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

(will be inserted by the editor)

Strong law of large numbers for a function

of the local times of a transient random walk in

_Z

I. M. Asymont · D. Korshunov (*)

the date of receipt and acceptance should be inserted later

Abstract For an arbitrary transient random walk(S_n)_n≥0inZd,d≥1, we prove a

strong law of large numbers for the spatial sum∑_x∈Zd f(l(n,x))of a functionf of the

local timesl(n,x) =∑ni=0I{Si=x}. Particular cases are the number of

(a) visited sites (first time considered by Dvoretzky and Erd˝os in [8]), which cor-responds to a function f(i) =_I{i≥1};

(b)α-fold self-intersections of the random walk (studied by Becker and K¨onig in

[1]), which corresponds to f(i) =iα_;

(c) sites visited by the random walk exactly jtimes (considered by Erd˝os and Taylor in [4] and by Pitt [13]), where f(i) =I{i= j}.

Keywords Transient random walk inZd·Local times·Strong law of large numbers Mathematics Subject Classification (2010) 60G50·60J55·60F15

1 Introduction and main results

LetX1,X2, . . . be a sequence of independent identically distributed random vectors

valued inZd,d≥1. Consider arandom walkgenerated byXn’s,S0:=0,Sn:=X1+

. . .+Xn, and the number of visits to a site x∈Zd up to timenwhich is called the local timeofx,

l(n,x):=

n

∑

I{Si=x}. (1)

I. M. Asymont

Financial University under the Government of the Russian Federation, Russia E-mail: [email protected]

D. Korshunov (*) Lancaster University, UK

Define random variables

Ln(α) :=

∑

lα_(n,x),

α≥0. (2)

In particular, theLn(0) =|{S0, ...,Sn}|represents the number of distinct sites visited by the random walk up to timen, called the rangeof (Sn)n≥1. The case α=1 is

trivial becauseLn(1) =n+1. The value ofLn(2)is the number of so called

self-intersectionsof a random walk. For an integerαthe value ofLn(α)is the number of

α-fold self-intersections up to timen.

It is known that for arecurrentrandom walk the quotientLn(0)/ntends to 0 as

n→∞(see, e.g. Spitzer [12, 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 Erd˝os [8, Theorem 3] for a simple random walk and in

ˇ

Cern´y [3] for a general one with zero drift and finite covariance matrix, it isn/logn

in 2 dimensions.

In present article we show that the law of large numbers forLn(α)with a non-zero limit and normalising sequencenholds in any dimensiondfor anytransientrandom walk, that is, when the probability of its return to the origin,

γ := P{Sn6=0 for alln≥1},

is strictly positive,γ>0. We assume in addition thatγ<1 which excludes a trivial

case where eitherl(n,x) =1 or 0 for allxwith probability 1, and henceLn(α) =n+1.

The result forLn(α)we are interested in follows from the following more general

result. Consider a function f:Z+→Rand a spatial sum Gn(f) :=

_∑

x∈Zd

f(l(n,x)).

In particular, for a power function f(i) =iα_{, we getL}

n(α) =Gn(f).

Theorem 1 Let the random walk(Sn)n≥0be transient and f :Z+→Rbe a function satisfying

∞

∑

f2(j)j(1−γ)j <∞. (3)

Then

Gn(f)

n →γ

2 ∞

∑

f(j)(1−γ)j−1 as n→∞ (4)

in mean square and with probability1.

The proof of Theorem 1 is given in Section 4. In Sections 2 and 3 we discuss an asymptotic behaviour of the expectation and variance ofGn(f)asn→∞respectively, needed further in the proofs.

Corollary 2 For anyα≥0, it holds that

Ln(α)

n →γ

2 ∞

∑

jα₍₁₋

γ)j−1 as n→∞ (5)

in mean square and with probability1.

The caseα=0 was considered by Spitzer in [12, Theorem 1.4.1] where

conver-gence in probability is proven. Before then a strong law of large numbers forα=0

was proven for a simple random walk by Dvoretzky and Erd˝os in [8]. In Becker and K¨onig [1] the strong convergence (5) is proven for allα≥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 cased≥3, however in the casesd∈ {1,2}

it is assumed there that either the stepsXi are square integrable or, for someη>0

andC<∞,

∞

∑

P{Sk=0} ≤

C

nη for alln. (6)

Corollary 3 Let J⊂N. Then, with probability1,

1

n

∑

x∈_Zd

I{l(n,x)∈J} →

_∑

j∈J

γ2(1−γ)j−1 as n→∞.

IfJis a singleton{j}, then we get the strong law of large numbers for the number of sites visited exactly j times up to timen. For these statistics, the last corollary generalises Theorem 12 in Erd˝os and Taylor [4] from a simple random walk ind≥

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 jby Pitt in [13]. Notice that, for an arbitraryJ, sayJthe 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 ofSn.

The growth condition (3) is satisfied for all subexponential functions f(i)of order

eo(i) as i→∞, and also for exponentially growing functions of order O(eci) with exponent coefficientc<λ∗/2 where

λ∗:=log

1 1−γ.

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

E|f(l(∞,0))|=

∞

∑

|f(k)|(1−γ)k−1γ < ∞,

Theorem 4 Let, for some C<∞andε>0,

(i)either the condition

n

∑

kP{Sk=0} ≤Cn1−η for all n (7)

hold for someη∈(0,1)and|f(i)| ≤Ceiλ∗/i2+εfor all i,

(ii)or the condition

n

∑

kP{Sk=0} ≤Clogn for all n (8)

hold and|f(i)| ≤Ceiλ∗_/ilog2+ε_{i for all i>0.} Then the convergence (4) holds with probability1.

For the proof, see Section 5. It is based on truncation technique and on a strong limit theorem for the maximal local time,l(n):=max{l(n,x), x∈_Zd_{}, see} Proposi-tion 8 there.

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

n

∑

kP{Sk=0}= n

∑

n

∑

P{Sk=0} ≤ C n

∑

j−η ≤ C

1−ηn

1−η_.

On the other hand, it follows from (7) that, for allm,

C(2m)1−η ≥ 2m−1

∑

kP{Sk=0} ≥m

2m−1

∑

P{Sk=0},

hence

∞

∑

P{Sk=0}=

∞

∑

n2j+1−1

∑

P{Sk=0} ≤ C21−η

∞

∑

(n2j)−η = 2C

2η₋₁n −η_.

Also notice that, ind≥3 dimensions, if a random walk is not concentrated in some 3-dimensional subspace, then the condition (7) is valid because_P{Sn=0}=O(1/nd/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 [6]. For the same reason, ind≥4 di-mensions, 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, thenGn(f)would require stronger normalisation than justn, in order to have a proper limit asn→∞. The answer may be conjectured as follows: letτ=inf{n≥1 :

Sn=S0}be the first return time to the origin, then

E|f(l(n,0))|=

n

∑

|f(k)|P{τe1+. . .+τek−1≤n}(1−γ) k−1

where_eτ1,τe2, . . . are independent copies ofτconditioned on{τ<∞}. For example, consider f such thatf(k)∼c1/(1−γ)k, then

Ef(l(n,0))∼c2

∞

∑

P{τe1+. . .+τek−1≤n} asn→∞.

As shown in [7, Theorem 4], in the case whereEX1=0,EkX1k2<∞andd≥3, we have an asymptotic relationP{eτ=n} ∼c3/n

d/2_asn→∞.

Hence, in the case d ≥5, Eeτ1<∞ and it follows from the renewal theorem that thenEf(l(n,0))∼c2n/Eτe1, which together with asymptotic size of the range— which is of orderO(n)—indicates that the right normalisation forGn(f)should be

n2.

In the casesd=3 andd=4,Eeτ1=∞and it follows from Erickson’s renewal theorem [5, Theorem 5] that thenEf(l(n,0))∼c4n1/2andc5n/logn respectively,

which in turn indicates that the right normalisation for Gn(f) should ben3/2 and

n2/lognrespectively.

2 Asymptotics for expectation ofGn(f)

In this section we discuss the asymptotic behaviour ofEGn(f)asn→∞. We prove the following result.

Lemma 5 Let f:Z+→Rbe a function satisfying

∞

∑

|f(j)|(1−γ)j<∞. (9)

Then

EGn(f)

n →γ

2 ∞

∑

f(j)(1−γ)j−1 as n→∞. (10)

Proof Following Dvoretzky and Erd˝os [8], we introduce

γn:=P{Sn6=Skfor all 0≤k≤n−1},

the probability that the site visited by random walk in thenth step has not been visited before then;γ0=1. As noticed in [8],

1−γn=P{Sn=Skfor some 0≤k≤n−1}

=_P{S_n−k=S0for some 0≤k≤n−1},

soγnequals the probability that the random walk does not return to the origin inn steps:

γn=P{Sk6=S0for all 1≤k≤n} = P{τ≥n+1}.

We observe the following monotone convergence

Consider the following spatial sum

Qn(j) :=

∑

I{l(n,x) = j},

which represents the number of sites visited exactly jtimes up to timen, hence

Gn(f) = n

∑

f(j)Qn(j). (12)

As Becker and K¨onig [1, Eq. (2.2)] do, we use the following equality, for j≥1:

EQn(j)

=

_∑

x∈Zd

P{l(n,x) =j}

=

_∑

x∈Zd

0≤k₁<...<k j≤n

P{Sk1=...=Skj=x,Sk6=xfor allk≤n,k6∈ {k1, . . . ,kj}}

=

_∑

0≤k1<...<kj≤n

γk1

j−1

∏

P{τ=ki+1−ki}

γn−kj, (13)

due to the Markov property of the random walk. In [1], the asymptotic behaviour of

EQn(j)asn→∞is argued by considering the generating function of{EQn(j),n≥ 1} and then referring to the Tauberian theorem, [9, Theorem XIII.5]. Notice that this approach requires the sequence{EQn(j),n≥1}to be ultimately increasing (see Sections 1.7.3 and 1.7.4 in [2]) which is not granted from the beginning and probably fails; at least such a discussion is missing in [1]. Notice that this problem can be fixed by first looking at the sum ofQj(n)over j≥ejfor someej(this is now monotonic in

n) and then looking at the differences. See also Pitt [13] for an alternative proof. Below we suggest another argument which does not require the Tauberian the-orem and is only based on the transience of the random walk. It follows from (13) that

EQn(j) =

∑

n₀,n j≥0

n₁,...,_{n j−1}≥1

n₀+...+n j=n

γn0γnj

j−1

∏

P{τ=ni}

=

_∑

n0,nj: 0≤n0+nj≤n−j+1

γn0γnjP{τ1+. . .+τj−1=n−(n0+nj)},

whereτ1,τ2, . . . are independent copies ofτ, the first return time to the origin. Thus

EQn(j) = n−j+1

∑

P{τ1+...+τj−1=n−s}

s

∑

γn0γs−n0,

hence

EQn(j)

n =

n

∑

P{τ1+...+τj−1=s}

1

n

n−s

∑

In view of the convergence (11), for any fixeds≥j−1, 1

n

n−s

∑

γn0γn−s−n0 →γ

2 _asn→

∞,

and, moreover,

1

n

n−s

∑

γn0γn−s−n0 ≤

n−s+1

n ≤ 1 for allnands≥1.

Therefore, by the dominated convergence theorem, asn→∞,

EQn(j)

n →γ 2

P{τ1+...+τj−1<∞}

=γ2

j−1

∏

P{τk<∞} = γ2(1−γ)j−1, owing to independence ofτk’s. In addition,

EQn(j)

n ≤P{τ1+...+τj−1<∞} = (1−γ)

j−1 _{for alln. (15)}

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

EGn(f)

n =

∞

∑

f(j)EQn(j)

n → γ

2 ∞

∑

f(j)(1−γ)j−1 asn→∞, which completes the proof of (10). Also notice that (15) implies an upper bound

EGn(f)≤n n

∑

f(j)(1−γ)j−1. (16)

u t

3 Estimation of variance ofGn(f)

The proof of the strong law of large numbers forLn(α)for a transient random walk

given by Becker and K¨onig in [1] is based on the following upper bound for the variance ofLn(α):

VarLn(α)≤Cn

_∑

x∈Zd

n

∑

P{Si=x}P{Sj=−x} for alln,

whereC=C(α)is a constant. Notice that the proof of this bound provided in [1]

starts with an analysis of some representation for the variance ofLn(α), which is only

available for integerα’s, implication of which is necessary for further arguments for

the strong law of large numbers forLn(α)in the case of a non-integerα.

For this reason we suggest below a different bound which works not only for

Lemma 6 For any non-decreasing function f with f(0) =0,

VarGn(f)≤EGn(f2) +4 n

∑

f(i)∆f(i)(1−γ)i−1

n

∑

r(n−r)P{Sr=0}

for all n where∆f(i):=f(i)−f(i−1)≥0. Proof In view of the representation (12),

Gn(f) =

∑

n

∑

f(i)_I{l(n,x) =i},

hence

VarGn(f)

=

_∑

x,y∈_Zd

n

∑

f(i)f(j)P{l(n,x) =i,l(n,y) =j} −P{l(n,x) =i}P{l(n,y) = j}

=

_∑

x∈Zd

n

∑

f2(i)_P{l(n,x) =i}+

_∑

x6=y n

∑

f(i)f(j)_P{l(n,x) =i,l(n,y) =j}

−

_∑

x,y∈_Zd

n

∑

f(i)f(j)P{l(n,x) =i}P{l(n,y) =j},

becauseP{l(n,x) =i,l(n,y) =j}=0 ifx=yandi6=j. Thus, due to f ≥0,

VarGn(f) ≤ EGn(f2) + n

∑

f(i)f(j)

_∑

x6=y

P{l(n,x) =i,l(n,y) =j} −

_∑

x,y

P{l(n,x) =i}P{l(n,y−x) =j}

=:EGn(f2) +Σn1−Σn2, (17) and it only remains to estimate the difference of sumsΣ_n1−Σ_n2on the right hand side.

Since f(0) =0, n

∑

f(i)f(j)_P{l(n,x) =i,l(n,y) =j}

=

n

∑

P{l(n,x) =i,l(n,y) =j}

i

∑

j

∑

∆f(i1)∆f(j1)

=

n

∑

∆f(i1)∆f(j1)

n

∑

n

∑

P{l(n,x) =i,l(n,y) =j}

=

n

∑

and similar equalities hold for ordinary sums. Therefore,

Σ_n1−Σ_n2=

n

∑

∆f(i)∆f(j)

∑

P{l(n,x)≥i,l(n,y)≥j}

−

_∑

x,y

P{l(n,x)≥i}P{l(n,y)≥j}

,

where∆f(i)≥0 for allibecause f is non-decreasing, and the tail probabilities do not decrease asngrows, which makes it possible to perform a required analysis of the double sum. Let us decompose the eventB=B(x,y,i,j):={l(n,x)≥i,l(n,y)≥j}

forx6=yas a union of four disjoint eventsB∩Bxy,B∩Byx,B∩BxyxandB∩Byxy, where

Bxyx:={Sn1 =x,Sn2=y,Sn3=xfor somen1<n2<n3≤n}, Bxy :={all visits toxoccur before all visits toy}.

Denote byτx(i)the time ofith visit toxby the random walk(Sn)n≥0. Then the event B∩Bxyimplies the event:

Bxy(i,j):={no visits toybeforeτx(i)

and not less than jvisits toyafterτx(i)}. Altogether these imply the following upper bound

P{B(x,y,i,j)} ≤P{B∩Bxyx}+P{B∩Byxy}

+_P{B∩Bxy(i,j)}+P{B∩Byx(j,i)}. (18) Let us estimate every probability on the right hand side here. Sinceτx(i)is a Markov time,

P{B∩Bxy(i,j)}

=

_∑

k≤n

P{τx(i) =k,not less than jvisits toywithin time interval[k+1,n]}

=

_∑

k≤n

P{τx(i) =k}P{l(n−k,y−x)≥j} ≤

_∑

k≤n

P{τx(i) =k}P{l(n,y−x)≥j},

because the event{l(n,y−x)≥j}can only increase asngrows. Therefore,

P{B∩Bxy(i,j)} ≤P{τx(i)≤n}P{l(n,y−x)≥j} =_P{l(n,x)≥i}P{l(n,y−x)≥j}.

Then summation over allx6=yimplies that

∑

x6=y

P{B∩Bxy(i,j)}+P{B∩Byx(j,i)}

≤

_∑

x6=y

P{l(n,x)≥i}P{l(n,y−x)≥j}+P{l(n,y)≥j}P{l(n,x−y)≥i}

≤

_∑

x,y

Together with non-negativity of increments of the functionf it implies that n

∑

∆f(i)∆f(j)

∑

P{B∩Bxy(i,j)}+P{B∩Byx(j,i)}

−

_∑

x,y

P{l(n,x)≥i}P{l(n,y)≥j}

≤ 0. (19)

Further, the eventBxyxmay be described as follows: firstly the sitexis visited at least once, sayt≥1 times, then the siteyis visited one or more times, says≥1 times, and then again the sitexis visited, which is followed by visits toxandyin an arbitrary order. Thus, fori≥j,

P{B∩Bxyx}

≤

_∑

t,s,k1<...<kt+s+1≤n

P{Sk1 =. . .=Skt =x,Skt+1 =. . .=Skt+s=y,

Skt+s+1 =x, there are no other visits toxandyup tokt+s+1, Sk=xat leasti−t−1 times pastkt+s+1}.

and similarly for j≥i P{B∩Bxyx}

≤

_∑

t,s,k1<...<kt+s+1≤n

P{Sk1 =. . .=Skt =x,Skt+1 =. . .=Skt+s=y,

Skt+s+1=x,there are no other visits toxandyup tokt+s+1, Sk=yat least j−stimes pastkt+s+1}.

Summing up for allxandywe arrive at the following upper bound

∑

P{B∩Bxyx}

≤Pi−1{τ<∞}

_∑

P{Sr2−Sr1 =z,Sr3−Sr2=−z} = (1−γ)i−1

_∑

z∈Zd,r1<r2<r3≤n

P{Sr2−Sr1=z,Sr3−Sr2=−z}

in the casei≥jand similarly with coefficient(1−γ)j−1in the case j≥i. Since

∑

z∈_Zd

P{Sr2−Sr1=z,Sr3−Sr2 =−z} = P{Sr3−Sr1=0},

we get, fori≥j,

∑

P{B∩Bxyx} ≤(1−γ)i−1

∑

(r3−r1)P{Sr3−Sr1=0}

= (1−γ)i−1

n

∑

which together with (17), (18) and (19) shows that the variance ofGn(f)does not exceed

EGn(f2) +4 n

∑

∆f(i)∆f(j)(1−γ)i−1

n

∑

r(n−r)P{Sr=0}.

The sum of∆f(j)from j=1 to iequals f(i), hence the desired upper bound for

VarGn(f). ut

4 Proof of Theorem 1

Without loss of generality we assumef(0) =0. Any function f:Z+→Rwithf(0) =

0 is decomposable into a difference of two non-decreasing functions, f = f1−f2,

where

f1(j) =

j

∑

(f(i)−f(i−1))+, f2(j) =

j

∑

(f(i)−f(i−1))−. (20)

Since

fk(j)≤ j

∑

|f(i)−f(i−1)| ≤ 2 j

∑

|f(i)|, k=1, 2,

we get the following upper bound

∞

∑

f_k2(j)(1−γ)j≤4

∞

∑

(1−γ)j

j

∑

|f(i)|

2

≤4

∞

∑

(1−γ)jj

j

∑

f2(i)

=4

∞

∑

f2(i)

∞

∑

(1−γ)jj ≤ 4

γ2

∞

∑

f2(i)(i+1)(1−γ)i.

Therefore, the condition (3) implies that

∞

∑

f_k2(j)(1−γ)j<∞, k=1, 2. (21)

Hence, without loss of generality we assume that f is a non-decreasing function sat-isfying (21) and f(0) =0.

The transience of the random walk(Sn)n≥0is equivalent to the convergence of

the series

∞

∑

The condition (21) allows us to apply the upper bound (16) to f2and to conclude that

EGn(f2)≤c1nfor somec1<∞. Since f is non-decreasing and f(0) =0,∆f(i)≤

f(i). Therefore, by Lemma 6,

VarGn(f)≤c1n+c2n

n

∑

f2(i)(1−γ)i−1

n

∑

rP{Sr=0}

≤c1n+c3n

n

∑

rP{Sr=0},

again by the condition (21). In view of (22),

an := 1

n

n

∑

rP{Sr=0} →0 asn→∞

and hence

Var Gn(f)

n ≤

c1

n +c3an→0 asn→∞, (23)

which is equivalent to the convergence(Gn(f)−EGn(f))/n→0 inL2. Together with

the convergence (10) this completes the proof ofL2-convergence stated in Theorem

1.

For the proof of the almost sure convergence, first let us notice that (22) yields

∞

∑

∑

∑

rP{Sr=0}

≤

∞

∑

P{Sr=0}r

∞

∑

1

n2 ≤ 2

∞

∑

P{Sr=0} < ∞.

Hence we can apply Lemma 7 proven below to the sequence{an}n≥1, so, for any

fixedδ >0, there is an increasing subsequence{nr}r≥1such that∑∞r=1anr<∞and

√

1+δnr−1≤nr+1≤(1+δ)nrfor allr.

Using Chebyshev’s inequality, the upper bound (23) and the convergence (10) we conclude that, for anyε>0,

∞

∑

Gnr(f)

EGnr(f)

−1 >ε ≤ ∞

∑

VarGnr(f)

ε2(EGnr(f))2

≤ C

ε2

∞

∑

(1/nr+anr)<∞.

Then it follows from the Borel–Cantelli lemma that

Gnr(f)

EGnr(f)

a.s.

→1 asr→∞. (24)

Further, for anynthere existsrsuch thatnr≤n≤nr+1and, hence Gnr(f)

EGnr+1(f)

≤ Gn(f) EGn(f)

≤ Gnr+1(f) EGnr(f)

It follows from (10) that

EGnr+1(f) EGnr(f)

∼ nr+1 nr

asr→∞.

Moreover,nr<nr+1≤(1+δ)nrfor allr. Then (24) and (25) imply that

1

1+δ ≤ lim infn→∞

Gn(f)

EGn(f)

≤lim sup n→∞

Gn(f)

EGn(f)

≤ 1+δ a.s.

Due to arbitrary choice ofδ >0, the a.s. convergenceGn(f)/EGn(f)→1 follows.

u t

In the last proof, we have made use of the following auxiliary result.

Lemma 7 Let vn≥0and ∑∞n=1

vn

n <∞. Then, for any fixed δ >0, there exists an

increasing subsequence{nr}r≥1such that∑∞r=1vnr <∞and

√

1+δnr−1≤nr+1≤ (1+δ)nrfor all r≥1.

Proof Let us fix an arbitraryb∈(1,2)and identify a K=K(b)such that [bK_]−

[bK−1_{]≥2. Forr≥1, choose}

nr∈[[bK+r−2] +1,[bK+r−1]]such thatvnr = min

[bK+r−2_]+1≤n≤[bK+r−1_]vn.

By this construction,

∞

∑

vn

n ≥vn1

[bK]

∑

1

n+vn2

[bK+1]

∑

1

n+. . .+vnr

[bK+r−1]

∑

1

n+. . .

Since

[bK+r−1]

∑

1

n →logb > 0 asr→∞,

the convergence of the series∑nv_nn guarantees convergence of the series∑rvnr. Also,

for allr,

nr+1 nr−1

≥ b

K+r−1

bK+r−2 = b and nr+1

nr

≤ b

K+r

bK+r−2 = b 2_,

5 Proof of Theorem 4

To prove Theorem 4, let us first consider the maximal local time,l(n):=max{l(n,x),x∈ Zd}. Theorem 13 in Erd˝os and Taylor [4] states a strong limit theorem forl(n): for a

simple random walk ind≥3 dimensions,

l(n)

logn →

1 log_1−γ1 =:

1

λ∗

asn→∞with probability 1. (26)

The proof in [4] is split into two parts, dealing with upper and lower bounds. There is some issue with the proof of the upper bound, that is,

lim sup n→∞

l(n)

logn ≤

1

λ∗

with probability 1. (27)

The proof suggested in [4] is based on the inequality for tails

P{l(n)>t} ≤nP{l(n,0)>t} for allt>0, (28)

and on the observation that the number of returns to the origin,l(n,0), is dominated by a geometrically distributed random variable with parameter 1−γ. Notice that

justification of (28) in [4] is not complete because it is based there on the assumption that all sites visited by the random walk—clearly not more thann—can be treated in the same way as the origin. This point requires further justification because the set of visited sites is random. The same issue occurs is the proof of Theorem 1 in Revesz [11]. Notice that this set is contained in the ball of radiusn, which leads to the coefficientndinstead ofnon the right hand side of (28) which in its turn leads to the constantd/λ∗on the right hand side of (27) instead of 1/λ∗.

The last issue may be resolved in different ways, particularly, we may condition on the non-zero value ofS1,

P{l(n)>t} ≤P{l(n,0)>t}+

_∑

x6=0

P{l(n−1,x)>t|S1=x}P{S1=x},

followed by an induction argument onn. Hence, the upper bound (28) holds for any transient random walk in any dimensions. Therefore,

P{l(n)>t} ≤nP{l(∞,0)>t} ≤ n(1−γ)t−1 = ne−(t−1)λ∗_,

which implies, for allε>0 andm∈ {0,1,2, . . .}, the following upper bound

P{C(n,ε,m)} ≤ 2

logn. . .log_(m−2)nlog1(m+−1ε )n

,

for the events

C(n,ε,m):=nl(2n)>1+ 1

λ∗

(logn+. . .+log_(m−1)n+ (1+ε)log(m)n) o

;

Therefore, the series ∑∞k=1P{C(2k,ε,m)} converges, hence by the Borel-Cantelli

lemma, only finitely many ofC(2k,ε,m) occur, with probability 1. For any n∈

[2k,2k+1)and the event

B(n,ε,m):=

n

l(n)>1+ 1

λ∗

(logn+. . .+log_(m−1)n+ (1+ε)log(m)n) o

,

we have inclusionB(n,ε,m)⊆C(2k,ε,m), and thus only finitely many ofB(2k,ε,m)

occur, with probability 1. In other words, we arrive at the following result.

Proposition 8 For allε>0and m∈ {0,1,2, . . .},

l(n)≤ 1

λ∗

logn+. . .+log_(m−1)n+ (1+ε)log(m)n

,

for all n≥N where N is finite with probability1.

Notice that, for a simple random walk inZd,d≥3, an upper a.s. boundλ∗−1(logn+ (1+ε)log logn) and—in the case ofd≥4—a lower a.s. bound λ∗−1(logn−(3+

ε)log logn)is derived by Revesz in [11] following a different technique; he has also proved thatl(n)≥λ∗−1(logn+ (1−2/(d−2)−ε)log logn)infinitely often a.s. The

maximal local time for a zero drift random walk onZwith finite variance—which is

clearly recurrent—was studied by Kesten in [10].

Let us proceed with the proof of Theorem 4, we start with the case (ii). In-troducing two non-decreasing functions, f1 and f2 as in (20), we notice that then

fk(i)≤Cee iλ∗/ilog2+εifork=1, 2, because n

∑

eiλ∗

ilog2+ε_i =O

enλ∗

nlog2+ε_n

asn→∞.

Hence, without loss of generality we assume that f is non-decreasing with f(0) =0. The function f satisfies the condition (9), but not (21), and this generates a cer-tain difficulty we need to overcome. To this end, let us introduce two sequences of truncated functions

fn(i):= f(i)Ii≤λ∗−1logn , f_n+(i):= f(i)_I

λ∗−1logn<i≤λ∗−1(logn+bn) ,

wherebn=log(2)n+2 log(3)nand make use of the following decomposition

Gn(f) =Gn(fn) +Gn(fn+) +Gn(f−fn−fn+). Since f satisfies the condition (9), we get equivalences

EGn(fn)∼EGn(f) ∼ nγ2

∞

∑

f(j)(1−γ)j−1 asn→∞, (29)

and, by (16), the following upper bound

EGn(fn2)≤c1 n

lognlog2+ε

(2) n

EGn(fn) ≤ c2 n2

Further, it follows from the condition (8) that

n

∑

r(n−r)P{Sr=0} ≤n n

∑

rP{Sr=0} ≤ c3nlogn.

In addition,

n

∑

fn(i)∆fn(i)(1−γ)i−1≤

λ∗−1logn

∑

f2(i)e−(i−1)λ∗

≤c4 eiλ∗

i2_log4_i

i=λ∗−1logn

≤c5 n

log2nlog4₍₂₎n.

Substituting the last three bounds into the right hand side of the inequality provided by Lemma 6, we derive that

VarGn(fn)≤c6n2/lognlog2(2)n. Choose a subsequencenr= [er/log

1/4_r

], then, by Chebyshev’s inequality, the last up-per bound and (29), we conclude that

∞

∑

Gnr(fnr)

EGnr(fnr)

−1 >

1

log1/8r

≤

∞

∑

log1/4rVarGnr(fnr)

(EGnr(fnr))2

≤c7

∞

∑

1

rlog3/2r < ∞,

which allows us to apply the Borel–Cantelli lemma, hence obtaining

Gnr(fnr)

EGnr(fnr)

a.s.

→1 asr→∞.

Similar to (25), ifnr≤n≤nr+1then Gnr(fnr)

EGnr+1(fnr+1)

≤ Gn(fn) EGn(fn)

≤ Gnr+1(fnr+1) EGnr(fnr)

.

In addition, by (29),

EGnr+1(fnr+1) EGnr(fnr)

∼ nr+1 nr

∼ e

r+1 log1/4(r+1)−

r

log1/4r _{→ 1 asr→∞.}

Therefore,

Gn(fn)

EGn(fn) a.s.

Further, it follows from (16) that, for allm≤n,

EGn(f

+ m)

n ≤

[λ∗−1(logm+bm)]

∑

f(j)(1−γ)j−1

≤c7

[λ∗−1(logm+bm)]

∑

1

jlog2+ε _j

=O

_b

m

logmlog2+ε

(2) m

= O

₁

lognlog1+ε

(2) n

,

ifm≥n/2. Applying Chebyshev’s inequality to non-negative random variablesGnk+1(f

+ nk)/nk+1

withnk=2k, we get the following series convergence

∞

∑

P

nG_nk+1(f_n+

k)

nk+1

≥ 1

logε/2_k

o

≤

∞

∑

c8/klog1+ε_k

1/logε/2_k < ∞,

which in its turn implies by the Borel–Cantelli lemma that

Gnk+1(f

+ nk)

nk+1

→0 ask→∞with probability 1.

In addition, forn_k≤n≤nk+1, Gn(fn+)

n ≤

Gnk+1(f

+ nk)

nk

= 2Gnk+1(f + nk)

nk+1

,

hence

Gn(fn+)

n →0 asn→∞with probability 1. (31)

Finally, sincel(n)is the largest local time,

{Gn(f−fn−fn+)>0} ⊆ {l(n)>λ

−1

∗ logn+log(2)n+2 log(3)n

},

which implies a.e. convergenceGn(f−fn−fn+)→0 asn→∞, due to Proposition 8 withm=3. Together with (30), (31) and (29) it implies the desired convergence (4) in the case (ii).

In the case (i), the proof requires some alterations. Consider a sequence of trun-cated functions

fn(i):= f(i)I

i≤λ −1 ∗ η

2 logn

,

and make use of the following decomposition

As above, the equivalences (29) hold and, by (16),

EGn(fn2)≤c9nη/2EGn(fn) ≤ c10n1+η/2. Further, it follows from the condition (7) that

n

∑

r(n−r)P{Sr=0} ≤n n

∑

rP{Sr=0} ≤ c11n2−η.

In addition,

n

∑

fn(i)∆fn(i)(1−γ)i−1≤

λ∗−1η

2 logn

∑

f2(i)e−(i−1)λ∗ _{≤ c}

12nη/2.

Substituting the last three bounds into the right hand side of the inequality provided by Lemma 6, we derive that

VarGn(fn)≤c13n2−η/2,

sinceη<1. Then similar to the case (ii) we deduce (30). Further, it follows from (16) that, for allm≤n,

EGn

(f−fm)

n ≤

n

∑

−1

∗ η

2 logm]

f(j)(1−γ)j−1

≤c14

∞

∑

−1

∗ η

2 logm]

1

j2+ε = O

1 log1+ε_n

,

ifm≥n/2. Again similar to the case (ii), we deduce from the last bound that

Gn(f−fn)

n →0 asn→∞with probability 1,

which together with (30) and equalityGn(f) =Gn(fn) +Gn(f−fn)implies (4) in

the case (i). The proof of Theorem 4 is complete. ut

Acknowledgments

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