On the invariant measure of the random difference equation <span class="mathjax-formula">$X_n=A_nX_{n-1}+B_n$</span> in the critical case

www.imstat.org/aihp 2012, Vol. 48, No. 2, 377–395

DOI:10.1214/10-AIHP406

© Association des Publications de l’Institut Henri Poincaré, 2012

On the invariant measure of the random difference equation X _n = A _n X _{n − 1} + B _n in the critical case ¹

Sara Brofferio

, Dariusz Buraczewski

and Ewa Damek

aLaboratoire de Mathématiques et IUT de Sceaux, Université Paris-Sud, 91405 Orsay Cedex, France. E-mail:sara.brofferio@math.u-psud.fr bInstytut Matematyczny, Uniwersytet Wroclawski, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland.

E-mail:dbura@math.uni.wroc.pl;edamek@math.uni.wroc.pl Received 9 August 2010; revised 25 October 2010; accepted 25 November 2010

Abstract. We consider the autoregressive model onR^ddefined by the stochastic recursionXn=AnX_n−1+Bn, where{(Bn, An)}

are i.i.d. random variables valued inR^d×R⁺. The critical case, whenE[logA₁] =0, was studied by Babillot, Bougerol and Elie, who proved that there exists a unique invariant Radon measureνfor the Markov chain{Xn}. In the present paper we prove that the weak limit of properly dilated measureνexists and defines a homogeneous measure onR^d\ {0}.

Résumé. Nous considérons le modèle autorégressif surR^ddéfini par récurrence par l’équation stochastiqueX_n=A_nX_n−1+B_n, où{(B_n, A_n)}sont des variables aléatoires à valeurs dansR^d×R⁺, indépendantes et de même loi. Le cas critique, c’est-à-dire lorsqueE[logA₁] =0, a été étudié par Babillot, Bougerol et Elie, qui ont montré qu’il existe une et une seule mesure de Radon νinvariante pour la chaîne de Markov{X_n}. Dans ce papier nous démontrons que la mesureν, convenablement dilatée, converge faiblement vers une mesure homogène surR^d\ {0}.

MSC:Primary 60J10; secondary 60B15; 60G50

Keywords:Random walk; Random coefficients autoregressive model; Affine group; Random equations; Contractive system; Regular variation

1. Introduction and the main result

We consider the autoregressive process onR^d: X^x₀=x,

(1.1) X^x_n=A_nX_n^x₋₁+B_n,

where the random pairs{(B_n, A_n)}n∈Nvalued inR^d×R⁺are independent, identically distributed (i.i.d.) according to a given probability measureμ. The Markov chain{X^x_n}occurs in various applications e.g. in biology and economics, see [1] and the comprehensive bibliography there.

It is convenient to defineX_n in the group language. LetGbe the “ax+b” group, i.e.G=R^d R⁺, with mul- tiplication by(b, a)·(b, a)=(b+ab, aa). The groupGacts onR^d by(b, a)·x=ax+b, for(b, a)∈Gand

1This research project has been partially supported Marie Curie Transfer of Knowledge FellowshipHarmonic Analysis, Nonlinear Analysis and Probability(contract number MTKD-CT-2004-013389). S. Brofferio was also supported by CNRS project ANR – GGPG (Réf. JCJC06 149094).

D. Buraczewski and E. Damek were also supported by MNiSW grants N201 012 31/1020 and N N201 393937.

x∈R^d. For eachn, we sample the random variables(B_n, A_n)∈Gindependently with respect to the measureμ, then X_n^x=(B_n, A_n)· · · · ·(B₁, A₁)·x.

The Markov chainX_n^xis usually studied under the assumptionE[logA₁]<0. Then, if additionallyE[log⁺|B₁|]<

∞, there is a unique stationary measureν[16], i.e. the probability measureνonR^dsatisfying

μ∗Gν(f )=ν(f ) (1.2)

for every positive measurable functionf. Here μ∗Gν(f )=

G

R^df (ax+b)ν(dx)μ(dbda).

One of the main results concerning the stationary measureνis Kesten’s theorem [16] (see also [13,15]) saying that if E[A^α₁] =1 (and some other assumptions are satisfied), then

ν

u: |u|> z

∼Cz^−α asz→ +∞

for a positive constantC.

Here we study the critical case, whenE[logA1] =0. ThenXn has no invariant probability measure. However, it was proved by Babillot, Bougerol and Elie [1] that if

• P[A₁=1]<1 andP[A₁x+B₁=x]<1 for allx∈R^d,

• E[(|logA₁| +log⁺|B₁|)^2+ε]<∞, for someε >0,

• E[logA₁] =0.

Then there exists a unique (up to a constant factor) invariant Radon measureν, i.e. a measure satisfying (1.2) (see also [2,3]). We will say thatμsatisfies hypothesis (H) if all the assumptions above are satisfied. For our purpose we will need an additional assumption, which will be called hypothesisM(δ):

• there existsδ >0 such thatE[A^δ₁+A⁻₁^δ+ |B₁|^δ]<∞.

The measureνappears in a natural way when problems related to the processX^x_nor to random walks on the group Gin the critical case are investigated. Let us mention two examples. Le Page and Peigné [17] proved the local limit theorem forX^x_n, saying that under some further assumptions, √

nE[f (X_n^x)]converges toν(f )for any compactly supported functionf. Elie [10] described the Martin boundary for the left random walk on the affine group with the measureνplaying the central role. Therefore, it is natural to ask about a quantified description of the measureν and the aim of this paper is to answer this question. Our main result is an analogue of Kesten’s theorem in the critical case.

Theorem 1.1. Assume that hypotheses(H),M(δ)are satisfied and the lawμ_A ofA₁is aperiodic.Then there exists a probability measure Σ on the unit ball S^d−1⊂R^d and a strictly positive number C₊ such that the measures δ_(0,z−1)∗Gνconverge weakly onR^d\ {0}toC₊Σ⊗^da_a asz→ +∞,that is

z→+∞lim

R^dφ z⁻¹u

ν(du)=C₊

R⁺

S^d−1

φ(aw)Σ (dw)da a for every functionφ∈C_c(R^d\ {0}).

In particular,for everyα < β

zlim→∞ν

u: αz <|u|< βz

=C₊logβ

α. (1.3)

The first estimate of the behavior of the measureν at infinity was given by Babillot, Bougerol and Elie [1], who proved, ford=1 and under some nondegeneracy hypotheses, that for everyα < β

ν

(αz, βz]

∼log(β/α)·L(z) asz→ +∞,

whereLis a slowly varying function.

The second author recently proved [4] that the functionL(z)is in fact constant, but in a more restrictive setting:

besides the hypotheses stated above, one assumes in [4] thatd=1, the closed semigroup generated by the support of μis the whole groupGand the measureμ_Ais spread-out. Moreover nondegeneracy of the limiting constantC₊was proved there only in the particular case whenB₁≥εa.s.

When the measureμ is related to a differential operator, stronger results have been obtained recently in [6,8].

Namely, let{μ_t}be the one parameter semigroup of probability measures, whose infinitesimal generator is a second- order elliptic differential operator onR^d×R⁺. Then there exists a unique Radon measureνthat isμ_t-invariant, for anyt. Moreover,νhas a smooth densitymsuch that

m(zu)∼C(u)z^−d asz→ +∞

for some continuous nonzero functionConR^d\ {0}.

In this paper we also describe the behavior of the measure νin the case when the measureμ_A is periodic. This situation has been quite neglected up to now, also in the contracting case. Even if we cannot obtain the convergence of the measureνat infinity, we still have a good estimation of the asymptotic of the measure of the ball of radiusz.

Theorem 1.2. Suppose that hypotheses(H)andM(δ)are satisfied.If the measureμ_A is periodic of periodp,i.e.

suppμA = {e^np}n∈Z,then the family of measuresδ_(0,z−1)∗Gνis weakly compact and there exists a positive constant C₊such that

zlim→∞

R^dφ z⁻¹u

ν(du)=C₊

k∈Z

φ e^pk

for any functionφbelonging toT,the subset ofCC(R^d\{0})consisting of radial functions such that

k∈Zφ(ae^pk)=

k∈Zφ(e^pk)for alla∈R⁺.In particular ν

u: |u| ≤z

∼C₊

p logz asz→ +∞.

The case when B₁ is positive is also of a particular interest in applications and generally allows to use more powerful techniques. It will be the subject of a forthcoming paper, where we prove that the moment hypothesis of Theorem1.1can be weakened.

Let us mention that Theorems1.1and1.2have been applied recently to study tails of fixed points of the so-called smoothing transform in a boundary case. However in this context ‘boundary case’ concerns probability measures having infinite mean. See [5] for more details.

The structure of the paper is the following. First we estimate the behavior ofν at infinity in Section2under the very mild hypothesis (H). In Theorem2.1, we show thatδ_(0,z−1)∗Gν(K)is smaller thanC_KL(z), for all compact sets Kand a slowly varying functionL, i.e. the family of measuresδ_(0,z−1)∗Gν/L(z)is weakly compact. We also prove that

R^d(1+ |u|)^−γν(du) <∞for anyγ >0 and we obtain some invariance properties of the accumulation points of δ_(0,z−1)∗Gν/L(z).

Next, as in [4], we reduce the problem to study asymptotic behavior of positive solutions of the Poisson equation.

More precisely, letμbe the law of−logA1. The mean ofμis equal to 0 and given a positiveφ∈Cc(R^d\ {0})we define the function onR:

f_φ(x)=δ_(0,e−x)∗Gν(φ)=

R^dφ e^−xu

ν(du). (1.4)

Thenf_φcan be considered as a solution of the Poisson equation

μ∗Rfφ(x)=fφ(x)+ψφ(x), x∈R, (1.5)

for a specific functionψφ. The functionψφ possesses some regularity properties and it is easier to study thanfφ. The main problem can be formulated as follows: given a functionψφ describe the behavior at infinity of positive

solutions of the Poisson equation. An answer to this rather classical question was given by Port and Stone [19], under the hypothesis thatμis spread out. However, their methods, slightly developed, work for general centered measureμ onR. Namely we can construct a classF(μ)of functionsψ, with well defined potential that can be used to describe solutions of the corresponding Poisson equation. All the details will be figured out in Section3.

The next step is to prove that the function ψφ belongs to F(μ). A priori, this is not true for all function φ.

However we are able to construct special functions that have the good properties and allow to deduce our main results (Section4).

Finally in Section5we prove that the limit ofν(αz <|u|< βz)is strictly positive and the only hypothesis needed for this result is condition(H).

2. The upper bound

The goal of this section is to prove a preliminary estimate of the measureνat infinity. We prove that, under the very mild hypothesis (H) on the measureμ, the tail measure of a compact setδ_(0,z−1)∗ν(K)is bounded by a slowly varying functionL(z), that is a function onR⁺such that limz→+∞L(az)/L(z)=1 for alla >0. Such functions grow very slowly, namely they are smaller thanz^γ for anyγ >0, in a neighborhood of+∞.

Theorem 2.1. If hypothesis (H)is fulfilled,there exists a positive slowly varying function L onR^∗₊ such that the normalized family of measures onR^d\ {0}

δ_(0,z−1)∗Gν

L(z) (2.1)

is weakly compact forz≥1.Thus(1+ |x|)^−γ∈L¹(ν)for anyγ >0.Furthermore,all limit measuresηare nonnull and invariant under the action ofG(μ_A),the closed sub-group ofR^∗₊generated by the support ofμ_A,that is

δ(0,a)∗Gη=η ∀a∈G(μA).

This theorem is a partial generalization of Proposition 5.2 in [1].

We first prove that theμ-invariance ofν implies that the accumulation points of the tail are invariant under the action ofG(μA), namely we have

Lemma 2.2. Suppose that there exists a functionL(z)such that the family(2.1)is weakly compact whenzgoes to +∞,then the accumulation pointsηare invariant under the action ofG(μA).

Proof. Letηbe a limit measure along a sequence{z_n}and fix a functionφ∈C_c¹(R^d\ {0}). We claim that the function h(y)=δ_(0,y)∗Gη(φ)= lim

n→∞

δ_(0,z−1

n y)∗Gν(φ) L(z_n)

onR⁺isμ_A-superharmonic. Indeed, observe that for all(b, a)∈Gthere is a compact setK=K(b)and a constant Csuch that

φ

z⁻¹(au+b)

−φ

z⁻¹(au)< Cz⁻¹b1K

z⁻¹(au)

for allz >1 andu∈R^d. Then

nlim→∞

|δ_(0,z−1

n )∗Gδ_(b,a)∗Gν(φ)−δ_(0,z−1

n )∗Gδ_(0,a)∗Gν(φ)|

L(z_n) ≤ lim

n→∞

C|z⁻_n¹b|ν(a⁻¹z_nK) L(z_n)

≤Cη a⁻¹K

· lim

n→∞z⁻_n¹b=0,

hence

G

h(ay)μ_A(da)=

G nlim→∞

δ_(0,z−1

n y)∗Gδ_(0,a)∗Gν(φ)

L(zn) μ(dbda)

=

G nlim→∞

δ_(0,z−1

n y)∗Gδ_(b,a)∗Gν(φ)

L(z_n) μ(dbda)

≤ lim

n→∞

δ_(0,z−1

n y)∗Gμ∗Gν(φ)

L(z_n) by Fatou’s Lemma

= lim

n→∞

δ_(0,z−1

n y)∗Gν(φ)

L(z_n) =h(y).

Sinceh is positive and continuous, then by the Choquet–Deny theoremh(ay)=h(y)for everya∈G(μ_A), that is

δ(0,a)∗Gη(φ)=η(φ).

Proof of Theorem2.1. Step 1.The first step is to prove that the tail of the measureνsatisfies a quotient theorem.

Namely that there exists a family of bounded compactly supported functions s such thatδ_(0,z−1)∗Gν(s)is strictly positive for allz≥1 and that for every compact setKthere is a positive constantCK such that

δ_(0,z−1)∗Gν(K)≤C_Kδ_(0,z−1)∗Gν(s) ∀z≥1. (2.2)

In other words we show the quotient family _δ^{δ(0,z−1)∗Gν}

(0,z−1)∗Gν(s) is weakly compact.

The proof of this property relies only on the fact that, by hypothesis (H), the support ofμcontains at least two ele- ments, one contracting and the other delatingR^d. Let call themg₊=(b₊, a₊)andg₋=(b₋, a₋)witha₊>1> a₋.

Given two real numbersαandβwe consider the annulus C(α, β)= u∈R^d|α≤ |u| ≤β

.

Observe that for all(b, a)∈Gthe following implication holds u∈C

α+ |b|

a ,β− |b| a

⇒ au+b∈C(α, β).

Using this remark and the fact thatνis invariant with respect toμ^∗n, one can verify that δ_(0,z−1)∗Gν

C(α, β)

≥μ^∗n(U )ν

C

(b,a)max∈U

αz+ |b| a , min

(b,a)∈U

βz− |b| a

(2.3) for anyUsubset ofGandn∈N.

First we prove that there exists a sufficiently largeR >0 such thatδ_(0,z−1)∗Gν(C(1/R, R))is strictly positive for allz≥1.

Fixz≥1 and taken∈Nsuch thata₊ⁿ⁻¹≤z≤a₊ⁿ. Clearly, ifgⁿ=(b(gⁿ), a(gⁿ))is thenth power of an element g=(b, a)∈Gthen

a gⁿ

=aⁿ and b gⁿ

=

n−1

i=0

aⁱb=aⁿ−1 a−1 b.

Consider theδ-neighborhood ofgⁿ Uδ

gⁿ

= (b, a)∈G|e^−δ< a⁻¹a gⁿ

<e^δandb−b

gⁿ< δ .

Observe thatμ^∗n(U_δ(g₊ⁿ)) >0 for allδ >0 and for(b, a)∈U_δ(gⁿ₊) z/R+ |b|

a ≤aⁿ₊/R+ |b(g₊ⁿ)| +δ e^−δa₊ⁿ ≤e^δ

1

R + |b₊| a₊−1+δ

=:α_R, Rz− |b|

a ≥Raⁿ₊⁻¹− |b(gⁿ₊)| −δ e^δa₊ⁿ ≥e^−δ

R

a₊− |b₊| a₊−1−δ

=:β_R.

Sinceνis a Radon measure with the infinite mass, its support cannot be compact. Thus, for a fixedδ, there exits a sufficiently largeRsuch that:ν(C(αR, βR)) >0.Then by (2.3):

δ_(0,z−1)∗Gν

C(1/R, R)

≥μ^∗n U_δ

g₊ⁿ ν

C(α_R, β_R)

>0 (2.4)

for allz≥1.

ForR >2 consider the compact setsK_±ⁿ =C(2a_±⁻ⁿ/R, a⁻_±ⁿR/2).Notice that forδ <log(4/3),(b, a)∈Uδ(g_±ⁿ) andz > zⁿ_±:=2R(|b(gⁿ_±)| +δ):

z/R+ |b|

a ≤ze^δ1/R+z⁻¹(|b(gⁿ_±)| +δ)

a_±ⁿ ≤z2a_±⁻ⁿ R

e^δ1+z⁻¹R(|b(g_±ⁿ)| +δ) 2

≤z2a⁻_±ⁿ R , zR− |b|

a ≥ze^−δR−z⁻¹(|b(g_±ⁿ)| +δ)

a_±ⁿ ≥za_±⁻ⁿR 2 ·2e^−δ

1−z⁻¹|b(gⁿ_±)| +δ R

≥za_±⁻ⁿR 2 . Thus by (2.3):

δ_(0,z−1)∗Gν

C(1/R, R)

≥μ^∗n U_δ

g_±ⁿ ν

C

z2a⁻_±ⁿ/R, za⁻_±ⁿR/2

=C_K⁻¹n

±δ_(0,z−1)∗Gν K_±ⁿ

for allz > zⁿ_±. Sinceδ_(0,z−1)∗Gν(C(1/R, R)) >0, the above inequality holds in fact for allz≥1, possibly with a bigger constantC_Kand sufficiently largeR. We may assume thatR >2 max{a₊,1/a₋}, then the family of setsK_±ⁿ coversR^d\ {0}.

Finally notice, that every functions∈Cc(R^d\ {0})and such thats(u)≥1C(1/R,R)(u)satisfies (2.2). Indeed, letK be a generic compact set inR^d\ {0}covered by a finite number of compacts{K_i}i∈Iof the typeK_±ⁿ, then

δ_(0,z−1)∗Gν(K)≤

i∈I

δ_(0,z−1)∗ν(K_i)≤

|I|max

i∈I C_Ki

δ_(0,z−1)∗Gν(s) for allz≥1.

Step 2.LetL(z)=δ_(0,z−1)∗Gν(s), so thatδ_(0,z−1)∗Gν/L(z)is weakly compact whenzgoes to+∞. It remains to prove thatLis a slowly varying function. Fixa∈G(μ_A)and observe that

L(az)

L(z) =δ_(0,a−1)∗Gδ_(0,z−1)∗Gν(s)

L(z) .

Let{z_n}n∈Nbe a sequence such thatδ(0, z⁻_n¹)∗Gν/L(z_n)converges to some limit measureη. Then by invariance of the limit measure

nlim→∞

L(az_n)

L(z_n) =δ_(0,a−1)∗Gη(s)=η(s)=1.

Since for any sequence, there exists a subsequence such that the conclusion above holds, ifμ_Aaperiodic,Lis slowly varying and the proof is completed.

Ifμ_Ais periodic, that isG(μ_A)= e^p, take any continuous compactly supported functions₀≥1C(1/R,R), i.e. a function satisfying (2.2), and define

s(u)=

R^∗₊1_[e^−p,e^p)(t)s0(u/t )dt

t . (2.5)

An easy argument shows that alsos is inC_c(R^d\ {0})and it is bigger than some multiple of 1C(1/R,R). We claim thatδ_(0,a−1)∗Gη(s)=η(s)for alla∈R^∗₊and not only fora∈G(μ_A)(and thusL(z)is slowly varying). In fact, let e^Kp∈G(μ_A)such that e^Kp> ae^pthen

δ_(0,a−1)∗Gη(s)=

R^d

R^∗₊1_[ae^−p,ae^p)(t)s0(u/t )dt t η(du)

=

R^d

R^∗₊

1_[ae−p,e^Kp)(t)−1_[aep,e^Kp)(t)

s₀(u/t )dt t η(du)

=

R^d

R^∗₊

1_[ae−Kp,e^p)(t)s₀(u/t )−1_[ae−Kp,e^−p)(t)s₀(u/t ) η(du)dt

t

=

R^d

R^∗₊1_[e^−p,e^p)(t)s0(u/t )dt

t η(du)=η(s),

sinceηisG(μ_A)-invariant.

3. Recurrent potential kernel and solutions of the Poisson equation for general probability measures

As it has been observed in theIntroduction, to understand the asymptotic behavior of the measureνone has to consider the function

f_φ(x)=

R^dφ ue^−x

ν(du) that is a solution of the Poisson equation

μ∗_Rf =f +ψ (3.1)

for a peculiar choice of the functionψ=ψ_φ=μ∗_Rf_φ−f_φ.

Studying solutions of such equation for a centered probability measure onRis a classical problem. Port and Stone in their papers [18,19] give an explicit formula describing all bounded from below solutions of (3.1) in terms of the recurrent potential kernelAof the functionψ. However, they suppose either that the measure is spread-out or, if not, that the Fourier transform ofψ is compactly supported. This second condition is too restrictive in our setting: such functions decay too slowly and the corresponding functionφwould not be aν-integrable. For this reason, the results of the previous paper [4] on the decay of the measureν were obtained under the hypothesis thatμis spread out. To avoid this restriction we need to generalize the technics used by Port and Stone [18] to a larger class of functions F(μ)associated to an arbitrary measureμ.

Let μ be a centered probability measure on R with finite second moment σ²=

Rx²μ(dx). We denote by μ(θ )=

Re^ixθμ(dx) its Fourier transform and given a function ψ ∈L¹(R) we define its Fourier transform by ψ (θ ) =

Re^ixθψ (x)dx.

LetF(μ)be the class of functionsψ, such that (1) ψ,x²ψandψare elements ofL¹(R),

(2) the function ₁^ψ(_{−μ(θ )}^{−θ )} is dθ-integrable outside any neighborhood of zero.

The second condition is satisfied e.g. when the measure μis aperiodic and ψhas a compact support or when the measure μ is spread-out (since is this case sup_|θ|>a|μ(θ )|<1). Thus, the set F(μ) contains the set of functions on which Port and Stone define the recurrent potential but it is, in many cases, bigger. In particular we will see in Lemma3.3, that if the measure has an exponential moment, thenF(μ) always contains some functions that decay exponentially. That will be sufficient to prove our main theorem in the next section.

LetJ (ψ )=

Rψ (x)dxandK(ψ )=

Rxψ (x)dx, then we have:

Theorem 3.1. Assume thatψ, g∈F(μ),gis positive and such thatJ (g)=1.Then:

• The recurrent potential Aψ (x):=lim

λ1

J (ψ )

∞ n=0

λⁿμ^∗n∗g(0)−^∞

n=0

λⁿμ^∗n∗ψ (x)

is a well defined continuous function.

• Aψ is a solution of the Poisson equation(3.1).

• IfJ (ψ )≥0,thenAψis bounded from below and

x→±∞lim Aψ (x)

x = ±σ⁻²J (ψ ). (3.2)

• IfJ (ψ )=0,thenAψ is bounded and has a limit at infinity

x→±∞lim Aψ (x)= ∓σ⁻²K(ψ ). (3.3)

The proof of this result is rather technical and follows the ideas of [18] and [19]. A sketch of the proof is proposed in theAppendixfor reader convenience.

A direct consequence of the previous theorem, is the following characterization of the bounded solutions of the Poisson equation:

Corollary 3.2. IfJ (ψ )=0, then every continuous solution of the Poisson equation bounded from below is of the form

f=Aψ+h,

wherehis constant ifμis aperiodic,and it is periodic of periodpif the support ofμis contained inpZ.Thus every continuous solution of the Poisson equation is bounded and the limit off (x)exists whenxgoes to+∞andx∈G(μ).

Conversely if there exists a bounded solution of the Poisson equation,then Aψ is bounded and J (ψ )=0. In particular the first part of corollary is valid.

Proof. LetJ (ψ )=0 and assume thatf is a continuous solution of the Poisson equation. Since μ∗f=f+ψ and μ∗Aψ=Aψ+ψ,

the function h=f −Aψ is μ-harmonic. It is bounded from below because both−Aψ andf are bounded from below. Therefore by the Choquet–Deny theorem [9],h(x+y)=h(x)for ally in the closed subgroup generated by the support ofμ.

Conversely, suppose that there exists a bounded solutionf0of the Poisson equation. ThenAψ−f0isμ-harmonic and bounded from below, and so the Choquet–Deny theorem implies thatAψ is bounded. Thus

xlim→∞

Aψ (x) x =0

and by (3.2), we deduceJ (ψ )=0.

As announced we need to construct a class of functions inF(μ)that will be used later on and that have the same type of decay at infinity asμ:

Lemma 3.3. LetY be a random variable with the lawμ,then the function r(x)=E

|Y +x| − |x|

is nonnegative and

r(θ )=C·μ(−θ )−1 θ²

forθ=0.Moreover ifE[e^δY+e^−δY]<∞,thenr(x)≤Ce^−δ1|x|forδ1< δ.

Hence r belongs toF(μ)and for every function ζ ∈L¹(R)such thatx²ζ is integrable the convolutionr∗_Rζ belongs toF(μ).

Proof. Observe that, sinceEY=0, forx≥0 we can write r(x)=E

(Y+x)−2(Y+x)1Y+x≤0−x

= −2E

(Y+x)1Y+x≤0

.

Proceeding analogously forx <0, we obtain r(x)=

−2E

(Y+x)1Y+x≤0

forx≥0, E

(Y+x)1Y+x>0

forx <0. (3.4)

Thus the functionris nonnegative.

The Fourier transform ofx can be computed in the sense of distributions. Leta(x)= |x| and observe thatr= (μ−δ₀)∗a. Thena(θ )=_θ^C2, hencer(θ )=C·^μ(−_θ^{θ )}2⁻¹.

To estimate the decay ofrwe use (3.4). Forx≥0, we write r(x)=2E

|Y+x|1Y+x≤0

=2

x+y≤0

|x+y|μ(dy)≤2

R|x+y|e^−δ0(x+y)μ(dy)≤Ce^−δ1x for some constantsδ1< δ0< δ.

Finally, ifψ=r∗ζ withζ andx²ζ inL¹(R), then it is easily checked that bothψandx²ψare integrable. Since ψ (θ ) =r(θ )ζ (θ )=C^{μ(−θ )−1}

θ² ζ (θ )andζ vanishes at infinity,ψ∈F(μ).

4. Proofs of Theorems1.1and1.2– Existence of the limit

Our aim is to apply the results of Section3and for this purpose we need to show thatψ_φ is sufficiently integrable.

The upper bound of the tail ofνgiven in Section2will guarantee integrability for positivex. To control the function for x negative we need to perturb slightly the measuresμandνin order to have more integrability near 0. This is included in the following lemma proved in [4] (Lemma4.1).

Lemma 4.1. For allx₀∈R^dthe translated measureν₀=δ_x0∗_R^dνis the unique invariant measure ofμ₀=δ_(x0,1)∗G

μ∗Gδ_(−x0,1)and it has the same behavior asνat infinity,that is:

x→+∞lim

R^dφ ue^−x

ν(du)−

R^dφ ue^−x

ν₀(du)

=0

for every functionφ∈C_c¹(R^d\ {0}).Furthermore there isx₀∈R^dsuch that the measureν₀satisfies

R^d

1

|u|^γν₀(du) <∞ for allγ∈(0,1). (4.1)

Using (4.1) we can guarantee that the functionψφdecays quickly at infinity, as it is proved in the following lemma.

Lemma 4.2. Assume that hypotheses(H)and M(δ) are satisfied.Furthermore assume that the function |u|^−γ is ν(du)-integrable for allγ∈(0,1).Letφbe a continuous function onR^dsuch that|φ(u)| ≤C(1+ |u|)^−β for some β, C >0.Thenf_φandμ∗f_φare well defined and continuous.Furthermore ifφis Lipschitz,then

R

G

R^d

φ

e^−x(au+b)

−φ

e^−xauν(du)μ(dbda)dx <∞ (4.2)

and

ψ_φ(x)≤Ce^−ζ|x| forζ <min{δ/4, β,1}. Proof. Ifζ <min{β,1}, then

f_φ(x)≤

R^d

φ

e^−xuν(du)≤

R^d

C

e^{−ζ x}|u|^ζν(du)≤Ce^{ζ x}. If we suppose alsoζ≤δ, we have that

μ∗fφ(x)≤

R

fφ(x+y)μ(dy)≤Ce^{ζ x}

R⁺a^−ζμA(da)≤Ce^{ζ x}.

Thus ψ_φ=μ∗_Rf_φ−f_φ is well defined, continuous and |ψ_φ(x)| ≤Ce^{ζ x}, that gives the required estimates for negativex. In order to prove (4.2) we divide the integral into two parts. For negativex we use the estimates given above:

₀

−∞

G

R^d

φ

e^−x(au+b)

−φ

e^−xauν(du)μ(dbda)dx

≤ 0

−∞

R^d

φ

e^−xuν(du)dx+ 0

−∞

G

R^d

φ

e^−xauν(du)μ(dbda)dx

≤ ₀

−∞

f_|φ|(x)dx+ ₀

−∞

μ∗f_|φ|(x)dx <∞.

To estimate the integral of|φ(e^−xau)−φ(e^−x(au+b))|forxpositive, we use the Lipschitz property ofφto obtain the following inequality for 0≤θ≤1

φ(s)−φ(r)≤C|s−r|^θ max

ξ∈{|s|,|r|}

1 (1+ξ )^{β(1−θ )}.

Again we divide the integral into two parts. First we consider the integral over the set where|au+b| ≥ ¹₂a|u|. We chooseθ <min{δ/2,1},γ <min{θ/2, β(1−θ )}. Then, in view ofM(δ), we have

|au+b|≥(1/2)|au|

φ e^−xau

−φ

e^−x(au+b)μ(dbda)ν(du)

≤

G

R^d

C|e^−xb|^θ

(1+ |e^−xau|)^{β(1−θ )}ν(du)μ(dbda)≤

G

R^d

C|e^−xb|^θ

|e^−xau|^γν(du)μ(dbda)

≤Ce^{−(θ−γ )x}

G

|b|^θ|a|^−γμ(dbda)

R^d|u|^−γν(du)

≤Ce^{−(θ−γ )x}

G

|b|^2θ+ |a|^−2γ

μ(dbda)≤Ce^{−γ x}.

If|au+b|<¹₂a|u|then|u| ≤^2|_a^b|. Therefore choosingθas above andγ <^δ₂−θ, in view of Proposition2.1, for the remaining part of the integral, applying againM(δ), we have

|au+b|≤(1/2)|au|

φ e^−xau

−φ

e^−x(au+b)μ(dbda)ν(du)

≤ |u|≤2|b|/a

e^−xb^θν(du)μ(dbda)

≤C

G

e^−xb^θ 1+2|b|

a γ

μ(dbda)≤Ce^{−θ x}

G

|b|^θ

1+2|b| a

γ

μ(dbda)

≤Ce^{−θ x}

G

|b|^θ+ |b|^{2(θ+γ )}+a^−2γ

μ(dbda)≤Ce^{−θ x}. That proves (4.2) and finally

ψ_φ(x)≤

G

R^d

φ e^−xau

−φ

e^−x(au+b)ν(du)μ(dbda) < Ce^−ζ|x|

forζ <min{δ/4, β,1}.

The following proposition contains key arguments of the proof of our main results.

Proposition 4.3. Assume that hypothesis(H)andM(δ)are satisfied.The family of measuresδ_(0,e−x)∗Gνis relatively compact in the weak topology onR^d\ {0}.

Suppose thatrbelongs toF(μ)and has an exponential decay.Letζbe a nonnegative Lipschitz function onR^d\{0}

such thatζ (u)≤e^{−γ|log|u||}forγ >0and set φ(u):=

Rr(t )ζ e^tu

dt. (4.3)

Then,the limit

x→+∞lim

R^dφ ue^−x

ν(du)

exists,it is finite and equal toη(φ)for any limit measureη.

Proof. Step 1.First we suppose thatμsatisfies (4.1). We are going to show that for functions of type (4.3) the limit

x→+∞lim

R^dφ ue^−x

ν(du)=T (φ):= −2σ⁻²K(ψ_φ)

exists and it is finite. To do this we will prove thatψ_φis an element ofF(μ)andJ (ψ_φ)=0. Thus, by Corollary3.2, the functionf_φ(x)=

R^dφ(ue^−x)ν(du)is the solution of the corresponding Poisson equation, it is bounded and it has a limit whenxconverge to+∞.

First observe that by Lemma3.3(in view ofM(δ) its assumptions are satisfied), forβ <min{δ, γ}, we have φ(u)≤C

Re^−β|t|e^{−γ|t+log|u||}dt≤C

Re^{−β(|t−log|u||)}e^−γ|t|dt

≤C

Re^{−β(−|t|+|log|u||)}e^−γ|t|dt=Ce^{−β|log|u||}.