Almost Sure Limit Inferior for Increments of Stable Subordinators

DOI: 10.4236/oalib.1100812   PDF        1,247 Downloads   1,513 Views  

Abstract

Let {X(t), 0≤t<∞} be a sequence of completely asymmetric stable process or stable subordinators defined on a common probability space (Ω,ζ,P). In this paper, for proper selection of norming constants, we study almost sure limit inferior for increments of stable subordinators of geometrically increasing subsequences. Also we obtain similar results to delayed sums and study the existence of moments for boundary crossing random variables.

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Vidyalaxmi, K. , Prakash, K. and Divanji, G. (2014) Almost Sure Limit Inferior for Increments of Stable Subordinators. Open Access Library Journal, 1, 1-8. doi: 10.4236/oalib.1100812.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Gut, A. (1986) Law of Iterated Logarithm for Subsequences. Probability and Mathematical Statistics, 7, 27-58.
[2] Hwang, K.S., Choi, Y.K. and Jung, J.S. (1997) On Superior Limits for the Increments of Gaussian Processes. Statistics and Probability Letter, 35, 289-296.
http://dx.doi.org/10.1016/S0167-7152(97)00025-4
[3] Lai, T.L. (1973) Limit Theorems for Delayed Sums. The Annals of Probability, 2, 432-440.
http://dx.doi.org/10.1214/aop/1176996658
[4] Mijhneer, J.L. (1975) Sample Path Properties of Stable Process. Mathematisch, Amsterdam.
[5] Mijnheer, J.L. (1995) On the Law of Iterated Logarithm for Subsequences for a Stable Subordinator. Journal of Mathematical Sciences, 76, 2283-2286.
http://dx.doi.org/10.1007/BF02362699
[6] Schwabe, R. and Gut, A. (1996) On the Law of the Iterated Logarithm for Rapidly Increasing Subsequences. Mathematische Nachrichten, 178, 309-332.
http://dx.doi.org/10.1002/mana.19961780115
[7] Vasudeva, R. and Divanji, G. (1988) Law of Iterated Logarithm for the Increments of Stable Subordinators. Stochastic Processes and Their Applications, 28, 293-300.
http://dx.doi.org/10.1016/0304-4149(88)90102-0
[8] Slivka, J. (1969) On the LIL. Proceedings of the National Academy of Sciences of the United States of America, 63, 2389-291.
[9] Slivka, J. and Savero, N.C. (1970) On the Strong Law of Large Numbers. Proceedings of the American Mathematical Society, 24, 729-734.

  
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