Consider two independent copies of IID random walk on ${\bf Z}$ starting from $0$, and let $N\_1(t)$ (resp. $N\_2(t)$) denote the number of times, up to time $t$, that the first (resp. second) walker has returned to $0$. We know that $N\_1(t),N\_2(t) \rightarrow \infty$ almost surely and that $N\_1(t)/t,N\_2(t)/t \rightarrow 0$ almost surely. Is it the case that $N\_1(t)/N\_2(t) \rightarrow 1$ almost surely? What if we replace random excursions from $0$ in ${\bf Z}$ by random excursions from $(0,0)$ in ${\bf Z}^2$?



Counting returns in null-recurrent random walk
You can easily derive everything from the first principles by the following back of envelope computation:


If we have a random walk starting *anywhere*, then after $t$ steps the expected number $EN(t)$ of returns is at most $\sum\_{m=1}^t \frac 1{\sqrt m}\approx \sqrt t$ and $E[N(t)^2]\le\sum\_{1\le m< M\le t} \frac 1{\sqrt m}\frac 1{\sqrt{M-m}}+\sum\_{1\le m\le t}\frac 1{\sqrt m}\approx t$ with approximate equality attained if we start at $0$. Note that it implies that $P(N(t)>c\sqrt t)>c$ for some fixed $t>0$ if we start at $0$. On the other hand, if we start above $C\sqrt t$ with large $C>0$, then the probability to ever reach $0$ is as small as we want, so the trivial Cauchy-Schwarz yields $EN(t)\le \sqrt{P(\exists \tau X(\tau)=0)E[N(t)^2]}<\frac{c^2}4\sqrt t$ if $C$ is large enough so $P(N(t)>\frac c2\sqrt t)\le c/2$.


Now just take $T>0$ and let $t$ be the first time after $T$ such that $X\_1(t)=0$. Then the probability that $|X\_2(t)|>C\sqrt t$ is some constant $p\_0$, say. From there and until the time $2t$ with probability $\ge c-\frac c2=\frac c2$, $X\_1$ acquires $>c\sqrt t$ returns but $X\_2$ less than $\frac c2\sqrt t$ returns. So, with fixed probability ($p\_0c/2$) they have noticeably different increments on the scale $\sqrt t$. Also, the probability that $N\_2(t)$ is much larger than $\sqrt t$ is small (we condition only on $X\_1$, so $X\_2$ still has the standard distribution). Thus, for every $M$, with constant probability, there exists $t>M$ such that $|N\_1(t)/N\_2(t)-1|+|N\_1(2t)/N\_2(2t)-1|>\delta>0$ for some fixed absolute $\delta$. That definitely rules out convergence to $1$ even in distribution, forget about a.s.


I tried to avoid any high-tech here and just answered the question as posed. I leave more advanced considerations to someone else.


In $\mathbb Z^2$, the situation is the same only with $\log t$ instead of the $\sqrt t$ everywhere.


I hope I haven't written any nonsense (lately I'm more prone to it than I should be).


