Renewal Theory in the Plane

Bickel, Peter J.; Yahav, Joseph A.

Ann. Math. Statist., Tome 36 (1965) no. 6, p. 946-955 / Harvested from Project Euclid

Résumé

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.

Publié le : 1965-06-14
Classification:

@article{1177700067,
     author = {Bickel, Peter J. and Yahav, Joseph A.},
     title = {Renewal Theory in the Plane},
     journal = {Ann. Math. Statist.},
     volume = {36},
     number = {6},
     year = {1965},
     pages = { 946-955},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1177700067}
}

Bickel, Peter J.; Yahav, Joseph A. Renewal Theory in the Plane. Ann. Math. Statist., Tome 36 (1965) no. 6, pp.  946-955. http://gdmltest.u-ga.fr/item/1177700067/