On a functional law of iterated logarithm for a multiparameter Brownian motion. (Russian) Zbl 0577.60032

Zinchenko, N. M.

Teor. Veroyatn. Mat. Stat. 29, 46-51 (1983).

Let \(W=W(t)\), \(t\in [0,1]^ q\), be a q-parametric Brownian motion. Let \(N\in {\mathbb{N}}^ q\), and \(\phi\) be a positive function on \(R^ q_+\), nondecreasing, with some limit properties. It is shown that the set of limit points of the family \[ f_ N(t)=W(N_ 1t_ 1,...,N_ qt_ q)/(\prod^{q}_{i=1}t_ i)^{1/2}\phi (N) \] in \(C_ o([0,1]^ q)\) with probability 1 coincides with the set of absolutely continuous functions from \(C_ 0([0,1]^ q)\) with Radon-Nikodym derivatives, the quadratic mean of which is bounded by some constant.

Reviewer: C.Baldauf

MSC:

60F17	Functional limit theorems; invariance principles
60G60	Random fields
60J65	Brownian motion

Keywords:

functional law of iterated logarithm; multiparameter Brownian; motion