On different versions of the law of iterated logarithm for R^∞ and 1_p valued wiener process

Abstract

In this article we extend the laws of iterated logarithm established by M. Csorgo and P. Revesz for the standard Wiener process to 1_p (p>1) and R_∞ valued Wiener processes {W_t; t≥0}. In particular one of the obtained results states that

$$\mathop {\lim }\limits_{T \to \infty } \mathop {sup}\limits_{0 < t < T - a_t } \mathop {\sup }\limits_{0 < s < a_t } b_t \parallel W_{t + s} - W_t \parallel = 1 a.e.,$$

, where $b_T = (2a_T [\log (T/a_T ) + \log log T])^{ - {\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}}$, while a_T is an nondecreasing function of T.