Collision Problems of Random Walks in Two-Dimensional Time*

Let (Xij: i > 0, i > 0} be a double sequence of independently, identically distributed random variables (i.i.d.) which takes values in the d-dimensional integer lattice Ed . The double sequence {Sm,,: m > 0, n > 0} defined by Sllan= xy=, zyW, Xgj is called the random walk in two-dimensional time generated by 41, or a two-parameter random walk, or simply a random walk when there is no danger of confusion. In this paper we study two different but closely related problems. The first one is the recurrence properties of the random walk when the distribution of X,, is symmetric and the second one is the collision problems of these random walks. To be more specific, one wants to know if the associated random walk would return to the origin infinitely often in certain time sets, and also whether two or more mutually independent random walks would meet infinitely often in certain time sets of interest. In this work after giving some notations and preliminary estimates in Section 1, we give, in Section 2, a necessary and sufficient conditron in terms of the characteristic function associated with a symmetric random walk so that it will return to the origin infinitely often when the time set is the positive integer lattice in the plane. In Section 3, we use the result of Section 2 in order to establish some criteria in terms of the characteristic functions associated with two or more mutually independent random walks with the same distribution so that they