J. Math. Kyoto Univ. (JMKYAZ) 24-3 (1984) 507-520 Another limit theorem for slowly incrasing occupation times By Yuji KASAHARA (Communicated by Prof. S. Watanabe, February 7, 1983) O. Introduction S. Kotani and the author [6] proved two limit theorems for occupation times of two-dimensional Brownian motion and in [5] we generalized one of them for a class of Markov processes. This article is its continuation and we will prove a gener- alization of the other theorem. 1. Main theorems Let B(t)=(B,(t), B 2 (1)) be a two-dimensional standard Brownian motion starting at (0, 0) and f(x), x e R 2 be a bounded measurable function vanishing outside a compact set. Define t(t)= lim (40 - ' 1 I, __ (B,(s))ds 0 t,n i ti B 1 (u)=i1 Then [6] proved the following two theorems: Theorem A. e22, (1 //1) f(B(s))ds ,) as co )o where . 1= (1/n) f(y)d y. Theorem B. If, in addition,.f= 0, then (1 0 - ) f(B(s))ds Ï C B 2 (t(a,)) as where C 2 = —(2/n2) 55 log Ix— yif(x)f(y)dxdy.