Geodesic
Convergence of the Increments of a Wiener Process
Abdelkader Bahram1 ; Abdelkader Bahram
1Université Djillali Liabes Faculté de Technologie Département d'informatique
Acta mathematica Universitatis Comenianae, Tome 83 (2014) no. 1, pp. 113-118 
Abdelkader Bahram; Abdelkader Bahram. Convergence of the Increments of a Wiener Process. Acta mathematica Universitatis Comenianae, Tome 83 (2014) no. 1, pp. 113-118. http://geodesic.mathdoc.fr/item/AMUC_2014_83_1_a8/
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     author = {Abdelkader Bahram and Abdelkader Bahram},
     title = { Convergence of the {Increments} of a {Wiener} {Process}},
     journal = {Acta mathematica Universitatis Comenianae},
     pages = {113--118},
     year = {2014},
     volume = {83},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2014_83_1_a8/}
}
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Voir la notice de l'article provenant de la source Comenius University

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 $\infty$.