Renewal Theory in the Plane

Abstract

This paper presents some generalizations of the elementary renewal theorem (Feller [9]) and the deeper renewal theorem of Blackwell [1], [2] to planar walks. Let $U\lbrack A\rbrack$ denote the expected number of visits of a transient 2-dimensional nonarithmetic random walk to a Borel set $A$ in $R^2$. Let $S(\mathbf{y}, a)$ denote the sphere of radius $a$ about the point $\mathbf{y}$ for a given norm $\| \cdot \|$ of the Euclidean topology. Then, the elementary renewal theorem for the plane, given in Section 2, states that $\lim_{a\rightarrow\infty} U\lbrack S(\mathbf{O}, a)\rbrack/a = 1/ \|E\lbrack\mathbf{X}_1\rbrack\|$, where $\mathbf{X}_1 = (X_{11}, X_{21})$ is the first step of the walk, if $E\lbrack\mathbf{X}_1\rbrack$ exists. Farrell has obtained similar results for nonnegative walks in [7]. Section 4 contains the main result of the paper, the generalization of the Blackwell renewal theorem in the case of polygonal norms for random walks which have both $E\lbrack X^2_{11}\rbrack$ and $E\lbrack X^2_{21}\rbrack$ finite and one of $E\lbrack X_{11}\rbrack, E\lbrack X_{21}\rbrack$ different from 0. The theorem states that $\lim_{a\rightarrow\infty} \{U\lbrack S(\mathbf{0}, a + \Delta,)\rbrack - U\lbrack S(\mathbf{0}, a)\rbrack\} = \Delta/\|E\lbrack\mathbf{X}_1\rbrack\|$ for every $\Delta \geqq 0$ and $\| \cdot \|$ specified above. This result is also established with no restrictions on $E\lbrack X^2_{11}\rbrack, E\lbrack X^2_{21}\rbrack$ under different regularity conditions, in particular, for the $L_\infty$ norm if both $E\lbrack X_{11}\rbrack$ and $E\lbrack X_{21}\rbrack$ are different from 0, and correspondingly for the $L_1$ norm if $\|E\lbrack X_{11}\rbrack| \neq \|E\lbrack X_{21}\rbrack|$. Farrell in [8] has obtained more general results for nonnegative walks under somewhat more restrictive regularity conditions and by a different method. The next section gives the Blackwell theorem for totally symmetric transient walks with finite step expectations, both of whose marginal walks are recurrent. We conclude with a discussion of extensions of these results to higher dimensions and some open questions.