SOME RESULTS ON INCREMENT OF A TWOPARAMETER WIENER PROCESS

In this paper we discuss a step further some convergence problems on inorement of a two-parameter Wiener Process. We give the following results: Let W(x, y)(0 \leqslant x, y< \infty) be a two-parameter Wiener Process and suppose that (1) \lambda_T is a noninoreasing function of T. (2) a_T / T is a nonincreasing function of T. Then (3) \beginaligned & \lim _T \rightarrow \infty \frac\log T a_T^-1+\log \left(1+\log b_T a_T^-\frac12\right)\log \log T=r, \quad(0 \leqslant r< \infty) . \\ & \lim _T \rightarrow \infty \sup _R \in L_F \delta_T|W(R)|=\lim _T \rightarrow \infty \sup _R \in L_T \delta_T|W(R)|=\sqrt\fracrr+1 \quad \text a.s. \\ & \varlimsup_T \rightarrow \infty \sup _R \in L_F^* \delta_T|W(R)|=\varlimsup_T \rightarrow \infty \sup _R \in \mathcalL_T \delta_T|W(R)|=1 \quad \text a.s. \\ & \varlimsup_T \rightarrow \infty \sup _R \in L_\mathbfF \lambda_T|W(R)|=\varlimsup_T \rightarrow \infty \sup _R \in L_\mathbfr \lambda_T|W(R)|=\sqrt\fracr+1r \text a.s. \endaligned .