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Statement

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

Theorem 10 (p. 157). There are constants $\lambda_d$ such that, with probability 1,

$$\text{(i)}\quad \frac1{\log N}\sum_{n=1}^N\frac{n^{-1/2}}{1+\varrho_1(n)}\longrightarrow\lambda_1,$$ $$\text{(ii)}\quad \frac1{(\log N)^2}\sum_{n=1}^N\frac1{1+\{\varrho_2(n)\}^2}\longrightarrow\lambda_2,$$ $$\text{(iii)}\quad \frac1{\log N}\sum_{n=1}^N\frac1{1+\{\varrho_d(n)\}^2}\longrightarrow\lambda_d \qquad(d=3,4,\ldots).$$

The exponent $-1/2$ in (i) is as printed. The exponent in the denominator of (iii) is faint in the scan and reads as $2$; with that exponent the normalization $\log N$ matches the order $1/n$ of the summands' means in dimension three and above.

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 10 and the start of its proof on p. 157, the end of the proof on p. 158. The edition read is identified on the source card.

Read depth. Claims checked: the three displays were read on the printed page. The planar proof on pp. 157--158 was read for the pointer below and not checked step by step. The argument that (i) fails as printed is this page's own and is not independently reviewed.

Proof pointer

Pages 157--158. The paper proves only the planar case (ii), saying the three cases are very similar and the method is that of its Theorem 5. Local estimates for $\mathbf P\{S_2(n)=P\}$, for $\lvert P\rvert<n^{1/2}/\log n$ ((5.12)) and for $\lvert P\rvert>n^{1/2}\log n$ ((5.13)), give $\mathbb E[1/(1+\varrho_2(n)^2)]=(2\log n/n)(1+o(1))$ ((5.14), p. 157). Summing, the normalized sum in (ii) has mean $1+o(1)$ (p. 158), so the paper's computation gives $\lambda_2=1$; its variance is $O(1/\log N)$. As for Theorem 5, the limit is first taken along a sparse sequence $r_k$ of exponential growth, whose exponent is faint in the scan (p. 158), and then extended to all $N$. Cases (i) and (iii) are left to the same method.

Case (i) as printed

Display (i) cannot hold with a finite $\lambda_1$. Since $\mathbb E[1/(1+\varrho_1(n))]\sim(\log n)/\sqrt{2\pi n}$, the expected value of its sum grows like $(\log N)^2/(2\sqrt{2\pi})$, not like $\log N$. The normalized sum also tends to infinity almost surely. For $n\ge M^2$ the $n$th summand is at least $\min(M,\sqrt n/\varrho_1(n))/(2n)$, so the almost-sure central limit theorem bounds the lower limit of the normalized sum below by $\mathbb E\min(M,1/\lvert Z\rvert)/2$ for a standard normal $Z$. This bound is unbounded in $M$, because $\mathbb E\lvert Z\rvert^{-1}=\infty$. The display is recorded as printed; the intended statement is not determined here.

Dependencies

The method of the paper's Theorem 5 (p. 149) and the local estimates for the planar walk; the remark on case (i) uses the local central limit theorem and the almost-sure central limit theorem for the simple walk on $\mathbb Z$.

Bears on

No problem page of this corpus.