Convergence of the Increments of a Wiener Process
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Abstract
Let $\lambda_{(t;\alpha) = (2at (\log(t/a_t) + \log \log t + (1- \alpha)\log\log a_t))^{-1/2}$ where $0 \leq \alpha \leq 1$ be a standard Wiener process. Suppose that at is a nondecreasing function of t such that $0 < a_t$ t and $a_t/t$ is nonincreasing. In this paper we study the almost sure behaviour of $\lim \sup _\to\infty\sup \lambda{(t_k,\alpha}|W_{(t_k+s)}-W(t_k)|$ where $\{t_k\}$ be some increasing sequence diverging to $\infty$.
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Bahram, A.
(2014).
Convergence of the Increments of a Wiener Process.
Acta Mathematica Universitatis Comenianae, 83(1), 113-118.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/82/36
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References
[1] Bahram, A.: Some results on increments of the Wiener Process. Acta Math.Univ. Comenianoe. Vol. LXXIV 2, 163-168 (2005).
[2] Billingsley,P.: Convergence of Probability Measure. Wiley, New York (1979).
[3] Gut, A.: The law of the iterated logarithm for subsequence, Probab. Math. Stat. 7, 27-38 (1986).
[4] Feller,W. :An introduction to probability theory and its applications. Vol 2,2nd. Willy, New York (1968).
[5] Komlos, J. Major, P and Tusnady, G.: An approximation of partial sums of independent r.v.'s and the sample df I, Z. Warrsch. Verw. Gebiete 32, 111-131 (1975).
[6] Komlos, J. Major, P and Tusnady, G.: An approximation of partial sums of independent r.v.'s and the sample df II, Z. Warrsch. Verw. Gebiete 34, 33 - 58 (1976).
[2] Billingsley,P.: Convergence of Probability Measure. Wiley, New York (1979).
[3] Gut, A.: The law of the iterated logarithm for subsequence, Probab. Math. Stat. 7, 27-38 (1986).
[4] Feller,W. :An introduction to probability theory and its applications. Vol 2,2nd. Willy, New York (1968).
[5] Komlos, J. Major, P and Tusnady, G.: An approximation of partial sums of independent r.v.'s and the sample df I, Z. Warrsch. Verw. Gebiete 32, 111-131 (1975).
[6] Komlos, J. Major, P and Tusnady, G.: An approximation of partial sums of independent r.v.'s and the sample df II, Z. Warrsch. Verw. Gebiete 34, 33 - 58 (1976).