Strong law of large numbers for a function of the local times of a transient random walk in $\mathbb Z^d$

TL;DR

This paper establishes a strong law of large numbers for sums involving functions of local times of a transient random walk in integer lattice spaces, unifying and extending previous results on visited sites and self-intersections.

Contribution

It proves a general strong law of large numbers for functions of local times, covering various specific cases like visited sites, self-intersections, and sites visited exactly j times.

Findings

Proves a strong law of large numbers for sums of functions of local times.

Unifies previous results on visited sites and self-intersections.

Extends the understanding of local time functionals in transient random walks.

Abstract

For an arbitrary transient random walk $(S_n)_{n\ge 0}$ in $\mathbb Z^d$, $d\ge 1$, we prove a strong law of large numbers for the spatial sum $\sum_{x\in\mathbb Z^d}f(l(n,x))$ of a function $f$ of the local times $l(n,x)=\sum_{i=0}^n\mathbb I\{S_i=x\}$. Particular cases are the number of (a) visited sites (first time considered by Dvoretzky and Erd\H{o}s), which corresponds to a function $f(i)=\mathbb I\{i\ge 1\}$; (b) $\alpha$-fold self-intersections of the random walk (studied by Becker and K\"{o}nig), which corresponds to $f(i)=i^\alpha$; (c) sites visited by the random walk exactly $j$ times (considered by Erd\H{o}s and Taylor and by Pitt), where $f(i)=\mathbb I\{i=j\}$.

Equations

$$l(n,x):=\sum _{i=0}^n\mathbb{I}\{S_i=x\}.$$

$$L_n(\alpha ):=\sum _{x\in {\mathbb{Z}}^d:l(n,x)>0}{l}^{\alpha }(n,x),\alpha \ge 0.$$

$$\gamma :=\mathbb{P}\{S_n\ne 0\text{mbox}foralln\ge 1\},$$

$$G_n(f):=\sum _{x\in {\mathbb{Z}}^d}f(l(n,x)).$$

$$\sum _{j=1}^{\infty }{f}^2(j)j(1-\gamma {)}^j$$

$$\frac{G_n(f)}{n}$$

$$\frac{L_n(\alpha )}{n}$$

$$\sum _{k=n}^{\infty }\mathbb{P}\{S_k=0\}$$

$$\frac{1}{n}\sum _{x\in {\mathbb{Z}}^d}\mathbb{I}\{l(n,x)\in J\}$$

$${\lambda }_{\ast }:=\log \frac{1}{1-\gamma }.$$

$$\mathbb{E}|f(l(\infty ,0))|$$

$$\sum _{k=1}^{n}k\mathbb{P}\{S_k=0\}$$

$$C(2m{)}^{1-\eta }\ge \sum _{k=m}^{2m-1}k\mathbb{P}\{S_k=0\}$$

$$\sum _{k=n}^{\infty }\mathbb{P}\{S_k=0\}$$

$$\mathbb{E}|f(l(n,0))|$$

$$\mathbb{E}f(l(n,0))$$

$$\sum _{j=1}^{\infty }|f(j)|(1-\gamma {)}^j$$

$$\frac{\mathbb{E}{G}_n(f)}{n}$$

$${\gamma }_n$$

$$1-{\gamma }_n$$

$${\gamma }_n$$

$${\gamma }_n-\gamma =\mathbb{P}\{n+1\le \tau <\infty \}$$

$$Q_n(j):=\sum _{x\in {\mathbb{Z}}^d}\mathbb{I}\{l(n,x)=j\},$$

$$G_n(f)$$

$$\mathbb{E}{Q}_n(j)$$

$$\frac{\mathbb{E}{Q}_n(j)}{n}$$

Full text

(a) visited sites (first time considered by Dvoretzky and Erdős in DE), which corresponds to a function $f(i)={\mathbb{I}}\{i\geq 1\}$;

(b) $\alpha$-fold self-intersections of the random walk (studied by Becker and König in BK), which corresponds to $f(i)=i^{\alpha}$;

(c) sites visited by the random walk exactly $j$ times (considered by Erdős and Taylor in ET and by Pitt Pitt), where $f(i)={\mathbb{I}}\{i=j\}$.

1 Introduction and main results

Let $X_{1}$, $X_{2}$, …be a sequence of independent identically distributed random vectors valued in ${\mathbb{Z}}^{d}$, $d\geq 1$. Consider a random walk generated by $X_{n}$’s, $S_{0}:=0$, $S_{n}:=X_{1}+\ldots+X_{n}$, and the number of visits to a site $x\in{\mathbb{Z}^{d}}$ up to time $n$ which is called the local time of $x$,

[TABLE]

Define random variables

[TABLE]

In particular, the $L_{n}(0)=|\{S_{0},...,S_{n}\}|$ represents the number of distinct sites visited by the random walk up to time $n$, called the range of $(S_{n})_{n\geq 1}$. The case $\alpha=1$ is trivial because $L_{n}(1)=n+1$. The value of $L_{n}(2)$ is the number of so called self-intersections of a random walk. For an integer $\alpha$ the value of $L_{n}(\alpha)$ is the number of $\alpha$-fold self-intersections up to time $n$.

It is known that for a recurrent random walk the quotient $L_{n}(0)/n$ tends to [math] as $n\to\infty$ (see, e.g. Spitzer (S, Ch. 1, Sect. 4, Theorem 1)), which assumes a slower growing normalising sequence for a proper limit in the law of large numbers. As shown in Dvoretzky and Erdős (DE, Theorem 3) for a simple random walk and in Černý Cerny for a general one with zero drift and finite covariance matrix, it is $n/\log n$ in 2 dimensions.

In present article we show that the law of large numbers for $L_{n}(\alpha)$ with a non-zero limit and normalising sequence $n$ holds in any dimension $d$ for any transient random walk, that is, when the probability of its return to the origin,

[TABLE]

is strictly positive, $\gamma>0$. We assume in addition that $\gamma<1$ which excludes a trivial case where either $l(n,x)=1$ or [math] for all $x$ with probability $1$, and hence $L_{n}(\alpha)=n+1$.

The result for $L_{n}(\alpha)$ we are interested in follows from the following more general result. Consider a function $f:{\mathbb{Z}}^{+}\to{\mathbb{R}}$ and a spatial sum

[TABLE]

In particular, for a power function $f(i)=i^{\alpha}$, we get $L_{n}(\alpha)=G_{n}(f)$.

Theorem 1

Let the random walk $(S_{n})_{n\geq 0}$ be transient and $f:{\mathbb{Z}}^{+}\to{\mathbb{R}}$ be a function satisfying

[TABLE]

Then

[TABLE]

in mean square and with probability $1$.

The proof of Theorem 1 is given in Section LABEL:sec:proof1. In Sections 2 and 3 we discuss an asymptotic behaviour of the expectation and variance of $G_{n}(f)$ as $n\to\infty$ respectively, needed further in the proofs.

The following corollaries are immediate.

Corollary 2

For any $\alpha\geq 0$, it holds that

[TABLE]

in mean square and with probability $1$.

The case $\alpha=0$ was considered by Spitzer in (S, Theorem 1.4.1) where convergence in probability is proven. Before then a strong law of large numbers for $\alpha=0$ was proven for a simple random walk by Dvoretzky and Erdős in DE. In Becker and König BK the strong convergence (5) is proven for all $\alpha\geq 0$ (up to a gap in the proof of Proposition 2.1, see a comment on it in the proof of Lemma 5 following equation (13)) without any further conditions in the case $d\geq 3$, however in the cases $d\in{\{1,2\}}$ it is assumed there that either the steps $X_{i}$ are square integrable or, for some $\eta>0$ and $C<\infty$,

[TABLE]

Corollary 3

Let $J\subset{\mathbb{N}}$. Then, with probability $1$,

[TABLE]

If $J$ is a singleton $\{j\}$, then we get the strong law of large numbers for the number of sites visited exactly $j$ times up to time $n$. For these statistics, the last corollary generalises Theorem 12 in Erdős and Taylor ET from a simple random walk in $d\geq 3$ dimensions to an arbitrary transient random walk; a general result for transient random walks on a countable Abelian group was proven by induction on $j$ by Pitt in Pitt. Notice that, for an arbitrary $J$, say $J$ the set of all odd numbers, Corollary 3 can be reduced to the singleton case, once we know the strong law of large numbers for the range of $S_{n}$.

The growth condition (3) is satisfied for all subexponential functions $f(i)$ of order $e^{o(i)}$ as $i\to\infty$, and also for exponentially growing functions of order $O(e^{ci})$ with exponent coefficient $c<\lambda_{*}/2$ where

[TABLE]

It is very likely that the condition (3) may be relaxed to the condition (9) below because under the latter condition we have

[TABLE]

and since the number of visited sites up to time $n$ is not greater than $n$, it clearly indicates that the family $\{G_{n}(f)/n,n\geq 1\}$ is stochastically bounded. But if we only assume (9), then it requires a much more delicate analysis compared to the estimation of the variance carried out in Lemma 6, as it happens when we prove a strong law of large numbers for a random walk where existence of the second moment of jumps essentially simplifies proving technique. In the result below we show how it can be done under some additional technical assumptions.

Theorem 4

Let, for some $C<\infty$ and $\varepsilon>0$,

(i)* either the condition*

[TABLE]

hold for some $\eta\in(0,1)$ and $|f(i)|\leq Ce^{i\lambda_{*}}/i^{2+\varepsilon}$ for all $i$,

(ii)* or the condition*

[TABLE]

hold and $|f(i)|\leq Ce^{i\lambda_{*}}/i\log^{2+\varepsilon}i$ for all $i>0$.

Then the convergence (4) holds with probability $1$.

For the proof, see Section LABEL:sec:max. It is based on truncation technique and on a strong limit theorem for the maximal local time, $l(n):=\max\{l(n,x),\ x\in{\mathbb{Z}}^{d}\}$, see Proposition LABEL:l:ln.limit.thm there.

Notice that the condition (7) is equivalent to (6). Indeed, on the one hand, it follows from (6) that

[TABLE]

On the other hand, it follows from (7) that, for all $m$,

[TABLE]

hence

[TABLE]

Also notice that, in $d\geq 3$ dimensions, if a random walk is not concentrated in some $3$-dimensional subspace, then the condition (7) is valid because ${\mathbb{P}}\{S_{n}=0\}=O(1/n^{d/2})$, due to an upper bound for the concentration function of a sum of random vectors, see e.g. Corollary of Theorem 6.2 in Esseen Esseen. For the same reason, in $d\geq 4$ dimensions, the condition (8) is valid for any random walk not concentrated in some $3$-dimensional subspace.

If the function $f$ grows faster than assumed in Theorem 4, say if the condition (9) fails, then $G_{n}(f)$ would require stronger normalisation than just $n$, in order to have a proper limit as $n\to\infty$. The answer may be conjectured as follows: let $\tau=\inf\{n\geq 1:S_{n}=S_{0}\}$ be the first return time to the origin, then

[TABLE]

where $\widetilde{\tau}_{1}$, $\widetilde{\tau}_{2}$, … are independent copies of $\tau$ conditioned on $\{\tau<\infty\}$. For example, consider $f$ such that $f(k)\sim c_{1}/(1-\gamma)^{k}$, then

[TABLE]

As shown in (DK, Theorem 4), in the case where ${\mathbb{E}}X_{1}=0$, ${\mathbb{E}}\|X_{1}\|^{2}<\infty$ and $d\geq 3$, we have an asymptotic relation ${\mathbb{P}}\{\widetilde{\tau}=n\}\sim c_{3}/n^{d/2}$ as $n\to\infty$.

Hence, in the case $d\geq 5$, ${\mathbb{E}}\widetilde{\tau}_{1}<\infty$ and it follows from the renewal theorem that then ${\mathbb{E}}f(l(n,0))\sim c_{2}n/{\mathbb{E}}\widetilde{\tau}_{1}$, which together with asymptotic size of the range—which is of order $O(n)$—indicates that the right normalisation for $G_{n}(f)$ should be $n^{2}$.

In the cases $d=3$ and $d=4$, ${\mathbb{E}}\widetilde{\tau}_{1}=\infty$ and it follows from Erickson’s renewal theorem (Erickson, Theorem 5) that then ${\mathbb{E}}f(l(n,0))\sim c_{4}n^{1/2}$ and $c_{5}n/\log n$ respectively, which in turn indicates that the right normalisation for $G_{n}(f)$ should be $n^{3/2}$ and $n^{2}/\log n$ respectively.

2 Asymptotics for expectation of $G_{n}(f)$

In this section we discuss the asymptotic behaviour of ${\mathbb{E}}G_{n}(f)$ as $n\to\infty$. We prove the following result.

Lemma 5

Let $f:{\mathbb{Z}}^{+}\to{\mathbb{R}}$ be a function satisfying

[TABLE]

Then

[TABLE]

Proof

Following Dvoretzky and Erdős DE, we introduce

[TABLE]

the probability that the site visited by random walk in the $n$th step has not been visited before then; $\gamma_{0}=1$. As noticed in DE,

[TABLE]

so $\gamma_{n}$ equals the probability that the random walk does not return to the origin in $n$ steps:

[TABLE]

We observe the following monotone convergence

[TABLE]

Consider the following spatial sum

[TABLE]

which represents the number of sites visited exactly $j$ times up to time $n$, hence

[TABLE]

As Becker and König (BK, Eq. (2.2)) do, we use the following equality, for $j\geq 1$:

[TABLE]

due to the Markov property of the random walk. In BK, the asymptotic behaviour of ${\mathbb{E}}Q_{n}(j)$ as $n\to\infty$ is argued by considering the generating function of $\{{\mathbb{E}}Q_{n}(j),n\geq 1\}$ and then referring to the Tauberian theorem, (Feller, Theorem XIII.5). Notice that this approach requires the sequence $\{{\mathbb{E}}Q_{n}(j),n\geq 1\}$ to be ultimately increasing (see Sections 1.7.3 and 1.7.4 in BGT) which is not granted from the beginning and probably fails; at least such a discussion is missing in BK. Notice that this problem can be fixed by first looking at the sum of $Q_{j}(n)$ over $j\geq\widetilde{j}$ for some $\widetilde{j}$ (this is now monotonic in $n$) and then looking at the differences. See also Pitt Pitt for an alternative proof.

Below we suggest another argument which does not require the Tauberian theorem and is only based on the transience of the random walk. It follows from (13) that

[TABLE]

where $\tau_{1}$, $\tau_{2}$, … are independent copies of $\tau$, the first return time to the origin. Thus

[TABLE]

hence

[TABLE]

In view of the convergence (11), for any fixed $s\geq j-1$,

[TABLE]

and, moreover,

[TABLE]

Therefore, by the dominated convergence theorem, as $n\to\infty$,

[TABLE]

owing to independence of $\tau_{k}$’s. In addition,

[TABLE]

Then the condition (9) makes it possible to apply dominated convergence again and to conclude that

[TABLE]

which completes the proof of (10). Also notice that (15) implies an upper bound

[TABLE]

∎

3 Estimation of variance of $G_{n}(f)$

The proof of the strong law of large numbers for $L_{n}(\alpha)$ for a transient random walk given by Becker and König in BK is based on the following upper bound for the variance of $L_{n}(\alpha)$:

[TABLE]

where $C=C(\alpha)$ is a constant. Notice that the proof of this bound provided in BK starts with an analysis of some representation for the variance of $L_{n}(\alpha)$, which is only available for integer $\alpha$’s, implication of which is necessary for further arguments for the strong law of large numbers for $L_{n}(\alpha)$ in the case of a non-integer $\alpha$.

For this reason we suggest below a different bound which works not only for $L_{n}(\alpha)$ with a non-integer $\alpha$, but also for $G_{n}(f)$ with a function $f$ other than power. This bound provides a straightforward way for proving the strong law of large numbers for $G_{n}(f)$ with $f$ satisfying the growth condition (3).

Lemma 6

For any non-decreasing function $f$ with $f(0)=0$,

[TABLE]

for all $n$ where $\Delta f(i):=f(i)-f(i-1)\geq 0$.