LAWS OF THE ITERATED LOGARITHM FOR THE LOCAL-TIMES OF RECURRENT RANDOM-WALKS ON Z(2) AND OF LEVY PROCESSES AND RANDOM-WALKS IN THE DOMAIN OF ATTRACTION OF CAUCHY RANDOM-VARIABLES

Laws of the iterated logarithm are obtained for the number of visits of a recurrent symmetric random walk on Z2 to a point in it's state space and for the difference of the number of visits to two points in it's state space. Following convention the number of visits is called the local time of the random walk. Laws of the iterated logarithm are also obtained for the local time of a symmetric Levy process, at a fixed point in it's space, as time goes to infinity and for the difference of the local times at two points in it's state space for Levy processes which at a fixed time are in the domain of attraction of a Cauchy random variable. Similar results are obtained for the local times of symmetric recurrent random walks on Z1 which are in the domain of attraction of Cauchy random variables. These results are related by the fact that the truncated Green's functions of all these processes are slowly varying at infinity and follow from one basic theorem.