Study of the Convergence of the Increments of Gaussian Process

Abdelkader Bahram¹, Shaban A. El-Shehawy²

¹Department of Mathematics, Djillali Liabes University, Sidi Bel Abbès, Algeria.
²Department of Mathematics, Faculty of Science, Menoufia University, Shebin El-Kom, Egypt.

DOI: 10.4236/am.2015.66085                

Abstract

Let be a Gaussian process with stationary increments . Let be a nondecreasing function of t with . This paper aims to study the almost sure behaviour of where

with and is an increasing sequence diverging to .

Keywords

Wiener Process, Gaussian Process, Law of the Iterated Logarithm, Regularly Varying Function

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Bahram, A. (2014) Convergence of the Increments of a Wiener Process. Acta Mathematica Universitatis Comenianae, 83, 113-118.
[2]	Hwang, K.S., Choi, Y.K. and Jung, J.S. (1997) On Superior Limits for the Increments of Gaussian Processes. Statistics and Probability Letters, 35, 289-296. http://dx.doi.org/10.1016/S0167-7152(97)00025-4
[3]	Choi, Y.K. (1991) Erdös-Réyi Type Laws Applied to Gaussian Process. Journal of Mathematics of Kyoto University, 31, 191-217.
[4]	Slepian, D. (1962) The One-Sided Barrier Problem for Gaussian Noise. Bell System Technical Journal, 41, 463-501. http://dx.doi.org/10.1002/j.1538-7305.1962.tb02419.x
[5]	Vasudeva, R. and Savitha, S. (1993) On the Increments of Weiner Process—A Look through Subsequences. Stochastic Processes and Their Applications, 47, 153-158. http://dx.doi.org/10.1016/0304-4149(93)90101-9
[6]	Fernique, X. (1975) Evaluations of Processus Gaussian Composes. Probability in Banach Spaces. Lecture Notes in Mathematics, 526, 67-83.