---

Sources. Erdős and Taylor (1960), Theorem 12, printed p. 160 (canonical PDF); Csáki, Földes and Révész, Heavy points of a d-dimensional simple random walk, arXiv:math/0504242v1 (12 April 2005), Theorem D and equation (1.13), p. 3.

Statement with the later source's normalization. For symmetric nearest-neighbor simple random walk $(S_j)$ on $\mathbb Z^d$, $d\ge3$, let

$$\gamma_d=\mathbb P(S_j\ne0\text{ for every }j\ge1),\qquad Q_d(t,n)=\#\{x:\#\{1\le j\le n:S_j=x\}=t\}.$$

Then, for every fixed positive integer $t$, almost surely,

$$\frac{Q_d(t,n)}n\longrightarrow \gamma_d^2(1-\gamma_d)^{t-1}.$$

Here $0<\gamma_d<1$. The 1960 scan instead prints $\gamma_d(1-\gamma_d)^{t-1}$ on the right, with the same meaning of $Q_d(t,n)$ and $\gamma_d$. The later primary paper explicitly gives the squared factor and attributes this theorem to Erdős and Taylor. This is a documented discrepancy between the sources; no author-issued erratum is being claimed.

Why the printed normalization cannot hold. This elementary obstruction is a complete deduction, separate from a proof of the corrected theorem. For every walk and every positive integer $M$,

$$\sum_{t=1}^{M}tQ_d(t,n)\le n,$$

because the left side counts only visits to sites with multiplicity at most $M$. If the original displayed limits held, their finitely many probability-one events for $1\le t\le M$ would imply

$$\sum_{t=1}^{M}t\gamma_d(1-\gamma_d)^{t-1}\le1.$$

But the infinite sum of these nonnegative terms equals $1/\gamma_d>1$: for $0<r<1$, summing $t=\sum_{j=1}^t1$ in the geometric series gives $\sum_{t\ge1}tr^{t-1}=(1-r)^{-2}$. Some finite partial sum therefore already exceeds one, a contradiction. Including the time-zero visit would replace the bound by $n+1$ and give exactly the same contradiction after division by $n$. Thus an initial-time convention cannot repair the missing factor. $\square$

Proof scope. The corrected strong law is recorded with a precise external source, not a complete rewritten proof. The original paper refers to a simplified version of its preceding multiplicity argument; the later paper states this fixed-$t$ theorem before proving a stronger uniform result. Neither full argument is reconstructed here. The corrected factor passes the necessary counting check above, but that check alone does not prove the strong law.

Depends on. The corrected statement uses transience and the fixed-multiplicity strong law cited above. The normalization obstruction uses only the deterministic visit count, finite intersections of probability-one events, and a geometric series.

Bears on. No problem page of this corpus. This theorem concerns $d\ge3$ and is not an input to the planar results recorded for Problems 1165 and 1166.