50,
60J65
Additional Information
R. P. Pakshirajan
Affiliation:
227, 18th Main, 6th Block, Koramangala, Bengaluru -560095, India
Email:
vainatheyarajan@yahoo.in
M. Sreehari
Affiliation:
6-B, Vrundavan Park, New Sama Road, Vadodara 390008, India
Email:
msreehari03@yahoo.co.uk
Keywords:
Chung’s LIL,
domain of non-normal attraction,
Skorohod embedding,
slowly varying function,
Karamata representation
Received by editor(s):
June 10, 2021
Accepted for publication:
December 23, 2021
Published electronically:
November 8, 2022
Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv