Limit theorem for reflected random walks

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Limit theorem for reflected random walks

Hoang-Long Ngo, Marc Peigné

To cite this version:

Hoang-Long Ngo, Marc Peigné. Limit theorem for reflected random walks. 2019. �hal-02303955v3�

Limit theorem for reflected random walks

Hoang-Long Ngo and Marc Peigné

AbstractLetξ_n,n ∈ Nbe a sequence of i.i.d. random variables with values inZ. The associated random walk onZisS(n)=ξ₁+· · ·+ξ_n+1and the corresponding

“reflected walk” onN0is the Markov chainX=(X(n))_n≥0given byX(0)=x∈N0

andX(n+1)=|X(n)+ξ_n+1|forn≥0. It is well know that the reflected walk(X(n))_n≥0 is null-recurrent when theξ_nare square integrable and centered. In this paper, we prove that the process(X(n))_n≥0, properly rescaled, converges in distribution towards the reflected Brownian motion onR⁺, whenE[ξ_n²]<+∞,E[(ξ_n⁻)³]<+∞and theξ_n are aperiodic and centered.

1 Introduction and Notations

Let (ξ_n)_n≥1 be a sequence of Z-valued, independent and identically distributed random variables, with common lawµdefined on a probability space(Ω,F,P). We denoteS =(S(n))_n≥0the classical random walks with stepsξ_k defined byS(0)=0 andS(n)=ξ₁+. . .+ξ_nfor anyn≥1.

Throughout this paper, we denoteN0 the set of non-negative integers and we consider thereflected random walk (X(n))_n≥0onN0defined by

X(n+1)=|X(n)+ξ_n+1|, forn≥0,

whereX(0)is aN0-valued random variables. When X(0)=xP-a.s., withx ∈N0, the process(X(n))_n≥0is also denoted by(X^x(n))_n≥0. It evolves as the random walk Hoang-Long Ngo

Hanoi National University of Education, 136 Xuan Thuy - Cau Giay - Hanoi - Vietnam, e-mail:

ngolong@hnue.edu.vn Marc Peigné

Institut Denis Poisson UMR 7013, Université de Tours, Université d’Orléans, CNRS France e-mail:

peigne@univ-tours.fr

1

x+S(n)as long as it stays non negative. Whenx+S(n)enters the set of negative integers, the sign of its value is changed; the same construction thus applies starting from|x+S(n)|, . . .and so on.

The process(X^x(n))_n≥0is a Markov chain onN0starting fromx. Several papers describing its stochastic behavior have been published; we refer to [16] where the recurrence of the reflected random walk is studied under some conditions which are nearly to be optimal. The reader may find also several references therein.

Firstly,(X^x(n))_n≥0has some similarities with the classical random walk onR; for instance, a strong law of large numbers holds, namely

n→lim+∞

X^x(n)

n =0 P-a.s.

whenE[|ξ_n|] <+∞andE[ξ_n] =0 (see Lemma 4 in section 3). Nevertheless, in contrast to what holds for the classical random walk onR, this does not yield to the recurrence of (X^x(n))_n≥0. In [16], it is proved that the process (X^x(n))_n≥0 is null-recurrent whenE[|ξ_n|^3/2] < +∞andE[ξ_n] =0 and that(X^x(n))_n≥0 may be transient whenE[|ξ_n|^3/2]= +∞, even ifE[|ξ_n|^3/2−]<+∞for any >0. The reader can find in [12] a necessary and sufficient condition for the recurrence of(X^x(n))_n≥0

(see Theorem 4.6) but this condition cannot be reduced to the existence of some moments.

Once the strong law of large number holds, it is natural to study the oscillations of the process around its expectation. Let us state our result.

Theorem 1Let(ξ_n)_n≥1be a sequence of Z-valued i.i.d. random variables such that A1.E[ξ_n²]=σ²<+∞and E[(ξ_n⁻)³]<+∞ ⁽1⁾;

A2.E[ξ_n]=0;

A3.The distribution of theξ_nis strongly aperiodic, i.e. the support of the distribution ofξ_nis not included in the coset of a proper subgroup ofZ.

Let(X(t))_t≥0be the continuous time process constructed from the sequence(X(n))_n≥0 by linear interpolation between the values at integer points. Then, asn →+∞, the sequence of stochastic processes(X_n(t))_n≥1, defined by

X_n(t):= 1 σ√

nX(nt), n≥1,0≤t ≤1,

weakly converges in the space of continuous functions on[0,1]to the absolute value (|B(t)|)_t≥0of the Brownian motion onR.

Let us insist on the fact that X^x(n)coincides with x+S(n)as long as it stays non-negative, but after it may differ drastically. The sequence of successive reflection times of(X^x(n))_n≥0 introduces some strong inhomogeneity on time and makes it necessary to adopt a totally different approach to prove an invariance principle as stated above.

1ξ_n⁻=max(0,−ξn)denotes the negative part ofξn

A model which is quite similar to(Xⁿ(x))_n≥0is the queuing process(W^x(n))_n≥0, also called theLindley process, corresponding to the waiting times in a single server queue. We think to(W^x(n))_n≥0 as an absorbing random walk onN0; asW^x(n), it evolves as the random walkx+S(n)as long as it stays non-negative and, when it attempts to cross 0 and become negative, the new value is reset to 0 before continuing.

We refer to [15] for precise descriptions and variations on this process and follow the same strategy to obtain the invariance principle.

The excursions of(W^x(n))_n≥0 and(X^x(n))_n≥0 between two consecutively times of absorption-reflection coincide with some parts of the trajectory of(S(n))_n≥0, up to a translation; thus, their study is related to the fluctuations of(S(n))_n≥0. Hence, as in [15], we introduce the sequence of strictly descending ladder epochs(`<sub>l</sub>)<sub>l≥0</sub>of the random walk(S(n))<sub>n≥0</sub>defined inductively by`₀ =0 and, for anyl ≥1,

`<sub>l+1</sub>:=min{n> `_l |S(n)<S(`_l)}.

WhenE[|ξ_n|]<+∞andE[ξ_n]=0,the random variables`<sub>1</sub>, `₂−`<sub>1</sub>, `₃−`<sub>2</sub>, . . .areP- a.s. finite and i.i.d. and the same property holds for the random variablesS(`₁),S(`₂)−

S(`<sub>1</sub>),S(`₃)−S(`<sub>2</sub>), . . .. In other words, the processes(`_l)_l≥0and(S(`<sub>l</sub>))<sub>l≥0</sub>are random walks onN0andZwith respective distributionL(`₁)andL(S(`₁)).

Let us briefly point out the main difference between(W^x(n))_n≥0 and(X(n))_n≥0. At an absorption time, the value of the processW^x(n)is reset to 0 before continuing as a classical random walk for a while: there is a total loss of memory of the past after each absorption. Rather, at a reflection time, the process X^x(n)equals the absolute value ofx+S(n). This value is the “new” starting point of the process, for a while, and has a great influence on the next reflection time; in other words, the process always captures some memory of the past at any time of reflection. This phenomenon has to be taken into account and requires a precise study of the sub-process(X(r_k))_k≥0 of (X(n))_n≥0 corresponding to these successive times (r_k)_k≥0 of reflection; our strategy consists in studying the spectrum of the transition probabilities matrixRof (X(r_k))_k≥0, acting on some Banach spaceB =B_αof functions fromN0 toCwith growth less than x^α at infinity, for someα > 0 to be fixed. In particular, in order to apply recent results on renewal sequences [9], we need precise estimates on the tail of distribution of the reflection times; this is the main reason of the restrictive assumptionE[(ξ_n⁻)³]<+∞instead of moment of order 2, as we could expect. More precisely, throughout the paper, we need the following properties to be satisfied:

(i) The operatorRacts onB_α.

This holds whenE[|S(`₁)|^1+α] <+∞and yields to the conditionE[(ξ_n⁻)^2+α] <

+∞(see Proposition 1).

(ii) The functionN0→N0,x7→ x,belongs toB_α; this imposes the conditionα≥1 (see Proposition 2).

Eventually, we fixα=1 from Section 1 on.

Notations.Throughout the text, we use the following notations. Letu=(u_n)_n≥0and v=(v_n)_n≥0be two sequences of positive reals; we write

• u ^c v(or simplyuv) whenu_n ≤cv_nfor some constantc>0andnlarge enough;

• u_n∼v_nwhenlimn→+∞ un vn =1.

• u_n≈v_nwhenlimn→+∞(u_n−v_n)=0.

AcknowledgmentH.-L. Ngo thanks the University of Tours for generous hospitality in the Instittue Denis Poisson (IDP) and financial support in May 2019. This article is a result of the research team with the title "Quantitative Research Methods in Economics and Finance", Foreign Trade University, Ha Noi, Vietnam.

M. Peigné thanks the Vietnam Institute for Advanced Studies in Mathematics (VIASM) and the Vietnam Academy of Sciences And Technology (VAST) in Ha Noi for their kind and friendly hospitality and accommodation in June 2018.

Thanks are also due to M. Pollicott who proposed to publish this article in the Chaire Jean Morlet Series.

Both authors thank the referee for many helpful comments that improved the text and some proofs. Duy Tran Vo also pointed them several misprints.

2 Fluctuations of random walks and auxiliary estimates

Let h be the Green function of the random walk(S(`_l))_l≥0, called sometimes the

“descending renewal function” ofS, defined by

h(x)=













+∞

Õ

l=0

P[S(`_l) ≥ −x] if x≥0,

0 otherwise.

The functionhis harmonic for the random walk(S(n))_n≥0killed when it reaches the negative half line(−∞; 0]; namely, for anyx≥0,

E[h(x+ξ₁);x+ξ₁>0]=h(x).

This holds for any oscillating random walk, possible without finite second moment.

Similarly, we denote ˜h the ascending renewal function of the random walk (S(n))_n≥0(i.e the descending renewal function of(−S(n))_n≥0).

Both functionshand ˜hare increasing,h(0)=h(0)˜ =1 andh(x)=O(x),h(x)˜ = O(x)asx→+∞(see [2], p. 648).

We have also to take into account the fact that the random walkSdoes not always start from the origin; hence, for anyx≥0, we setτ^S(x):=inf{n≥1 :x+S(n)<0};

it holds

[τ^S(x)>n]=[L_n ≥ −x],

whereL_n=min(S(1), . . . ,S(n)). The following result is a combination of Theorem 2 and Proposition 11 in [7] and Theorem A in [13] (see also Theorems II.6 and II.7 in [14]).

Lemma 1For anyx≥0, 1.

P[τ^S(x)>n] ∼c₁h(x)

√n as n→+∞, wherec₁ = E[−S_`1]

σ√

2π . Moreover, there exists a constantC₁ >0such that for any x≥0andn ≥1,

P[τ^S(x)>n] ≤C1

h(x)√ n . 2. For anyx,y≥0,

P[τ^S(x)>n,x+S(n)=y] ∼ 1 σ√

2π

h(x)h(y)˜

n^3/2 as n→+∞, and there exists a constantC₂>0such that, for any anyx,y≥0andn ≥1,

P[τ^S(x)>n,x+S(n)=y] ≤C₂h(x)h(y)˜ n^3/2 .

These assertions yield a precise estimate of the probabilityP[τ^S(x)=n]itself, and not only the tail of the distribution of τ^S. As a direct consequence, the sequence of descending ladder epochs(`_l)_l≥1 of the random walk (S(n))_n≥0 satisfies some renewal theorem [7]. Let us state these two consequences which enlighten the next section where similar statements concerning the successive epochs of reflections of the reflected random are proved.

Corollary 1For anyx≥0, P[τ^S(x)=n] ∼c₁

2 h(x) 1

n^3/2 as n→+∞, and there exists a constant C₃ >0such that, for anyx≥0andn≥1,

P[τ^S(x)=n] ≤C₃h(x) n^3/2. Furthermore,

+∞

Õ

l=0

P[`_l =n] ∼ 1 c₁π

√1

n as n→+∞.

2.2 Conditional limit theorems

The following statement corresponds to Lemma 2.3 in [2]; the symbol “⇒” means

“weak convergence".

Lemma 2AssumeE(ξ_i²)<+∞andE(ξ_i)=0. Then, for anyx≥0, L

S([nt]) σ√

n

0≤t≤1|min{S(1), . . . ,S(n)} ≥ −x

⇒ L(L⁺) asn→+∞, whereL⁺is the Brownian meander.

In particular, for any bounded and Lipschitz continuous functionφ:R→R,

n→lim+∞E

φ

x+S(n) σ√

n

τ^S(x)>n

=∫ ₊∞ 0

φ(z)ze^−z2/2dz.

This Lemma is useful in the sequel to control the fluctuations of the excursions of the process(X(n))_n≥0between two successive times of reflection. In order to control also the higher dimensional distributions of these excursions, we need some invariance principle for random walk bridges conditioned to stay positive. The following result corresponds in our setting to Corollary 2.5 in [5].

Lemma 3For any bounded, Lipschitz continuous functionφ:R→R, anyx,y≥0, and anyt>s>0,

nlim→+∞E

φ

x+S([ns]) σ√

n

τ^S(x)>[nt],x+S([nt])=y

=∫ ₊∞ 0

2φ(u√

s)exp − u² 2^s_t^t−s_t

! u² q2π^s3

t³ (t−s)³

t³

du.

3 On the sub-process of reflections

We present briefly some results from [8] and [16]. The reflected timesr_n,n≥0,of the random walk(X(n))_n≥0are defined by: for anyx≥0,

r0=r0(x)=0 and r_n+1=inf{m>rn | X(r_n)+ξ_rn+1+· · ·+ξ_m<0}.

Notice that these random variables areN0∪{+∞}-valued stopping times with respect to the filtration(G_n)_n≥0.

WhenE[|ξ_n|]< +∞andE[ξ_n] = 0, the random walk(S(n))_n≥0 is oscillating, hence ther_n,n≥0, are all finiteP-a.s. andS(n)/nconvergesP-a.s. towards 0. The strong law of large numbers is still true for the reflected random walk(X^x(n))_n≥0on N0but does not derive directly.

Lemma 4If E[|ξ_n|]<+∞andE[ξ_n]=0, then, for anyx∈N0,

n→lim+∞

X^x(n)

n =0 P-a.s.

Proof For anyn ≥1, there exists a (random) integerk_n ≥ 1 such thatr_kn ≤ n <

r_kn+1. It holds

X^x(n)=X^x(r_kn)+ ξ_rkn+1+· · ·+ξ_n=X^x(r_kn)+S(n) −S(r_kn), so that

0≤ X^x(n)

n = X^x(r_kn) n +S(n)

n −S(r_kn)

n ≤ max{|ξ1|, . . . ,|ξ_n|}

n +S(n)

n −S(r_kn) n . The first term on the right hand side convergesP-a.s. towards 0 sinceE[|ξ_n|]<

+∞.

By the strong law of large number, the second term tendsP-a.s. to 0.

At last, the same property holds for the last term, since

S(r_kn) n

=

S(r_kn) r_kn

×r_kn

n ≤

S(r_kn) r_kn

.

It follows from Lemma 2.3 in [17] that the sub-process of reflections(X(r_k))_k≥0 is a Markov chain onN0with transition probabilityRgiven by: for allx,y∈N0,

R(x,y)=

(0 ify=0 Íx

w=0U^∗(−w)µ^∗(w−x−y) ify≥1, (1) whereµ^∗is the distribution ofS(`1)andU^∗=

+∞

Õ

n=0

(µ^∗)^?ndenotes its potential.

SetC :=sup{y≥1 : µ(−y)>0}. The support ofµ^∗equalsZ⁻ =Z∩ (−∞,0) when C = +∞, otherwise it is {−C, . . . ,−1}; furthermore,U^∗(−w) > 0 for any w≥0. Then,R(x,y)>0 if and only ify∈Sr, whereSr =N0\ {0}whenC= +∞ andSr = {1, . . . ,C}otherwise. Consequently, the setSr is the unique irreducible and ergodic class of the Markov chain(X(r_k))_k≥0and this chain is aperiodic onSr.

The measureνonN0defined by ν(x)= ⁺

∞

Õ

y=1

1

2µ^∗(−x)+µ^∗ (−x−y,−x)+1

2µ^∗(−x−y) µ^∗(−y),

is, up to a multiplicative constant, the unique stationary measure for(X(r_k))_k≥0; its support equalsSr(see Theorem 3.6 [17]).

Notice that this measureνis finite whenE[ξ_n]=0 andE|S(`₁)|^1/2 <+∞(and in particular whenE[ξ_n]=0 andE[|ξ_n|^3/2]<+∞[16]).In this case, we normalizeν it in such a way it is a probability measure.

3.1 On the spectrum of the transition probabilities matrixR

Let us recall some spectral properties of the matrixR=(R(x,y))_x,y∈N0. By Property 2.3 in [8], the matrixRis quasi-compact on the spaceL^∞(N0)of bounded functions onN0, with 1 as the unique (and simple) dominant eigenvalue; in particular, the rest of the spectrum ofRis included in a disc with radius<1.

It is of interest in the next section to letRact on a bigger space thanL^∞(N0). For instance, following [8], we may fixK >1 and consider the Banach space

L_K(N0):={φ:N0→C:kφk_K :=sup

x≥0

|φ(x)|/K^x <+∞}

endowed with the normk · k_K. By Property 2.3 in [8], ifÕ

x≥0

K^xµ(x)<+∞thenR acts as a compact operator onLK(N0).

In this article, we only assume that µ has a finite moment of order 2 and its negative part has moment of order 3. Consequently, we consider a smaller Banach spaceB_αadapted to these hypotheses and defined by: forα >0 fixed,

B_α:=n

φ:N0→C:|φ|_α:=sup

x≥0

|φ(x)|

1+x^α <+∞o .

Endowed with the norm| · |_α, the spaceB_αis a Banach space onC.

Proposition 1Fixα > 0 and assumeE[ξ²_n]+E[(ξ_n⁻)^2+α] < +∞and E[ξ_n] = 0.

Then, the operatorRacts onB_αandR(B_α) ⊂ L^∞(N0). Furthermore, 1.Ris compact onB_αwith spectral radius1;

2.1is the unique eigenvalue ofR with modulus1, it is simple with corresponding eigenspaceC1;

3. the rest of the spectrum ofRonB_αis included in a disc with radius<1.

Let Π be the projection from B_α onto the eigenspace C1 corresponding to this spectral decomposition, i.e. such thatΠR=RΠ=Π. In other words, there exists a bounded operatorQonB_αwith spectral radius<1 such thatRmay be decomposed as follows:

R=Π+Q, ΠQ=QΠ=0 with Π(·)=ν(·)1.

In the next section, we require thatB_αdoes contain the descending and ascending renewal functionshand ˜hof the random walkS. This imposes in particular thatα is greater or equal to 1.

Proof (1) By (1), for anyφ∈ B_αandx≥0, Rφ(x)=Õ

y≥1 x

Õ

w=0

U^∗(−w)µ^∗(w−x−y)φ(y)

withU^∗(−w)=⁺

∞

Õ

n=0

P[S(l_n)=−w]=P h

∪_n≥0[S(l_n)=−w]i

≤1.Therefore,

|Rφ(x)| ≤Õ

y≥1

Õx w=0

µ^∗(w−x−y)|φ(y)|

≤Õ

y≥1

µ^∗((−∞,−y))|φ(y)|

≤©

­

« Õ

y≥1

(1+y^α)µ^∗((−∞,−y))ª

®

¬

|φ|_α.

By Theorem 1 in [6], the conditionE[(ξ⁻_n)^2+α]<+∞impliesE|S(`₁)|^1+α <+∞;

hence, Õ

y≥1

(1+y^α)µ^∗((−∞,−y)) ≤E[|S(`<sub>1</sub>)|]+E|S(`₁)|^1+α <+∞.

Consequently,

|Rφ|_α≤ |Rφ|∞ ≤

E[|S(`<sub>1</sub>)|]+E|S(`₁)|^1+α

|φ|_α (2)

which proves thatRacts onB_αwhenE[(ξ_n⁻)^2+α]<+∞. More precisely, the operator Ris bounded fromB_αintoL^∞(N0)and since the canonical injectionL^∞(N0),→ B_α is compact, the operatorRis compact onB_α.

Let us now check thatRhas spectral radiusρ_α =1 onB_α. On the one hand, the equalityR1 =1, with1 ∈ B_α, yields ρ_α ≥ 1. On the other hands, R is a power bounded operator onB_α, which readily impliesρ_α ≤1; indeed, for anyn≥1,

|Rⁿφ(x)| ≤

+∞

Õ

z=0

Rⁿ⁻¹(x,z)|Rφ(z)| ≤ |Rφ|∞ +∞

Õ

z=0

Rⁿ⁻¹(x,z)=|Rφ|∞,

which yields, combining with (2),

|Rⁿφ|_α≤ |Rⁿφ|∞ ≤

E[|S(`<sub>1</sub>)|]+E|S(`₁)|^1+α

|φ|_α.

Consequently, denotingkRⁿk_αthe norm ofRⁿonB_α, it holds sup

n≥0

kRⁿk_α ≤

E[|S(`<sub>1</sub>)|]+E|S(`₁)|^1+α

<+∞.

This achieves the proof of assertion 1.

(2) Let us control the peripherical spectrum ofRinB_α. Letθ∈Randφ∈ B_α such thatRφ=e^iθφ.

By (2), the functionRφis bounded, so isφ. Furthermore, the operatorRbeing positive, it holds|φ| ≤ R |φ|. Consequently, the function|φ|∞− |φ|is super-harmonic and non-negative, hence constant since the Markov chain(X(r_n))_n≥0is irreducible and recurrent on this set.

Without loss of generality, we may assume|φ|=1 onSr, i.eφ(x)=e^iϕ(x)for any x∈Sr, withϕ:Sr→R. EqualityRφ=e^iθφmay be rewritten as: for anyx∈Sr,

Õ

y∈Sr

ei(ϕ(y)−ϕ(x))R(x,y)=e^iθ.

Recall that R(x,y) > 0 for any x,y ∈ Sr; thus, by convexity, e^{i(ϕ(y)−ϕ(x))} = e^iθ for any x,y ∈ Sr. Thus, e^iθ = 1 and the function φ is harmonic on Sr, hence constant. Eventually, the function φ is constant onN0: this is the consequence of equalityRφ(x)=e^iθφ(x)=φ(x), valid for anyx∈N0, combined with the facts that R(x,y)>0 if and only ify∈Sr and thatφis constant onSr.

(3) Assertion 3 is a consequence of assertion 2 and the compactness ofRonB_α.

3.2 A Renewal limit theorem for the times of reflections

In this section, we prove the analogous of Corollary 1 for the process(r_n)_n≥0. Let us introduce some notations and conventions.

From now on, we focus on the process(X(n))_n≥0and denote ((N⁰)^⊗N,(P(N⁰))^⊗N,(X(n))_n≥0,(Px)_x∈N0, θ)

the canonical space associated to this process, that is the space of trajectories of the Markov chain (X(n))_n≥0. In particular, Px,x ∈ N0, denotes the conditional probabilitywith respect to the event[X(0)=x]andExthe correspondingconditional expectation. The operatorθis the classicalshift transformationdefined by: for any (x_k)_k≥0 ∈ (N⁰)^⊗N,

θ((x_k)_k≥0)=((x_k+1)_k≥0. Forn ≥1 andx,y≥0, set

R_n(x,y):=Px[r₁ =n,X(n)=y], and

Σ_n(x,y):=

+∞

Õ

k=1

Px[r_k =n,X(n)=y].

We are interested in the behavior asn→+∞of these quantities. It has been already studied in [15] (see Lemma 7) for the Lindley process. For the reflected random walk, the argument is more complicated since the position at timerk may vary, so that the excursions of the random walk(X(n))_n≥0between two successive reflection

times are not independent. This explain why we focus here on the reflection process and it is of interest to express quantitiesR_n(x,y)andΣ_n(x,y)in terms of operators and product of operators related to this sub-process.

We consider the linear operatorsRn:L^∞(N0) →L^∞(N0),n≥0, defined by: for anyφ∈L^∞(N0)andx≥0,

Rnφ(x)=Õ

y≥1

Rn(x,y)φ(y)=Ex[r₁=n;φ(X(n))].

In particular,R_n(x,y)=R_n1_{y}(x). The quantityΣ_n(x,y)is also expressed in terms of theRkas follows:

Σ_n(x,y)=

+∞

Õ

k=1

Px[r_k =n,X(n)=y]

=⁺

∞

Õ

k=1

Õ

j1+···+jk=n

Px[r₁ =j₁,r₂−r₁=j₂, . . . ,r_k−r_k−1= j_k,X(n)=y]

=

+∞

Õ

k=1

Õ

j1+···+jk=n

R_j1. . .R_jk1_{y}(x) (3)

Firstly, let us check that theR_nact onB_α.

Lemma 5There exists a positive constantC4such that, for anyn≥1andα >0,

|R_n|_α≤C₄E(ξ_n⁻)^2+α n^3/2 . Proof For anyφ∈ B_αandx≥0,

|R_nφ(x)| ≤Õ

y≥1

|φ(y)|Px[r₁ =n,X(n)=y]

=Õ

y≥1

Õ

z≥0

|φ(y)|P[τ^S(x) ≥n−1,x+S(n−1)=z,z+ξ_n=−y]

=Õ

y≥1

Õ

z≥0

|φ(y)|P[τ^S(x) ≥n−1,x+S(n−1)=z]P[ξ_n=−y−z].

Hence, by Lemma 1,

|R_nφ(x)|

1+x^α 1 n^3/2

Õ

y≥1

Õ

z≥0

|φ(y)| h(x)

1+x^αh(z)˜ P[ξ₁ =−y−z].

Sinceh(x)=O(x)and ˜h(z)=O(z),

|R_nφ(x)|

1+x^α |φ|_α n^3/2

Õ

y≥1

Õ

z≥0

(1+y^α)h(z)˜ P[ξ₁=−y−z]

|φ|_α n^3/2

Õ

y≥1

Õ

z≥0

(1+y^α)zP[ξ₁=−y−z]

= |φ|_α n^3/2

Õ

t≥1

Õt y=1

(1+y^α)(t−y)P[ξ₁=−t]

|φ|_α n^3/2

Õ

t≥1

t^2+αP[ξ₁=−t],

which achieves the proof.

Hence,Õ

n≥1

|R_n|_α<+∞; in particular, the sequence(ÍN

n=1R_n)_N≥1converges inB_α. Note that its limit equalsRinB_α; indeed,

Õ

n≥1

R_nφ(x)=Õ

n≥1

Ex[φ(X(n)),r₁=n]=Ex[φ(X(r₁))]=Rφ(x).

We can writeR=Í

n≥1R_nand, for anyz∈D:={z∈C:|z| ≤1}, we set R(z)=Õ

n≥1

zⁿR_n.

Proposition 2Fixα >0and assumeE[ξ_n²]+E[(ξ_n⁻)^2+α]<+∞andE[ξ_n]=0. The sequence(R_n)_n≥0 is anaperiodic renewal sequence of operators, i.e. it satisfies the following properties (see [9]):

(R1). The operator R = R(1)has a simple eigenvalue at1 and the rest of its spectrum is contained in a disk of radius<1.

(R2).For anyn≥1, setr_n :=νR_n1=Í

x≥1ν(x)Px(r₁ =n);hence, ΠR_nΠ=r_nΠ,

whereΠdenotes the eigenprojection of Rfor the eigenvalue 1.

(R3).There exists a constantC>0such that |R_n|_α≤ ^C

n^3/2. (R4).Í

j>nr_j ∼ ^√c

n withc=c₁ν(h), wherec₁is the positive constant given by Lemma 1 andhis the descending renewal function of the random walkS.

(R5).The spectral radius of R(z)is strictly less than1forz∈D\ {1}. Proof (R1) is a direct consequence of Proposition 1.

(R2) Recall thatΠφ=ν(φ)1for anyφ∈ B_α. Hence, settingg_n(x):=Px(r₁=n), it holdsR_nΠφ=ν(φ)g_n, thus

ΠRnΠφ=ν(φ)Π(g_n)=Õ

x≥1

ν(x)Px(r₁=n)ν(φ)1,

which is the expected result.

(R3) follows from Lemma 5.

(R4) Thanks to Lemma 1, Õ

j≥n

r_j =Õ

x≥1

Õ

j≥n

ν(x)Px[r₁=j]=Õ

x≥1

ν(x)Px[r₁ ≥n] ∼c₁ν(h)

√n as n→+∞.

Notice that 0< ν(h)<+∞sinceE[|S(`₁)|]<+∞; indeed, 1≤h(x)=O(x)and Õ

x≥1

xν(x) Õ

x≥1

Õ

y≥1 x+y

Õ

w=x

µ^∗(−w)µ^∗(−y)x=Õ

y≥1

Õ

w≥1

µ^∗(−w)µ^∗(−y)

w

Õ

x=(w−y)∨0

x

≤Õ

y≥1

Õ

w≥1

ywµ^∗(−w)µ^∗(−y)=©

­

« Õ

y≥1

yµ^∗(−y)ª

®

¬

2

=(E[|S(`1)|])²<+∞.

(R5) The argument is the same as the one used to control the peripherical spectrum ofRin Proposition 1. For anyz ∈D\ {1}, the operatorsR(z)are compact onB_α, with spectral radiusρ_z≤1.

If ρ_z = 1, there exist θ ∈ R and φ ∈ B_α such that R(z)φ = e^iθφ. Hence

|φ| =|R(z)φ| ≤ R |φ|and sinceR(B_α) ⊂ L^∞(N0), the function|φ|is bounded on N0, thus constant onSr.

Without loss of generality, we may assume |φ| =1 onSr, i.eφ(x)=e^iϕ(x)for anyx ∈Sr, withϕ:Sr → R. EqualityR(z)φ=e^iθφmay be rewritten as: for any x∈Sr,

Õ

n≥1

Õ

y∈Sr

zⁿe^iϕ(y)Px(r₁=n;X(n)=y)=e^iθe^iϕ(x). By convexity, sinceÍ

n≥1Í

y∈SrPx(r₁ =n;X(n)=y)=1, we obtain: for alln ≥1 andx,y∈Sr,

zⁿe^iϕ(y)=e^iθe^iϕ(x).

Setting x = y, it yields zⁿ = e^iθ, so that zⁿ does not depend onn. Finallyz =1.

Thus,ρ_z <1 whenz∈D\ {1}.

By (R5), for|z| < 1, the operatorT(z) := (I− R(z))⁻¹ is well defined in B_α; a direct formal computation yieldsT(z) = Í+∞

n=0T_nzⁿ, where theT_n are bounded operators onB_αdefined by:

T₀=I and T_n=

+∞

Õ

k=1

Õ

j1+···+jk=n

R_j1· · ·R_jk for n≥1.

The so-calledrenewal equationT(z):=(I−R(z))⁻¹is of fundamental importance to understand the asymptotics of the T_n, several functional analytic tools can be brought into play. Such sequences of operators(R_n)_n≥0 and(T_n)_n≥0 have been the object of many studies, related to renewal theory in a non-commutative setting. We

refer to the paper [9], which fits perfectly here. The following statement is analogous of the last assertion of Corollary 1 for the reflected random walk.

Corollary 2The sequence(√

nT_n)_n≥1converges inB_αtowards the operator_πc1¹_ν(h)Π. Proof Apply Theorem 1.4 in [9] withβ=1/2 and`(n)=c=c1ν(h).

As a direct consequence, by equality (3), it holds

n→lim+∞

√nΣ_n(x,y)= ν(y) πc₁ν(h).

In the next section, we have to consider and study some modifications of the Σ_n(x,y)which we introduce now. For anyx≥0 and 0<s<t <1,

bΣ_n(x,t,s):=nÕ

l≥0

Px[r_l=[ns],r_l+1>[nt]],

and

eΣ_n(x,t,s):=n²

+∞

Õ

l=0

Px[r_l =[ns],r_l+1=[nt]].

These quantities appear in a natural way to control the finite distribution of the process(X_n(t))_n≥0.

4 Proof of Theorem 1

From now on, we fixα=1; this implies thath∈ B_α, which is necessary from now on (see Lemmas 7 and 9).

4.1 One-dimensional distribution

We fix a bounded and Lispchitz continuous functionφ:R→R.

Lemma 6For anyt ∈ [0,1]andx≥0, it holds

n→lim+∞Ex[φ(X_n(t))]=∫ +∞ 0

φ(u)2e^−u2/2t

√2πt du=E[φ(|B_t|)], whereBis a standard Brownian motion.

Proof We fixt ∈ (0,1)and decompose the expectationE

φX([nt]) σ√

n as follows:

Ex

φ

X([nt]) σ√

n

≈

[nt]−1

Õ

k=0

Õ

l≥0

Ex

hφ

X([nt]) σ√

n

;

r_l =k,X(k)+ξ_k+1 ≥0, . . . ,X(k)+ξ_k+1+· · ·+ξ_[nt]≥0i

=

[nt]−1

Õ

k=0

Õ

y≥0

Σ_k(x,y)E hφ

y+ξ_k+1+. . .+ξ_[nt]

σ√ n

;

y+ξ_k+1 ≥0, . . . ,y+ξ_k+1+· · ·+ξ_[nt]≥0i

=

[nt]−1

Õ

k=0

Õ

y≥0

Σ_k(x,y)E

φ

y+S([nt] −k) σ√

n

|τ^S(y)>[nt] −k

×Pτ^S(y)>[nt] −k.

For eachk=2, . . . ,[nt] −4 and anys∈ [^k_n,^k+1_n ), f_n(s)=nÕ

y≥0

Σ_[ns](x,y)E

φ

y+S([nt] − [ns]) σ√

n

|τ^S(y)>[nt] − [ns]

×Pτ^S(y)>[nt] − [ns],

and f_n(s)=0 on[0,²_n)and[^[nt_n^]−1,t). Hence, Ex

φ

X([nt]) σ√

n =∫ t 0

f_n(s)ds+O 1

√n

.

Now, let us set : forn ≥1 and anyy∈N0,

a_n(y)=Σ_[ns](x,y)Pτ^S(y)>[nt] − [ns], bn(y)=E

φy+S([nt] − [ns]) σ√

n

|τ^S(y)>[nt] − [ns]

.

For anyn≥1, it holds Õ

y≥0

a_n(y)=nÕ

l≥0

Px[r_l =[ns],r_l+1 >[nt]]=:bΣ_n(x,t,s),

and|b_n(y)| ≤ |φ|∞. The two following lemmas allow us to control the behavior as n→+∞of the integral

∫ t 0

f_n(s)ds; the proof of Lemma 7 is postponed to the last section, the one of 8 is straightforward.

Lemma 7For each0<s<t <1,

n→lim+∞bΣ_n(x,t,s)= 1 πp

s(t−s) .

Moreover, there exists a positive constantC₅such that

bΣ_n(x,t,s) ≤C5

1+x

ps(t−s) for all 0<s<t <1 and x∈N. Lemma 8Let(a_n(y))_y∈

N^k0,(b_n(y))_y∈

N^k0 be arrays of real numbers for some integer k≥1. Suppose that

• a_n(y) ≥0;

• lim

n→+∞

Õ

y∈N^k0

a_n(y)=A;

• lim

n→+∞b_n(y)=Bfor ally∈N^k₀;

• sup

n≥1,y∈N^k0

|b_n(y)|<+∞.

Then

n→lim+∞

Õ

y≥0

a_n(y)b_n(y)=AB.

Lemmas 2, 7 and 8 combined altogether yield: for anys∈ (0,t),

n→lim+∞f_n(s)= 1 π

1 ps(t−s)

∫ ₊∞ 0

φ(z√

t−s)ze^−z2/2dz.

Moreover,

sup

n

|fn(s)| ≤C5

1+x ps(t−s)

|φ|∞=: ˆf(s).

Since ˆf ∈L¹[0,t], the Lebesgue dominated convergence theorem yields

n→lim+∞E

φ

X([nt]) σ√

n = lim

n→+∞

∫ t 0

f_n(s)ds

= 1 π

∫ t 0

1 ps(t−s)

∫ ₊∞ 0

φ(z√

t−s)ze^−z2/2dz

ds

=∫ ₊∞ 0

φ(u)2e^−u2/2t

√2πt du,

where the last equation follows from the identity ([11], p. 17)

∫ ₊∞ 0

√1 texp

−αt− β t

dt=

rπ αe⁻²

√αβ (α, β >0) (4)

and some change of variable computation. We achieve the proof of Lemma 6 by noting that, sinceφis Lipschitz continuous (with Lipschitz coefficient[φ]),

Ex

φ

X([nt]) σ√

n −Ex[φ(X_n(t))]

≤ [φ]Ex

X([nt]) σ√

n −X_n(t)

≤ 1

σ√

n[φ]E|ξ_[nt]+1|

→0 asn→+∞.

(5)

4.2 Two-dimensional distributions

The convergence of the finite-dimensional distributions of(X_n(t))_n≥1 is more del- icate. We detail the argument for two-dimensional ones, the general case may be treated in a similar way.

Let us fix 0<s<t,n≥1 and denote

κ=κ(n,s)=min{k>[ns]:X(k−1)+ξ_k <0}.

We decomposeEx

φ₁

X([ns]) σ√

n

φ₂

X([nt]) σ√

n as

[nt]

Õ

k=[ns]+1

Ex

φ₁

X([ns]) σ√

n

φ₂

X([nt]) σ√

n

1_{κ=k}

| {z }

A1(n)

+Ex

φ₁

X([ns]) σ√

n

φ₂

X([nt]) σ√

n

1{κ>[nt]}

| {z }

A2(n)

.

The term A₁(n)deals with the trajectories of the processX which reflect between [ns]+1 and[nt]whileA₂(n)concerns the others trajectories.

4.2.1 Estimate ofA₁(n)

As in the previous section, we decomposeA₁(n)as

A₁(n)=

[ns]−1

Õ

k1=0 [nt]

Õ

k2=[ns] +∞

Õ

l=0

Õ

y≥1

Õ

z≥1

Õ

w≥0

Ex

"

φ₁

X([ns]) σ√

n

φ₂

X([nt]) σ√

n

;

r_l =k₁,X(k₁)=z,z+ξ_k1+1≥0, . . . ,z+ξ_k1+1+· · ·+ξ_k2−2 ≥0, z+ξ_k1+1+· · ·+ξ_k2−1=w,w+ξ_k2 =−y

#

=

[ns]−1

Õ

k1=0 [nt]

Õ

k2=[ns]

+∞

Õ

l=0

Õ

y≥1

Õ

z≥1

Õ

w≥0

Ex

"

φ₁z+ξ_k1+1+· · ·+ξ_[ns]

σ√ n

×φ₂

y+ξ_k2+1+· · ·+ξ[nt]

σ√ n

;

r_l =k₁,X(k₁)=z,z+ξ_k1+1≥0, . . . ,z+ξ_k1+1+· · ·+ξ_k2−2 ≥0, z+ξ_k1+1+· · ·+ξ_k2−1=w,w+ξ_k2 =−y

#

=

[ns]−1

Õ

k1=0 [nt]

Õ

k2=[ns]

+∞

Õ

l=0

Õ

y≥1

Õ

z≥1

Õ

w≥0

Ex

"

φ₂

y+ξ_k2+1+· · ·+ξ_[nt]

σ√ n

#

Px[r_l =k₁,X(k₁)=z]

×Ex

"

φ1

z+ξ_k1+1+· · ·+ξ_[ns]

σ√ n

,

z+ξ_k1+1 ≥0, . . . ,z+ξ_k1+1+· · ·+ξ_k2−2 ≥0,

z+ξ_k1+1+· · ·+ξ_k2−1=w,w+ξ_k2 =−y

# .

Using the fact that theξ_k are i. i. d., we obtain A1(n)=

[ns]−1

Õ

k1=0

Õ

z≥1

Σ_k1(x,z)

[nt]

Õ

k2=[ns]

Õ

y≥1

Õ

w≥0

Ey

"

φ2

X([nt] −k2) σ√

n #

×E

"

φ₁z+S([ns] −k₁) σ√

n

|τ^S(z)>k₂−k₁−1,z+S(k₂−k₁−1)=w

#

×P[τ^S(z)>k2−k1−1,z+S(k₂−k1−1)=w]P[ξ₁=−w−y].

For any 2 ≤ k1 < [ns] −6 and [ns] ≤ k2 ≤ [nt] and any s1 ∈ [^k_n¹,^k1_n⁺¹) and s2 ∈ [^k_n²,^k2_n⁺¹), we write

f_n(s₁,s₂)=n²Õ

z≥1

Σ_[ns1](x,z)Õ

y≥1

Õ

w≥0

Ey

"

φ₂

X([nt] − [ns₂]) σ√

n #

×E

"

φ₁

z+S([ns] − [ns₁]) σ√

n

|τ^S(z)>[ns₂] − [ns₁] −1,

z+S([ns₂] − [ns₁] −1)=w

#

×Pτ^S(z)>[ns₂] − [ns₁] −1,z+S([ns₂] − [ns₁] −1)=w

P[ξ1 =−w−y], and fn(s₁,s2)=0 for the others values ofk1, such that 0≤k1 ≤ [ns]. Hence,

A₁(n)=∫ s 0

ds₁

∫ t s

ds₂ f_n(s₁,s₂)+O 1

√n

.

It follows from Lemma 3 that, for eachz,w≥0,

n→lim+∞E hφ₁

z+S([ns] − [ns₁]) σ√

n

|τ^S(z)>[ns₂] − [ns₁] −1, z+S([ns₂] − [ns₁] −1)=wi

=∫ ₊∞ 0

2φ₁(u√

s₂−s₁)exp − u² 2_s^s₂⁻₋^s_s¹_1s^s₂²₋⁻_s^s₁

! u² q2π^(s−s1)3

(s2−s1)³ (s2−s)³ (s2−s1)³

du

= 2

√2π

∫ ₊∞ 0

φ₁(v)exp − v² 2^(s−s_s^1)(s2−s)

2−s1

! v² q(s−s1)³(s2−s)³

(s2−s1)³

dv.

By Lemma 6,

n→lim+∞Ey

φ2

X([nt] − [ns₂]) σ√

n =∫ +∞

0

φ2(u)2e^{−u2/2(t−s2)} p2π(t−s₂)du.

We set

a_n(x,y,z,w)=n²Σ_[ns1](x,z)

×Pτ^S(z)>[ns₂] − [ns₁] −1,z+S([ns₂] − [ns₁] −1)=w

×P[ξ₁=−w−y], b_n(y,z,w)=Ey

"

φ2

X([nt] − [ns₂]) σ√

n #

×E

"

φ₁

z+S([ns] − [ns₁]) σ√

n

|τ^S(z)>[ns₂] − [ns₁] −1,

z+S([ns₂] − [ns₁] −1)=w

#

Note thatÍ

z≥1Í

y≥1Í

w≥0an(x,y,z,w)=eΣ_n(x,s2,s1). The behavior asn→+∞of the quantityeΣ_n(x,s2,s1)is given by the following Lemma, whose proof is postponed to the last section.

Lemma 9For all0<s<t <1, it holds

n→lim+∞eΣ_n(x,t,s)= 1 2πp

s(t−s)³ .

Moreover, there exists a positive constantC₆ such that, for all0 < s <t <1and n≥0,

eΣ_n(x,t,s) ≤C6

1+x πp

s(t−s)³ .

By Lemmas 9 and 8, we get limn→+∞ f_n(s₁,s₂)= f(s₁,s₂)where f(s₁,s₂)= 1

π²√ s₁

∫ ₊∞ 0

φ₁(v)exp − v² 2^{(s2−s)(s−s}_s2−s1 ¹⁾

! v² p(s−s₁)³(s₂−s)³

dv

×

∫ ₊∞ 0

φ₂(u)e^{−u2/2(t−s2)}

√t−s₂ du.

Moreover, following the argument in the proof of Lemma 6, we can show that the sequence (|f_n|)_n≥1 is uniformly bounded by a function which is integrable with respect to Lebesgue measure on [0,s] × [s,t]. Hence, using again the Lebesgue dominated convergence theorem, we get