Stable laws and spectral gap properties for affine random walks

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www.imstat.org/aihp 2015, Vol. 51, No. 1, 319–348

DOI:10.1214/13-AIHP566

Stable laws and spectral gap properties for affine random walks ¹

Zhiqiang Gao

^a

, Yves Guivarc’h

^b

and Émile Le Page

^c

aSchool of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, 100875 Beijing, China.

E-mail:gaozq@bnu.edu.cn

bIRMAR, CNRS Rennes I Université de Rennes-1, Campus de Beaulieu, 35042 Rennes Cedex, France. E-mail:yves.guivarch@univ-rennes1.fr cLaboratoire de Mathématiques de Bretagne Atlantique, UMR CNRS 6205, Université de Bretagne Sud, Campus de Tohannic, BP 573, 56017

Vannes, France. E-mail:emile.le-page@univ-ubs.fr

Received 22 October 2012; revised 17 April 2013; accepted 5 May 2013

Abstract. We consider a general multidimensional affine recursion with corresponding Markov operator P and a uniqueP- stationary measure. We show spectral gap properties on Hölder spaces for the corresponding Fourier operators and we deduce convergence to stable laws for the Birkhoff sums along the recursion. The parameters of the stable laws are expressed in terms of basic quantities depending essentially on the matricial multiplicative part ofP. Spectral gap properties ofP and homogeneity at infinity of theP-stationary measure play an important role in the proofs.

Résumé. Nous considérons une relation de récurrence affine multidimensionelle à coefficients aléatoires et nous supposons que l’opérateur de MarkovP associé a une unique probabilité stationnaire. Nous montrons la propriété de trou spectral pour les opérateurs de Fourier correspondants sur certains espaces de fonctions Holdériennes, et nous en déduisons la convergence vers des lois stables pour les sommes de Birkhoff le long des trajectoires. Les paramètres des lois stables obtenues s’expriment à l’aide de quantités dépendant essentiellement de la partie multiplicative deP. La preuve est basée sur les propriétés spectrales de l’opérateur de Markov associé et l’homogénéité à l’infini de la mesure stationnaire.

MSC:Primary 60B20; secondary 60E07; 60F05 Keywords:Stable laws; Spectral gap; Affine recursions

1. Introduction and main results

We consider the vector space V =R^d endowed with the scalar product x, y =_d

i=1x_iy_i and the norm |x| = (_d

i=1|x_i|²)^1/2. We denote byH=V Gthe affine group ofV, withG=GL(d,R), i.e. the set of mapshof the form hx=gx+b (b∈V , g∈G). Letμ be a probability measure onH andx∈V. We denote by Pthe product measureμ^⊗NonΩ=H^Nand we consider the recurrence relation with random coefficients:

X^x₀=x, X^x_n=M_nX^x_n₋₁+Q_n (n≥1), (1)

where(Q_n, M_n)∈H are i.i.d. random variables with generic copy(Q, M)and with lawμ. Letμ¯ be the projection ofμonG, i.e. the law ofM, and let[suppμ¯]be the closed subsemigroup generated by the support ofμ. We will¯

1Partially supported by National Nature Science Foundation of China (No. 11101039, 11271045) and the Research Fund for the Doctoral Program of Higher Education (Grant No. 20100003110004).

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denote byP the corresponding Markov operator onC_b(V ), the space of continuous bounded functions onV: P ϕ(x)=

ϕ(gx+b)dμ(h), ϕ∈C_b(V ).

We observe that ifM_n=Id (resp.Q_n=0), thenX^x_n is an additive (resp. multiplicative) random walk onV (resp.

V \ {0}) (cf. [12,23,36]). Basic aspects of these special processes continue to hold in the general case of X^x_n, and give a heuristic guide for the study of the affine random walkX^x_n. On the other hand, independently of any density condition forμ, the conjunction of these two different processes give rise to new properties, in particular spectral gap properties forP (cf. [5,21]) and homogeneity at infinity for theP-stationary measure (cf. [6,17,22]).

For a positive Radon measureρ onV we denoteρP the new measure obtained fromρby the dual action ofP. Our hypothesis will imply that the above recursion (1) has a unique stationary measureηwhich satisfiesηP=ηand has an unbounded support. The probability measureηis the limit distribution ofX^x_n. A remarkable property ofηis its

“homogeneity at infinity,” a property which was first observed in [31] for the tails ofη, extended to the general case in [34] and further developed in [1,6,13], under special conditions. See [17] for a survey of [34] as well for a precise description of the homogeneity property ofη, proved in a special case in [6] and in a generic case in [22].

In this paper we are interested in the limit behavior of the sumS_n^x=_n

k=0X_k^x, conveniently normalized. Ford=1 this question is connected with the slow diffusion behavior of a simple random walk onZin a random medium (see [33,41]). The similar problem for a finitely supported random walk onZin a random medium is connected to the study of a recurrence relation of the form (1) (see [14,26]). More generally, the Eq. (1) is of fundamental interest for the study of generalized autoregressive processes (cf. [4,31]). In particular Eq. (1) is a basic model in collective risk theory ([13]); in the context of extreme value theory, the corresponding convergence problem for normalized sample autocorrelations of a GARCH model is considered in [37].

Ford=1, and under aperiodicity conditions, the limit behavior ofS_n^x is described in [21]. Ford >1, it turns out that, in the generic case considered below, the limits are stable laws of general type and that the multiplicative part of the recursion plays a dominant role in the asymptotics. Ford≥1, in the case whereM_n takes values in the similarity group ofV, the limit behavior ofS^x_n is described in [5]; the homogeneity at infinity result of [6] plays an essential role in the proof, and [5] contains a detailed description of the limit laws which turned out to be semi-stable in the sense of P. Lévy (see [36], p. 204). For other situations where stable laws appear naturally in limits theorems in sums of non i.i.d. random variables we refer to ([36], pp. 321–323) and [2]. Here we consider relation (1) in the case where[suppμ¯]is “large,” a case which is generic and opposite to the case of [5]. We will need the detailed information on the stationary lawηofP given in [22] and summarised in Theorem2.4below; also as in [5,21], a basic role will be played by the spectral properties of the Fourier operatorsPv (v∈R)defined byPvϕ=P (Xvϕ), whereXv(x)=eⁱ^v,x. Furthermore, the homogeneity at infinity of η plays an essential role and implies that the dominant eigenvalue ofP_v has an asymptotic expansion at 0 in terms of fractional powers of|v|. These properties allow us to develop a detailed analysis and to prove limit theorems. More generally, it turns out that, in the context of random walks associated with nonabelian semigroup actions, spectral gap properties are valid in certain functional spaces for large classes of random walks. Usually, such properties are studied in the context of the so called “Doeblin condition” (see [1,9] for example). Here instead, our study is based on the Ionescu-Tulcea and Marinescu theorem ([28]). This allows us to get spectral gap properties without density condition onμor μ. See [7,8,10,11,15,16,19]¯ for different classes of situations where analogous ideas are used. HereV can be considered as a boundary (see [12]) for the random walk onH defined byμ, and we will use spectral gap properties forP_v(v∈V )in Banach spaces of Hölder functions with slow growth at infinity. In [8] and [11] the relevant spaces areL²-spaces, while in [7,10,16], they are of mixed type. This type of analysis is not restricted to homogeneous spaces of Lie groups as shown in [38]

for certain classes of Lipschitz maps instead of affine maps. Here we follow the general line of [5,21]. With respect to these papers, new arguments are needed for the analysis of relation (1), in the generic case considered below (see [22]).

The asymptotics of products of random matrices (see [3,18,23]) will play an important role, and we need to give corresponding notations. We say that a semigroupΓ ⊂Gisstrongly irreducibleif no finite union of proper subspaces ofV isΓ-invariant. Also we say thatg∈Gisproximalifghas a dominant eigenvalueλ(g)∈Rwhich is the unique eigenvalue ofgsuch that|λ(g)| =limn→∞|gⁿ|^1/nwhere|g| =sup{|gx|: |x| =1}. We say thatΓ satisfies condition i-pifΓ is strongly irreducible and contains a proximal elementγ. It is proved in [39] that conditioni-pforΓ and its Zariski closureZc(Γ )are equivalent. SinceZc(Γ )is a closed Lie subgroup ofGwith a finite number of connected

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components, conditioni-pcan be checked in examples (see Section5for some examples). Under this condition, the limit setL(Γ )⊂P^d⁻¹is the uniqueΓ-minimal subset of the projective spaceP^d⁻¹andL(Γ )is the closure of the set of attracting fixed points of the proximal elements inΓ.

Fors≥0, we denote κ(s)= lim

n→∞

E|M_n· · ·M₁|^s1/n

, s_∞=sup

s≥0;κ(s) <∞ .

Forg∈G, we writev(g)=sup(|g|,|g⁻¹|). IfE(logv(M)) <+∞, we know that the Lyapunov exponent L(μ)¯ = lim

n→∞

1 nE

log|M_n· · ·M₁|

is well defined,L(μ)¯ =κ(0₊)ifs_∞>0. If conditioni-pis satisfied ands_∞>0, then logκ(s)is strictly convex on [0, s_∞), hence if lims→s_∞κ(s) >1, there exists a uniqueα∈(0, s_∞)withκ(α)=1.

Our hypothesis here is the following conditionC(see [22]):

C₁ [suppμ¯]satisfies conditioni-p, C₂ s_∞>0,L(μ) <¯ 0, lims→s_∞κ(s) >1, C₃ E(v(M)^α⁺^δ+ |Q|^α⁺^δ) <∞ for someδ >0, C₄ suppμhas no fixed point inV.

ConditionCwill be assumed in our results (compare with condition (H) of [5]), except if the contrary is specified.

We observe that conditioni-pfor[suppρ]is valid on an open dense set in weak topology of measuresρonG. It follows that conditionC is open in the weak topology of probability measures onH. ConditionsC₁andC₃are used to prove homogeneity at infinity ofη, a property which depends on the spectral gap properties of twisted convolution operators defined byμ¯ on the projective space ofV (cf. [22]). ConditionC₂plays the basic role in the homogeneity at infinity ofη.

A real numbert∈Rdefines a dilation onV which is denoted byv→t.v, and we extend this notation to the action ofRon measures onV. A Radon measureρonV is said to beα-homogeneous if for anyt >0,t.ρ=t^αρ.

LetP be the Markov operator onV defined by P ϕ(v)=

ϕ(gv)dμ(g),¯ ifϕ∈C_b(V ).

We observe thatP can be interpreted as the linearisation ofP at infinity. We denote by^sthes-homogeneous measure onR^∗₊defined by^s(dt )=_ts^dt+1. It is proved in Theorem C of [22] that ifd >1 and conditionCis valid, there exists c >0 and a probability measureσ_α on the unit sphereS^d⁻¹ such that the following vague convergence is valid on V \ {0}:

t→lim0₊t⁻^α(t.η)=cσ_α⊗^α=Λ. (2)

HereΛis defined by the above convergence, isα-homogeneous, and we haveΛP=Λ. We observe that the equation ΛP =Λis a limiting form of the stationarity equationηP=η. The proof is based on the general renewal theorem of [32] and on the spectral gap property of the operator on the projective space defined by twisted convolution withμ¯ (see [20,22]).

More generally, ifηis a probability measure such that the above convergence (2) is valid, we will say thatηis α-homogeneous at infinity. A probabilityηonV is said to be stable if for every integernthere exists a similarityh_n of the formhn(x)=anx+bn(an>0, bn∈V )such that thenth convolution power ofηis the push forward ofηby h_n. Ifa_n=n^1/α, we say thatηisα-stable.

Due to Theorem C of [22], ifsuppμ¯ has no invariant convex cone inV, thenΛis symmetric andσα⊗^α is the unique Radon measure defined by the following conditions:

σ_α is a probability measure onS^d⁻¹, σα⊗^α

P =σα⊗^α, t.

σα⊗^α

=t^α

σα⊗^α

, for allt >0.

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See [22] for more detail. In Section5below we give information onσ_α and examples of the typical situations which can occur. In any caseΛgives zero measure to any affine subspace, the projection ofσ_αon the projective spaceP^d⁻¹ is uniquely defined by the above condition and its support is equal to the limit setL([suppμ¯])inP^d⁻¹.

We will writeg^∗ for the transposed map ofg∈G,μ¯^∗for the push-forward ofμ¯ byg→g^∗. Also forx∈V, we writex^∗for the linear formx^∗(y)= x, y. The exponential eⁱ^x,y will be denoted byXx(y)and the characteristic function of a probability measureπonV will be defined by

π (x)=

VXx(y)dπ(y).

Coming back to the affine situation, we will write m=

xdη(x), m_α=κ(α₋).

The calculation of the limit law ofS_n^xwill involve considering the companion recursion:

W₀=0, W_n=M_n^∗(W_n₋₁+v), (3)

wherev∈V\ {0}is a fixed vector. We will denote byT_vthe corresponding transition operator, i.e.

T_v(ϕ)(x)=

ϕ

g^∗(x+v) dμ(g).¯

Then as above, the unique stationary measureη_vofT_vsatisfies the weak convergence onV\ {0}:

t→lim0₊t⁻^α(t.η_v)=Δ_v=0, (4)

andη_v, Δ_vsatisfy

ηt v=t.ηv, Δt v=t.Δv fort∈R^∗, ΔvP_∗=Δv, Δt v=t^αΔv fort >0, where, as above,P_∗is associated withμ¯^∗.

In order to state our first main result, we need to define a kind of Fourier transformΛofΛ. Ifα∈(0,2], we define Λas follows:

Λ(y)= Xy(x)−1

dΛ(x), if 0< α <1, Λ(y)= Xy(x)−1−i x, y

1+ |x, y|²

dΛ(x), ifα=1, Λ(y)= Xy(x)−1−ix, y

dΛ(x), if 1< α <2, Λ(y)= −1

4

y, x²dσ2(x), ifα=2.

The function exp(Λ)is the Fourier transform of the limit law of the normalized sum ofη-distributed i.i.d. random variables andΛsatisfies

Λ(ty)=t^αΛ(y) fort >0, P_∗Λ=Λ, and ReΛ(y) <0 fory=0.

We will use also the functionΛ¹defined byΛ¹(y)=Λ(y)1¯ _[1,∞)(|y|), wherey=y/|y|denotes the projection of y∈V \ {0}onS^d⁻¹.

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The Fourier transform of the limit law ofS_n^xforα∈(0,2]will be shown to be equal to e^C^α^(v)=Φ_α(v)where the functionC_α(v)is defined by

C_α(v)=

αm_αΔ_v(Λ¹), ifα∈(0,1)∪(1,2];

m1Δv(Λ¹)+iγ (v), ifα=1, (5)

with

γ (v)=

y+v, x

1+ |y+v, x|²− v, x

1+ |x|² − y, x 1+ |y, x|²

dΛ(x)dηv(y). (6)

(See the proof of Proposition2.6.) We have that fort >0

Cα(tv)=t^αCα(v) ifα=1, and C1(tv)=t C1(v)+i v, β(t )

, whereβ(t)=

( ^{t x}

1+|t x|² −₁_+|^{t x}_x_|2)dΛ(x). Hence e^C^α^(v) is the Fourier transform of an infinitely divisible probability measure which belongs to anα-stable convolution semigroup (see [27,29,40]).

Ifα >2, the following covariance formqofηwill enter in the formulas below, q(x, y)=

x, ξ−my, ξ−mdη(ξ ).

We will writez=E(M)for the averaged operator ofMifα >1. One sees easily that the operatorEMonV exists and has spectral radius less thanκ(α)=1, hence in particularI−z^∗is invertible.

We have the following limit theorem for the partial sumsS_n^x.

Theorem 1.1. Assume that the probability measureμonH=VGsatisfies conditionCabove.Then ifdimV >1, we have for anyx∈V,

(1) Ifα >2, √¹

n(S_n^x−nm)converges in law to the normal law onV with the Fourier transform Φ₂₊(v)=exp

−q(v, v)/2−q v,

I−z^∗₋1

z^∗v . (2) Ifα∈(0,2),lettn=n⁻^1/αand

d_n=

⎧⎨

⎩

0, α∈(0,1); nδ(t_n), α=1;

nt_nm, α∈(1,2), withδ(t)=

V t x

1+|t x|²dη(x)fort >0.

Then(t_nS_n^x−d_n)converges in law to theα-stable law with the Fourier transformΦ_α(v)=exp(Cα(v)),withC_α(v) given above.

Furthermore ifα=1,then for some constantK>0, δ(t)≤

K|t||log|t||, for|t| ≤¹₂; K|t|, for|t|>¹₂.

(3) Ifα=2,then √ ¹

nlogn(S_n^x−nm)converges in law to the normal law with Fourier transform Φ2(v)=exp

C2(v)

, whereC2(v)= −1

4 v, w2

+2v, wηv

w^∗ dσ2(w).

(4) In all cases,the limit laws are fully nondegenerate.

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The proof of Theorem1.1is based on the method of characteristic functions. The characteristic function ofS_n^xcan be expressed in terms of iterates of the Fourier operatorP_v defined above. This operator acts as a bounded operator on a certain Banach spaceBθ,ε,λ(defined below) of unbounded functions onV and has “nice” spectral properties on Bθ,ε,λ. MoreoverP₀=P and the spectral properties ofP_v allow to control the perturbationP_v ofP as well as its dominant eigenvaluek(v). Theorem1.1follows from the asymptotic expansion ofk(v)atv=0, which is based on the homogeneity at infinity ofηandη_v. The spectral properties ofP_vfollow from a theorem of Ionescu-Tulcea and Marinescu based on certain functional inequalities proved below which are consequences of the conditionL(μ) <¯ 0.

We denote byr(U )the spectral radius of a bounded linear operatorU. The spectral properties ofPvare described by the:

Theorem 1.2. Ifv∈V,the operatorPvonBθ,ε,λdefined byPvf=P (Xvf )has the following properties:

(1) P_vis a bounded operator with spectral radius at most1, (2) Ifv=0,r(P_v) <1,

(3) Ifv=0andπ0is the projection onC1defined byπ0ϕ=η(ϕ)1,we have for anyϕ∈Bθ,ε,λ: P₀ϕ=π₀ϕ+Qϕ,

whereQπ₀=π₀Q=0andr(Q) <1.

(4) Ifv is small,P_v has a unique eigenvaluek(v)with|k(v)| =r(P_v).Furthermore there exists a one dimensional projectionπ_vand a bounded operatorQvsuch thatQvπ_v=π_vQv=0,r(Qv) <|k(v)|and

P_vϕ=k(v)π_vϕ+Qvϕ, for anyϕ∈Bθ,ε,λ. Furthermorek(v), πv,Qvdepend continuously onv.

These spectral properties will allow us to reduce the study of the iterated operatorP_vⁿto the study of its dominant eigenvaluekⁿ(v); hencek(v)plays here the role of a characteristic function for the convolution operatorP defined by μonCb(V ).

The asymptotic behavior ofk(v)atv=0 is given by the Theorem 1.3. Letv∈V\ {0}and letCα(v)be given by(5).

(1) If0< α <1,then

t→lim0₊

k(tv)−1

t^α =C_α(v).

(2) Ifα=1,then

t→lim0₊

k(tv)−1−iv, δ(t )

t =C₁(v).

(3) If1< α <2,then

t→lim0₊

k(tv)−1−iv, t m

t^α =C_α(v).

(4) Ifα=2,then

tlim→0

t²|log|t|| =2C2(v).

(5) Ifα >2,then

tlim→0

t² =C₂₊(v),

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with

C2+(v)= −1

2q(v, v)−q v,

I−z^∗₋1

z^∗v .

As in [21] and [5], the proof of Theorem1.3is based on an intertwining relation between the families of operators P_vandT_vand on the homogeneity at infinity ofη,η_vproved in [22]; this relation allows us to expressk(v)in terms of the stationary measureηand an eigenfunctional forTv.

Remark 1.4.

(a) We may observe that,if we add stronger moment conditions(of order greater than4),part1of Theorem1.1,i.e.

convergence to a normal law,follows from the main result of[25],which is valid also for more general Lipschitz maps ofV into itself.

(b) Forα∈ [0,2],the limit law ofS_n^xis a multidimensionalα-stable law(see e.g. [27,29,36])whereα-stability holds with respect to the action of the dilation groupR^∗₊.In particular the limit law is infinitely divisible and belongs to a convolution semigroup ofR^d.This remarkable fact follows from the homogeneity ofΔvwith respect tov,hence from the formula forC_α(v).

(c) It follows from Theorem2.4below that the negative definite functionC_α satisfiesReC_α(v) <0 for v nonzero.

In Section5,we obtain more detailed information on the functionC_α.In particular,the functionC_α depends continuously onμin a natural weak topology which guarantees continuity of moments of orderα.Also,givenμ,¯ the magnitude ofC_α is closely related to the magnitude of the moment of orderαforQ.It follows that,for the stable limiting laws of the theorem,various situations occur,as in the case of sums ofη-distributed i.i.d.random variables onV:symmetric,nonsymmetric,supported on a proper convex cone.

(d) The fact that the stability group here isR^∗₊,ifαbelongs to[0,2]instead of a more complex one as in[5],is a consequence of the following property depending on conditioni-pandd≥2 (see[23,24]):the closed subsemi- group ofR^∗₊generated by the moduli of the dominant eigenvalues for the proximal elements in[suppμ¯]is equal toR^∗₊.This can be compared with the situation of[5]where semi-stable laws in the sense of[36],p. 204,appear as limits.As already mentioned conditionCis generically satisfied byμ,and like in the caseα >2of the main theorem in[5],our limit theorem is essentially not changed under perturbation ofμ.This open the possibility of getting convergence to stable laws in natural multidimensional stochastic systems.

(e) The theorem gives the convergence of normalized 1-marginals of S_n^x.A natural question is the existence of a functional limit theorem,i.e.the convergence towards a stable stochastic process with continuous time(cf. [36, 40]).

We note that closely related limit theorems forS_n^xhave been obtained recently in the reference [9], under a stronger hypothesis than here. In [9],μ¯ dominates a density onG and[suppμ¯] has no invariant convex cone, hence the limiting law is symmetric. Furthermoreα=2 is excluded and the caseα=1 is treated under symmetry restrictions.

The method is based on a renewal theorem of [1] for a Markov chain which satisfies Harris condition.

2. Homogeneity at infinity ofμ-stationary measures

The following proposition gives the existence and elementary properties of the stationary law ofX^x_n in our context.

The first part is well known.

Proposition 2.1. Assume thatμsatisfies conditionC.Let

R_n=Q1+

n−1

k=1

M1· · ·MkQk+1.

ThenR_nconverges a.e.to

R=Q1+ ∞ k=1

M1· · ·MkQk+1

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and the law of X_n^x converges to the law η ofR.Furthermore,η has no atom,gives measure zero to every affine subspace andE(|R|^θ)=

|x|^θdη(x) <∞ifθ < α.

Proof. The proofs of convergence are based on known arguments (see [4,31]), hence we give only a sketch in our setting. Ifs < α, we have by definition ofκ(s):

E

|M₁· · ·M_k|^s

=E

|M_k· · ·M₁|^s

≤C

κ(s)+k

for some C >0, any integer k >0 and 0< < κ(α)−κ(s). AlsoE(|Qk|^s)=E(|Q1|^s)≤E(|Q1|^α)^s/α <∞. It follows ifm > n,

E

|R_m−R_n|^s

≤C E

|Q1|^αs/α m−1

k=n

κ(s)+k

<∞.

Hence limm,n→∞E(|R_m−R_n|^s)=0. The convergence a.e. of R_n to R follows. The same calculation shows E(|R|^θ) <∞ ifα≤1 andθ < α. If α >1 and θ∈ [1, α[, we use Minkowski inequality in L^θ(Ω)and the inde- pendence ofM1· · ·M_k₋1, Q_k to get that:

E

|R|^θ

≤CE

|Q1|^θ 1+

∞ k=1

κ(θ )+k/θ

θ

<∞,

ifsatisfiesκ(θ )+ <1.

The fact thatηhas no atom is proved as follows.

LetA⊂V be the set of atoms ofη. ThenAis countable and

x∈Aη({x})≤1. It follows that, for every >0, the set{x∈A;η(x)≥}is finite; in particular, sup_x_∈_Aη({x})=cis attained. Let A0= {x∈A;η(x)=c}. Since ηP=η, we havehA0=A0ifh∈suppμ. Then the barycenter ofA0is asuppμ-invariant point, which is excluded by conditionC₄.

Assume now that there exists an affine subspaceWof positive dimension such thatη(W ) >0, and letWbe the set of affine subspaces of minimum dimensionrwithη(W ) >0. Ifr=0, the contradiction follows from above. Ifr >0, we observe that for anyW, W∈W withW =W, we haveη(W∩W)=0 since dim(W ∩W) <dimW. Then as above sup_W_∈Wη(W )=cis attained. IfW0= {W∈W: η(W )=c}, we havehW0=W0for anyh∈suppμ.

LetΓ be the closed subgroup ofH generated bysuppμ, hencehW0=W0for anyh∈Γ. Then the subsetΓ₀of Γ, which leaves invariant anyW ∈W0, is a finite index subgroup ofΓ. SinceL(μ) <¯ 0,[suppμ¯]has an element gwith|g|<1. Assumeh∈ [suppμ]has linear partgand observe thathhas a unique fixed pointx∈V which is attracting. SinceΓ0has finite index inΓ, we can findp∈Nsuch thath^p∈Γ0. Then for anyy∈W withW∈W0, we have

nlim→∞h^pny=x.

Sinceh^pny∈W, we getx∈W, hence

x∈

W∈W0

W=∅.

It follows that Γ leaves invariant the nontrivial affine subspace

W∈W0W. If dim

W∈W0W =0, we have constructed a point invariant under Γ, which contradicts conditions C₄. If dim

W∈W0W > 0, the direc- tion of this affine subspace is a proper suppμ-invariant linear subspace, which contradicts condition¯ i-p for

suppμ.¯

Forκ(s)we have the following proposition (see [20]):

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Proposition 2.2. Assume[suppμ¯]satisfies conditions i-p.Thenlogκ(s)is strictly convex on[0, s_∞[.Ifs_∞= ∞, we have:

slim→∞

logκ(s)

s = lim

n→∞

1 nsup

log|g|:g∈ [suppμ¯]ⁿ

= lim

n→∞

1 nsup

logr(g): g∈ [suppμ¯]ⁿ .

In particular,the conditionκ(s) <1on]0,∞[is equivalent tor(g)≤1on[suppμ¯],and iflims→s_∞κ(s)≥1there exists a uniqueα∈ ]0, s_∞]such thatκ(α)=1.

Remark 2.3. Regularity properties ofκ(s),not used here,are proved in[20].In particular,κ(s)is analytic on[0, s_∞[. It is known (see [20,22]) that sinceμ¯ satisfies conditioni-pandκ(s) <∞, there exists a unique probability measure ν_s onP^d⁻¹such that thes-homogeneous Radon measureν_s⊗^s onP^d⁻¹×R^∗₊=(V \ {0})/{±Id}satisfies

ν_s⊗^sP¯=κ(s)ν_s⊗^s,

where, by abuse of notation,P is the Markov operator defined byμ¯ on(V\ {0})/{±Id}. Ifx¯∈P^d⁻¹corresponds to x∈V, we denote|gx¯| =^|^gx_|_x_|^| and we consider the operatorρ_s(μ)¯ onP^d⁻¹defined by

ρ_s(μ)(ϕ)(¯ x)¯ =

ϕ(g· ¯x)|gx¯|^sdμ(g),¯

wherex¯→g· ¯x is the projective map defined byg∈G. Thenν_s is the unique probability measure onP^d⁻¹such that ρ_s(μ)ν¯ _s =κ(s)ν_s. Furthermore, suppν_s is equal to the limit set of [suppμ¯] andν_s gives zero measure to any projective subspace (see [20,22]). In the corresponding situation for the unit sphere, either there exists a unique probability measureσ_s on the unit sphere which satisfies the above equation or there exist two such measures with disjoint supports which are extremal and symmetric to each other (see [22], Theorem 2.17), if[suppμ¯]preserves a convex cone. The following consequence of the general renewal theorem of [32] and of the spectral gap property of the operatorρ_s(μ)¯ is proved in [22], Theorem C, and plays an essential role here.

Theorem 2.4. Ifd >1and conditionCholds,we have the following weak convergence:

t→lim0₊t⁻^α(t.η)=c

σ_α⊗^α

=Λ,

wherec >0,σ_αis a probability measure onS^d⁻¹which has projectionν_αonP^d⁻¹andΛsatisfiest.Λ=t^αΛift >0, ΛP¯=Λ.The above convergence is valid for any functionf with aΛ-negligible set of discontinuities and such that for someε >0

sup

x=0

|x|⁻^α|log|x||¹⁺^εf (x)<∞. (7)

In particular there existsA >0such that forklarge enough, 1

A2⁻^kα≤η

x∈V; |x| ≥2^k

≤A2⁻^kα. (8)

AlsoΛ(W )=0for any proper affine subspaceW⊂V. In the special case of the recurrence relation

W_n=M_n^∗(W_n₋₁+v) (n≥1),

the corresponding measure onH is denoted byμ^∗_v. The corresponding transition operator onV is denoted byT_v. Then we have the

(10)

Proposition 2.5. Assume condition C holds true for μ.Then conditionC is satisfied by the measureμ^∗_v onH,if v=0.

The sequence Z_n^∗=

n k=1

M₁^∗· · ·M_k^∗

convergesP-a.e.to Z^∗=

∞ k=1

M₁^∗· · ·M_k^∗,

whereZis defined by theP-a.e.convergent series_∞

k=1M_k· · ·M₁. The lawη_vofZ^∗vis the uniqueμ^∗_v-stationary measure andη_vsatisfies

|x|^θdηv(x) <∞ forθ∈ [0, α[,

|x|^αdηv(x)= ∞.

For anyt∈R^∗,we haveη_{t v}=t.η_v.Ifα >1,for allx∈V the mapv→η_v(x^∗)is a linear form.

The Radon measure Δ_v= lim

t→0₊t⁻^α(t.η_v)

isα-homogeneous,satisfiesΔt v=t^αΔvfort >0,ΔvP_∗=Δv,Δ₋vis symmetric ofΔv. The functionC_α(v)satisfies forv=0, ReC_α(v) <0and fort >0,

C_α(tv)=t^αC_α(v) ifα=1, and C1(tv)=t C1(v)+i v, β(t )

, whereβ(t)=

( ^{t x}

1+|t x|² −₁_+|^{t x}_x_|2)dΛ(x).

Proof. We observe that|M^∗| = |M|, hence

nlim→∞

EM_n^∗· · ·M₁^∗^s1/n

= lim

n→∞

E

|M₁· · ·M_n|^s1/n

=κ(s).

One verifies easily that conditioni-pfor [suppμ¯], which is valid, remains valid for[suppμ¯]^∗= [suppμ¯^∗]. If suppμ¯^∗had a fixed pointx∈V, theng^∗(x+v)=x for anyg∈suppμ. Since¯ vis nonzero, we havex=0. Also this impliesg₁^∗(g₂^∗)⁻¹x=xfor anyg1, g2∈suppμ, hence¯ x is invariant under the subgroup generated bysuppμ.¯ This contradicts irreducibility of[suppμ¯].

As in the proof of Proposition2.1, one sees that the condition

nlim→∞

EM₁^∗· · ·M_n^∗^θ1/n

=κ(θ ) <1 forθ < αimplies the convergence

nlim→∞

n k=1

M₁^∗· · ·M_k^∗=^∞

k=1

M₁^∗· · ·M_k^∗=Z^∗.

Since the mapg→g^∗is continuous, this gives the convergence ofZ_n=_n

k=1M_k· · ·M₁toZ=_∞

k=1M_k· · ·M₁. The second assertion onηvfollows from inequality (8) of Theorem2.4applied toμ^∗_v, since Proposition2.5implies that condition C forμandμ^∗_vare equivalent.

The third assertion on linearity ofη_vwith respect tovfollows from the relations Z^∗(tv)=t Z^∗(v), Z^∗(v+w)=Z^∗(v)+Z^∗(w).

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The last assertions follow from Theorem2.4, the relationη_{t v}=t.η_vfort∈R^∗and the definition ofC_α(v).

We recall that the characteristic functionη_vof the measureη_vis defined byη_v(x)=η_v(Xx)andw^∗= w,·. In the proof of Theorem1.3, we shall need the following formula for the quantityC_α(v). We denote byC_α(v)the following quantity:

Cα(v)=

⎧⎪

⎪⎨

⎪⎪

⎩

(Xv(x)−1)η_v(x)dΛ(x), if 0< α <1;

((Xv(x)−1)ηv(x)−i ^v,x

1+|x|²)dΛ(x), ifα=1;

((Xv(x)−1)η_v(x)−iv, x)dΛ(x), if 1< α <2;

−¹₄

(v, w²+2v, wη_v(w^∗))dσ2(w), ifα=2.

(9)

Proposition 2.6. The formulaC_α(v)=C_α(v)with the definition(5)is valid.

Proof. We start as in the proof of Proposition 5.19 in [5]. By definition ofΛ, we have C_α(v)=

(Λ(y+v)−Λ(y))dηv(y), ifα∈(0,1)∪(1,2];

(Λ(y+v)−Λ(y))dηv(y)+iγ (v), ifα=1,

whereγ (v)is given by (6). We follow the argument in [5], but we use in an essential way the information of [22] (see Theorems 2.6, 2.17), and in particular Theorem2.4above.

We define fors < αthe Radon measureΛ_s by Λ_s=cσ_s⊗^s,

wherec is given by Theorem2.4andσs is a probability measure onS^d⁻¹, depending continuously ons in weak topology, such that

Λ_sP¯=κ(s)Λ_s, and lim

s→α₋σ_s=σ_α,

andσα given by Theorem2.4. The existence and continuity ofσs fors < αfollow from the discussion of stationary measures given before Theorem2.4, which is based on ([22], Theorem 2.17). Hence we have the weak convergence:

s→limα₋Λ_s=Λ_α=Λ.

We define alsoΛ_sfors < α,s=1, Λ_s(y)= Xy(x)−1

dΛs(x), if 0< s <1, Λs(y)= Xy(x)−1−ix, y

dΛs(x), if 1< s <2.

ThenΛs depends continuously on(s, y)in[0, α] ×V\ {0}andΛs satisfies:

P_∗Λ_s(x)=

Λ_s g^∗x

dμ(g)¯ =κ(s)Λ_s(x), and Λ_s(tx)=t^sΛ_s(x), fort >0.

Fors < α, we define

Cs(v)= Λs(y+v)−Λs(y) dηv(y) and we observe that by dominated convergence,

s→limα₋C_s(v)= Λ(y+v)−Λ(y) dηv(y).

(12)

Hence lims→α₋C_s(v)=C_α(v)ifα=1, while lims→α₋C_s(v)=C_α(v)−iγ (v)ifα=1. On the other hand,Z₀^∗v= _∞

k=0M₀^∗· · ·M_k^∗vsatisfiesZ₀^∗v=M₀^∗(Z^∗v+v), where Z^∗=^∞

k=1

M₁^∗· · ·M_k^∗

andM₀^∗is a copy ofM^∗independent ofZ. It follows:

E Λ_s Z^∗₀v

=E

Λ_s g^∗

Z^∗v+v dμ(g)¯

=κ(s)E Λ_s

Z^∗v+v , hence

C_s(v)=E Λ_s

Z^∗v+v

−E Λ_s Z^∗v

= 1

κ(s)−1

E Λ_s Z^∗v

.

By Proposition2.2, the function logκ(s)is convex, henceκ(s)has a left derivativeκ(α₋)ats=α:

m_α= lim

s→α₋

1−κ(s) α−s .

In order to get the value ofC_α(v), we need to evaluate lims→α₋(α−s)E(Λ_s(Z^∗v)).

For this purpose we will use Theorem2.4, we write F_s,v(t)=

|x|≥t

Λ_s(x)¯ dηv(x)

and we observe that|Fs,v(t)| ≤sup_x_¯_∈S^d−1|Λs(x)¯ |is bounded byK <+∞on[0, α]by definition ofΛs. Also for t≥0:

t^αF_s,v(t)=

|x|≥1

Λ_s(x)¯ dη^t_v(x)

withη^t_v=t^α(t⁻¹.η_v). Hence, using the convergence ofη^t_vtoΔ_vfort→ +∞given by Theorem2.4and the fact that Λ¹is bounded byK <+∞withΔ_v-negligible discontinuities, we get fortlarge,

t^αF_s,v(t)=Δ_v Λ¹_s

+c_s(t),

whereΛ¹_s(x)=Λ_s(x)1¯ _[1,∞)(|x|)andc_s(t)=o(1)ast→ +∞uniformly in s∈ [0, α]. We note that uniformity of o(1)is valid since the functionΛs(x)¯ is continuous and bounded on[0, α] ×S^d⁻¹, henceΛ¹_s(x)is bounded by the Δ_v-integrable functionK1_[1,∞)(|x|). By definition ofF_s,v:

E Λ_s Z^∗v

=

|y|^sΛ_s(y)¯ dηv(y)=

V

0<t≤|y|st^s⁻¹dt

Λ_s(y)¯ dηv(y)

= _∞

0

sF_s,v(t)t^s⁻¹dt.

Letρbe a positive increasing function on[0, α)such that

s→limα₋ρ(s)= +∞, lim

s→α₋(α−s)ρ^s(s)=0, lim

s→α₋ρ^s⁻^α(s)=1.

Stable laws and spectral gap properties for affine random walks