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Statement

Setting (p. 1504). $X_n$ is a symmetric transient random walk in $\mathbb Z^d$, $d\ge3$, started at the origin and not supported on a proper subgroup; $\mu_\infty^X(A)=\sum_{j\ge0}\mathbf 1_A(X_j)$ counts time zero; $G$ is the Green function and $\gamma_d$ the probability of no return to the origin, so $G(0)=1/\gamma_d$. For $y\in\mathbb Z^d$, $t_y=\mathbf P(T_y<\infty)$ with $T_y=\inf\{s>0:X_s=y\}$.

Equation (4.1) (p. 1513). For $0\ne y\in\mathbb Z^d$,

$$\mathbf P\bigl(\mu_\infty^X(\{0,y\})>u\bigr) =\bigl(1-\gamma_d/(1+t_y)\bigr)^u,\qquad u=1,2,\ldots. \tag{4.1}$$

The range is printed as $u=1,2,\ldots$; Lemma 2.1, from which it is read off, holds for $u=0,1,\ldots$, and at $u=0$ both sides equal 1. No moment condition is used. The paper derives (4.1) in the course of proving (1.5) of Theorem 1.2.

Proof pointer

P. 1513. For $A=\{0,y\}$ the Green matrix $G_A$ has diagonal entries $G(0)$ and off-diagonal entries $G(y)$, so its eigenvalues are $G(0)\pm G(y)$ with eigenvectors proportional to $(1,1)$ and $(1,-1)$. From $G(y)=t_yG(0)$, $\Lambda_A=G(0)(1+t_y)=(1+t_y)/\gamma_d$ and $1-1/\Lambda_A=1-\gamma_d/(1+t_y)$. In the notation of Lemma 2.1 the weights are $h_1=1$ and $h_2=0$, so (2.1) reduces to the single geometric term (4.1).

Read depth

Claims checked: the statement and its derivation on p. 1513 were read clause by clause on the page image of the print and followed. Nothing here is independently reviewed.

Dependencies

Lemma 2.1.

Used in

Hao, Li, Okada and Zheng quote (4.1) for simple random walk, for $u\in\mathbb N$, as equation (3.5) in the proof of their Lemma 3.2, Hao–Li–Okada–Zheng, Lemma 3.2. From it and the bound $t_y\le1-\gamma_d$ they define, in their (3.7),

$$\delta=\inf_{y\in\mathbb Z^d\setminus\{0\}} \frac{-2\log\bigl(1-\gamma_d/(1+t_y)\bigr)}{-\log(1-\gamma_d)}-1>0,$$

the constant their Lemma 3.2 uses.

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

Problem 1165, indirectly. The problem concerns planar simple random walk, which this paper does not treat. Hao, Li, Okada and Zheng use (4.1) in their Lemma 3.2, an input to their favorite-count law for dimensions $d\ge3$, the transient companion of their planar result on the problem's question. The paper itself proves nothing about the problem.