Stochastic Processes and their Applications 25 (1987) 185-202 North-Holland 185 STABILITY AND INSTABILITY OF LOCAL TIME OF RANDOM WALK IN RANDOM ENVIRONMENT Mikl6s CSORGt~* Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada KIS 5B6 Lajos HORVATH** Bolyai Institute, Szeged University, Aradi V~rtamik Tere 1, 1-1-6720 Szeged, Hungary Pill RI~VI~SZ*** Institute for Statistics and Probability, Technical University Vienna, Wiedner Haupstrasse 8-10, A-1040 Wien, Austria, and Mathematical Institute, H-1053 Budapest, Redltanoda u. 13-15, Hungary Received 29 October 1986 Revised 24 March 1987 In this paper we study ratios of local times of a random walk in random environment. Strong and weak limit theorems are obtained. AMS 1980 Subject Classification: Primary 60J15, Secondary 60J55. random walk * random environment * local time * Wiener process * laws of iterated logarithm 1. Introduction Let {Xi}i~-~o be a sequence of independent identically distributed random vari- ables (i.i.d.r.v.'s) with t O, ifx~0, P{Xo<~X}= F(x), if0<x<l, I1, ifx~>l. We assume that P{Xo = 0} = P{Xo = 1} = 0. The sequence X = {Xi} i~-oo will be called a random environment. For any fixed realization of this random environment we can define a random walk So, $1,.,. by So = 0 and Px{S.+l=i+llS.=i}=Xi, Px{Sn+l=i-llSn=i}=l-X~, n =0, 1, 2,... and i=0, +1, +2, .... * Research partially supported by a NSERC Canada grant at Carleton University. ** Research done while at Carleton University, also supported by NSERC Canada grants of M. Cs6rg6 and D.A. Dawson and by an EMR Canada grant of M. Cs6rg6. *** Research done while at Carleton University, supported by a NSERC Canada grant of M. Cs6rg6. 0304-4149/87/$3.50 O 1987, Elsevier Science Publishers B.V. (North-Holland)