---

Statement

Setting (pp. 1504, 1506). $X_n$ is a symmetric transient random walk in $\mathbb Z^d$, $d\ge3$, started at the origin and not supported on a proper subgroup, with Green function $G$; $\mu_\infty^X(A)=\sum_{j\ge0} \mathbf 1_A(X_j)$ counts time zero; $(f,g)_A=\sum_{x\in A}f(x)g(x)$; and $G_A(x,y)=G(x-y)$ on $A$.

Lemma 2.1 (p. 1507). Let $\{X_n\}$ be a symmetric transient random walk in $\mathbb Z^d$ and let $A$ be a finite set in $\mathbb Z^d$ which contains the origin. Then

$$\mathbf P(\mu_\infty^X(A)>u)=\sum_jh_j \left(\frac{\lambda_j-1}{\lambda_j}\right)^u,\qquad u=0,1,\ldots, \tag{2.1}$$

where $\lambda_1>\lambda_2\ge\cdots\ge\lambda_{|A|}\ge\frac12$ are the eigenvalues of the symmetric matrix $G_A$, with orthonormal eigenvectors $\phi_j$, and $h_j=(1,\phi_j)_A\,\phi_j(0)$.

The paper presents (2.1) (p. 1506) as the random-walk counterpart of the Ciesielski–Taylor representation for the total occupation measure of Brownian motion. The hypotheses include no moment condition.

Proof pointer

Pp. 1507–1509. The paper computes the moments of $\mu_\infty^X(A)$ by grouping repeated indices, sums them into the generating function $\mathbb E(e^{\zeta\mu_\infty^X(A)})$ expressed through powers of $\tilde G_A=G_A-I$ (2.4), and expands in the eigenbasis of $G_A$ to get a sum of geometric generating functions (2.10); analyticity in $\zeta$ then identifies the point probabilities (2.14). The Fourier representation of $G$ gives every eigenvalue at least $\frac12$, and Perron–Frobenius makes the top eigenspace one-dimensional with a strictly positive eigenvector (p. 1508).

Read depth

Claims checked: the statement was read clause by clause on the page image of the print; the proof was read for its structure. Nothing here is independently reviewed.

Dependencies

External: spectral theory of symmetric matrices and the Perron–Frobenius theorem, as cited by the paper.

Source. E. Csáki, A. Földes, P. Révész, J. Rosen and Z. Shi, Frequently visited sets for random walks, Stochastic Process. Appl. 115 (2005), 1503–1517, doi:10.1016/j.spa.2005.04.003; the edition read is named on the source card.

Bears on

No Erdős problem directly. Its two-point case is equation (4.1).