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 $Ulbrack A brack$ 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 $lim_{a ightarrowinfty} Ulbrack S(mathbf{O}, a) brack/a = 1/ |Elbrackmathbf{X}_1 brack|$, where $mathbf{X}_1 = (X_{11}, X_{21})$ is the first step of the walk, if $Elbrackmathbf{X}_1 brack$ 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 $Elbrack X^2_{11} brack$ and $Elbrack X^2_{21} brack$ finite and one of $Elbrack X_{11} brack, Elbrack X_{21} brack$ different from 0. The theorem states that $lim_{a ightarrowinfty} {Ulbrack S(mathbf{0}, a + Delta,) brack - Ulbrack S(mathbf{0}, a) brack} = Delta/|Elbrackmathbf{X}_1 brack|$ for every $Delta geqq 0$ and $| cdot |$ specified above. This result is also established with no restrictions on $Elbrack X^2_{11} brack, Elbrack X^2_{21} brack$ under different regularity conditions, in particular, for the $L_infty$ norm if both $Elbrack X_{11} brack$ and $Elbrack X_{21} brack$ are different from 0, and correspondingly for the $L_1$ norm if $|Elbrack X_{11} brack| eq |Elbrack X_{21} brack|$. 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.