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Statement

Setting (pp. 137, 153). The walk is the symmetric nearest-neighbor walk $S_d(0)=0,S_d(1),\ldots$ on $\mathbb Z^d$, and $\varrho_d(n)=\lvert S_d(n)\rvert$ is its Euclidean distance from the origin at time $n$. All logarithms are natural.

Theorem 8 (p. 154), the paper's starting point. For the walk in $d$-space, $\mathbf P\{\varrho_d(n)>c\sqrt{(2/d)\,n\log\log n}\text{ i.o.}\}$ is $0$ or $1$ according as $c>1$ or $c\le1$. The paper calls this a result that must be well known, though not found stated in the literature, provable by modifying the proof for $d=1$, and gives no proof.

Theorem 9 (p. 154). Let $d\ge3$, $c<1$, and $\tau_d(n)=\inf_{m\ge n}\varrho_d(m)$. Then

$$\mathbf P\Bigl\{\tau_d(n)>c\sqrt{\tfrac2d\,n\log\log n}\text{ i.o.}\Bigr\}=1.$$

The print writes the infimum as $\inf_{m\ge n}\varrho_d(n)$ [sic]; the infimum over $m\ge n$ of $\varrho_d(m)$ is the intended quantity, the smallest distance the walk attains from time $n$ on.

Remark after the theorem (p. 154). Since $\tau_d(n)\le\varrho_d(n)$, Theorem 8 gives probability $0$ for every $c>1$. The paper asserts without proof that the case $c=1$ also has probability one, in the sharper form that for any $a<1$,

$$\mathbf P\Bigl\{\tau_d(n)>\Bigl[\tfrac nd\bigl(2\log\log n+a\,l_3(n)\bigr)\Bigr]^{1/2}\text{ i.o.}\Bigr\}=1,$$

where $l_3(n)=\log\log\log n$; the subscript of $l$ is faint in the scan and reads as $3$. It adds (p. 155) that it has no necessary and sufficient conditions for upper bounds on $\tau_d(n)$ matching the Erdős and Feller tests for the line.

Source. P. Erdős and S. J. Taylor, Some problems concerning the structure of random walk paths, Acta Math. Acad. Sci. Hungar. 11 (1960), 137--162: the walk on p. 137, $\varrho_d$ on p. 153, Theorem 8, Lemmas 1 and 2, Theorem 9 and the remark on p. 154, the proof on pp. 155--156. The edition read is identified on the source card.

Read depth. Claims checked: Theorems 8 and 9, Lemmas 1 and 2 and the remark were read clause by clause on the printed pages. The proof on pp. 155--156 was read for the pointer below and not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pages 155--156. Fix $c<c_{13}<c_{14}<1$. By looking at one coordinate axis, the walk at time $n$ is beyond $c_{14}\sqrt{(2/d)n\log\log n}$ with probability at least a negative power of $\log n$ ((5.3)); from there, Lemma 2 makes the chance of ever entering the ball of radius $c_{13}\sqrt{(2/d)n\log\log n}$ at most about $(c_{13}/c_{14})^{d-2}$, giving (5.5). Independence comes from the increments over the disjoint windows $[n_k,n_{k+2}]$ along $n_k=[e^{k^{1+\delta}}]$, so Borel--Cantelli gives infinitely many good windows ((5.9)). Theorem 8 controls the starting point $S_d(n_k)$ ((5.10)), and Lemma 1, the Dvoretzky--Erdős lower bound $\varrho_d(n)\ge n^{1/2}(\log n)^{-2}$ eventually, controls the walk after $n_{k+2}$ ((5.11)).

Dependencies

Theorem 8 (stated without proof). Lemma 1 (p. 154): for $d\ge3$, $\mathbf P\{\varrho_d(n)<n^{1/2}(\log n)^{-2}\text{ i.o.}\}=0$, a special case of the rate-of-escape theorem of A. Dvoretzky and P. Erdős, Some problems on random walk in space, Proc. Second Berkeley Symp., 353--367, dated 1950 in the paper's references. Lemma 2 (p. 154): for $d\ge3$ and $0<\lambda<1$, a walk started at distance $R$ from the origin enters the sphere of centre the origin and radius $\lambda R$ with probability $\lambda^{d-2}(1+o(1))$ as $R\to\infty$; the paper takes this from Dvoretzky's Brownian-motion result through the walk--Brownian motion connection.

Bears on

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