Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions

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Spectral gap properties for linear random walks and Pareto’s asymptotics for aﬀine stochastic recursions

Yves Guivarc’H, Emile Le Page

To cite this version:

Yves Guivarc’H, Emile Le Page. Spectral gap properties for linear random walks and Pareto’s asymptotics for aﬀine stochastic recursions. Annales de l’Institut Henri Poincaré (B) Probabilités et Statis- tiques, Institut Henri Poincaré (IHP), 2016, 52 (2), pp.503-574. �hal-00691516v6�

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Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions

Y. Guivarc’h, É. Le Page

Abstract

LetV=R^dbe the Euclideand-dimensional space,µ(respλ) a probability measure on the linear (resp affine) groupG=GL(V) (respH=Aff(V)) and assume thatµis the projection ofλonG.

We study asymptotic properties of the iterated convolutionsµⁿ∗δ_v (respλⁿ∗δ_v) ifv∈V, i.e asymptotics of the random walk onV defined byµ(respλ), if the subsemigroupT ⊂G (resp.

Σ⊂H) generated by the support ofµ(respλ) is “large”. We show spectral gap properties for the convolution operator defined byµon spaces of homogeneous functions of degrees≥0 onV, which satisfy Hölder type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernelΣ^∞₀ µ^k∗δv, which imply its asymptotic homogeneity. Under natural conditions theH-spaceV is aλ-boundary; then we use the above results and radial Fourier Analysis onV \ {0} to show that the uniqueλ-stationary measureρ onV is "homogeneous at infinity" with respect to dilationsv→t v(fort>0), with a tail measure depending essentially of µandΣ. Our proofs are based on the simplicity of the dominant Lyapunov exponent for certain products of Markov-dependent random matrices, on the use of renewal theorems for “tame”

Markov walks, and on the dynamical properties of a conditionalλ-boundary dual toV.

Résumé

SoitV l’espace Euclidien de dimension d, µ(resp.λ) une probabilité sur le groupe linéaire (resp.affine)G =GL(V) (resp. H=Aff(V)) et supposons queµsoit la projection deλsurG.

Nous étudions certaines propriétés asymptotiques des convolutions itérées deµ(resp.λ) appli- quées à un vecteur non nulv∈V, c’est à dire de la marche aléatoire surV définie parµ(resp.

λ), si le semigroupe T ⊂G (resp.Σ⊂H) engendré par le support deµ(resp.λ) est « grand ».

Nous montrons des propriétés d’isolation spectrale pour l’opérateur de convolution défini par µsur des espaces de fonctions homogènes de degrés≥0 surV, qui satisfont des conditions du type de Hölder. Comme conséquence de notre analyse nous obtenons des asymptotiques précises pour le noyau potentielΣ^∞₀ µ^k∗δ_v, qui impliquent son homogénéité à l’infini.Sous des conditions naturelles, leH-espaceV est uneλ-frontière ; nous utilisons alors les résultats précédents et l’analyse de Fourier radiale surV\ {0} afin de montrer que l’unique mesureλ- stationnaire est homogène à l’infini, par rapport aux dilatationsv→t v ( pourt >0),avec une mesure de queue qui dépend essentiellement deµetΣ. Nos preuves sont basÈes sur la sim- plicité de l’exposant de Lyapunov dominant de certains produits de matrices en dépendance markovienne, sur l’utilisation de théorèmes de renouvellement pour certaines marches marko- viennes et sur les propriétés dynamiques d’uneλ-frontière duale deV.

Key words and phrases: spectral gap, renewal theorem, Pareto asymptotics, random matrices, affine random recursions.

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1 Introduction, statement of results

We consider the d-dimensional Euclidean space V =R^d, endowed with the natural scalar product (x,y)→ 〈x,y〉, the associated normx→ |x|, the linear groupG=GL(V), and the affine groupH=Aff(V). Letλbe a probability measure onHwith projectionµ onG, such that suppλhas no fixed point inV. We denote by

T =[suppµ],(resp.Σ=[suppλ])

the closed subsemigroup ofG(respH) generated by suppµ(resp suppλ). Under natural conditions, including negativity of the dominant Lyapunov exponentL_µcorresponding toµ, for anyv∈V, the sequence of iterated convolutionsλⁿ∗δ_v converges weakly to ρ; the probability measure ρ is the unique probability which solves the convolution equationλ∗ρ=ρ, and suppρis unbounded if [suppµ] contains an expanding element.

Then, an important property ofρis the existence ofα>0 such thatR

|x|^sdρ(x)< ∞for s<αandR

|x|^sdρ(x)= ∞fors≥α, if suppλis compact. One of our main results below (Theorem C) gives theα-homogeneity ofρat infinity, i.e. Pareto’s asymptotics ofρ(see [47], p. 74).

In general, for the asymptotic behaviour ofλⁿ∗δ_v and the “shape at infinity” ofρthere are four cases:

1. The “contractive” case where the elements of suppµhave norms less than 1, ρ exists and is compactly supported.

2. The “expansive" case whereL_µ>0 andρdoes not exist.

3. The ‘critical” case whereL_µ=0 andρdoes not exist.

4. The “weakly contractive” case whereLµ<0 andρexists with unbounded support.

Heuristically, cases 3, 4, mentioned above, can be considered as transitions between the cases 1, 2, which appear to be extreme cases. In this paper we are mainly interested in case 4 and in the shape at infinity ofρ; in the corresponding analysis we use the approach of [35], based on the associated linear random walk, we develop methods and prove results which are of independent interest for products of random matrices.

An important tool here is the “Radon transform” ofρ, i.e. the function onV defined by ρ(vb )=ρ{x∈V;〈x,v〉 >1},

which allow us to transfer the shape problem forρinto an asymptotic problem forρ.b

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In [35] the shape problem was connected to the study of a Poisson equation on theG- spaceV\{0} satisfied byρband by a convolution operator associated withµ; the measure λwas assumed to be supported on the positive matrices or to have a density onH. In the first case an important result was the validity of Pareto’s asymptotics for the projection ofρon the positive directions.We observe that special cases of the above problem and various consequences of Pareto’s asymptotics have been considered in the litter- ature (see for example [10], [20], [38]), especially ifµhas a density onG, a condition which implies spectral gap properties for the convolution operators associated toµor λin suitable Hilbert spaces. In contrast,our basic hypothesis which involves only T andL_µ, implies that the above operators satisfy Doeblin-Fortet inequalities (see [33]), hence also spectral gap properties in spaces of Hölder functions. Here our main result, partly contained in Theorem C below, describes the general case and the homogeneity at infinity stated in the theorem gives new results even ford =1 (see [20]) or for the multidimensional situations considered in [35],[38].

We observe that, more generally, the homogeneous behaviour at infinity of certain invariant measures is of interest for various questions in Probability Theory and Mathem- atical Physics (see [10], [11], [12], [13], [35], [47]) but also in some geometrical questions such as dynamical excursions of geodesic flow and winding around cusps in hyperbolic manifolds (see [1], [44], [49]), or analysis of theH-space (V,ρ) as aλ-boundary and its dynamical consequences (see [3], [16], [17], [30]).

Hence, following [35], we start with the linear situation i.e. theG-action onV\ {0} and we consider a probability measureµonG. As in [17] we assume thatT satisfies the so- called i-p condition (i-p for irreducibility and proximality), i.e.T is strongly irreductible and contains at least one element with a unique simple dominant eigenvalue; ifd=1, we assume furthermore thatT is non arithmetic, i.e. T is not contained in a subgroup ofR^∗of the form {±aⁿ;n∈Z} for somea>0. We observe that ford>1 condition i-p is satisfied byT if and only if it is satisfied by the algebraic subgroupZ c(T),which is the Zariski closure ofT, hence condition i-p is satisfied ifT is “large” (see [22], [45]). On the other hand, the set of probability measuresµonGsuch that the associated semigroup T satisfies condition i-p is open and dense in the weak topology henceµis ’“generic”

ifd>1 andT satisfies i-p. Also, ifd>1, an essential aperiodicity consequence of condition i-p is the density inR^∗₊ of the multiplicative subgroup generated by moduli of dominant eigenvalues of the elements ofT (see [2], [25], [28]). We denote by|g|the norm ofg∈Gand we write

γ(g)=sup(|g|,|g⁻¹|), Iµ={s≥0 : Z

|g|^sdµ(g)< ∞},

we denote ]0,s_∞[ the interior of the intervalI_µ. For simplicity of exposition, and since

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linear maps commute with the symmetryv → −v, it is convenient to deal with theG- factor space ˘V ofV\ {0} by symmetry, instead ofV\ {0} itself. We use the polar decom- position

V˘=P^d⁻¹×R^∗₊,

and the corresponding functional decompositions, whereP^d⁻¹is the projective space ofV.

We consider the convolution action ofµon continuous functions onV \ {0} which are homogeneous of degrees≥0, i.e. functionsf which satisfy f(t v)= |t|^sf(v) (t∈R). This action reduces to the action of a certain positive operatorP^s onC(P^d⁻¹), the space of continuous functions on the projective spaceP^d⁻¹. More precisely, if f(v)= |v|^sϕ( ¯v) withϕ∈C(P^d⁻¹), ¯v∈P^d⁻¹, thenP^sϕis given by

P^sϕ(x)= Z

|g x|^sϕ(g·x)dµ(g),

wherex∈P^d⁻¹,x→g·xdenotes the projective action ofg onx, and|g x|is the norm of any vectorg v with|v| =1 and ¯v =x. Also forz =s+i t ∈C, withs ∈I_µ andt ∈R, we writeP^zϕ(x)=R

|g x|^zϕ(g·x)dµ(g). By dualityP^z acts also on measures onP^d⁻¹ and for a measureνwe denote byP^zνthe new measure obtained fromν. The space of endomorphisms of a Banach spaceB will be denoted by EndB. Forε>0 letH_ε(P^d⁻¹) be the space ofε-Hölder functions onP^d⁻¹, with respect to a certain natural distance.

We denote byℓ^s (respℓ) thes-homogeneous (resp. Haar) measure onR^∗₊and we write ℓ^s(d t)= d t

t^s⁺¹, ℓ(d t)=d t

t , h^s(v)= |v|^s.

An s-homogeneous Radon measureηon ˘V =P^d⁻¹×R^∗₊is written asη=π⊗ℓ^swhereπ is a bounded measure onP^d⁻¹. Fors∈I_µwe define the function

k(s)= lim

n→∞

µZ

|g|^sdµⁿ(g)

¶1/n

,

whereµⁿis then⁻^thconvolution power ofµon the groupGand we observe that logk(s) is a convex function onI_µ. A key tool in our analysis ford>1 is the

Theorem. A.Assume d >1and the subsemigroup T ⊂GL(V)generated by suppµsat- isfies condition i-p. Then, for any s∈Iµthere exists a unique probability measureν^s on P^d⁻¹, a unique positive continuous function e^s∈C(P^d⁻¹)withν^s(e^s)=1such that

P^sν^s=k(s)ν^s,P^se^s=k(s)e^s.

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For s∈I_µ, ifR

|g|^sγ^τ(g)dµ(g)< ∞for someτ>0and ifε>0is sufficiently small, the action of P^son H_ε(P^d⁻¹)has a spectral gap:

P^s=k(s)(ν^s⊗e^s+U^s),

where the operatorν^s⊗e^sis the projection onCe^sdefined byν^s,e^sand U^sis an operator with spectral radius less than 1 which satisfies U^s(ν^s⊗e^s)=(ν^s⊗e^s)U^s=0. Furthermore the function k(s)is analytic, strictly convex on]0,s_∞[and the functionν^s⊗e^sfrom]0,s_∞[ to End H_ε(P^d⁻¹)is analytic. The spectral radius of P^z is less than k(s)if s=Rez∈[0,s_∞[ and t=Imz6=0.

We observe that, since condition i-p is open, the last property is robust under perturbation ofµin the weak topology. Ifd =1,k(s) is equal toR

|x|^sdµ(x), hencek(s) is the Mellin transform ofµ(see [52]) and the above statements are also valid ifT is non arithmetic. However, the last property is not robust ford=1.

Ifs=0, P^s reduces to the convolution operator byµonP^d⁻¹and convergence to the unique µ-stationary measureν⁰ =ν was studied in [17] using proximality of theT- action onP^d⁻¹. In this case, spectral gap properties forP^z, if Rez=s is small, were first proved in [40] using the simplicity of the dominantµ-Lyapunov exponent (see [28]).

Limit theorems of Probability Theory for the productSn=gn···g₁of the random i.i.d.

matricesg_k, distributed according toµ, are consequences of this result and of radial Fourier analysis onV\ {0} used in combination with boundary theory (see[4], [6], [16], [21], [29], [40]). Ifµhas a density with compact support, Theorem A is valid for anys∈R.

In general and ford>1, it turns out that the functionk(s), as defined above, looses its analyticity at somes₁<0. For a recent detailed study of the operatorsP^z(s=Rez small) and their equicontinuous extensions in a geometrical setting which allows the algebraic groupZ c(T) to be reductive and defined over a local field of any characteristic, we refer to the forthcoming book [4]. We observe that condition i-p used here ands small imply that the basic assumptions of ([4], chap 8) are satisfied. Also, for s>0, the properties described in the theorem were considered in [29]; they are basic ingredients for the study of precise large deviations ofS_n(ω)v([41]).

The Radon measureν^s⊗ℓ^son ˘V satisfies the convolution equation µ∗(ν^s⊗ℓ^s)=k(s)ν^s⊗ℓ^s

and the support ofν^s is the uniqueT-minimal subset ofP^d⁻¹, the so-called limit set Λ(T) ofT (see [2], [4], [23]). The functione^s is an integral transform of the twistedµ- eigenmeasure^∗ν^s. Fors>0 andσa probability measure onP^d⁻¹not concentrated on a proper subspace,|g|^s is comparable toR

|g x|^sdσ(x); the uniqueness properties ofe^s

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andν^s are based on this geometrical fact. The proof of the spectral gap property depends on the simplicity of the dominant Lyapunov exponent for the product of random matricesS_n=g_n···g₁with respect to a natural shift-invariant Markov measureQ^s on Ω=G^N, which is locally equivalent to the product measureQ⁰=µ^⊗^N. A construction of a kernel-valued martingale (based on^∗ν^s) plays an essential role in the proof of simplicity and in the comparison of|Sn(ω)|with|Sn(ω)v|. Fors=0 this study corresponds to [28].

In order to develop probabilistic consequences of Theorem A we endowΩ=G^N with the shift-invariant measureP=µ^⊗^N (respQ^s). We know that ifR

logγ(g)dµ(g) (resp R|g|^slogγ(g)dµ(g)) is finite the dominant Lyapunov exponentLµ(respLµ(s)) ofSn= g_n···g₁with respect toP(respQ^s) exists and

L_µ= lim

n→∞

1 n

Z

log|S_n(ω)|dP(ω), L_µ(s)= lim

n→∞

1 n

Z

log|S_n(ω)|dQ^s(ω).

Ifs∈]0,s_∞[,k(s) has a continuous derivativek^′(s) andLµ(s)=^k_k(s)^′^(s). By strict convexity of logk(s), if lim

s→s_∞k(s)≥1 ands_∞>0, we can defineα>0 byk(α)=1.

We consider the potential kernelU on ˘V defined byU(v,·)=^∞Σ

0µ^k∗δ_v. Then we have the following multidimensional extensions of the classical renewal theorems (see [15]), which describes the asymptotic homogeneity ofU(v,·). We recall that forv ∈V˘ and A⊂V˘,U(v,A) is the mean number of visits ofS_n(ω)vtoAforn≥0.

Theorem. B. Assume T satisfies condition i-p,R

logγ(g)dµ(g)< ∞and L_µ>0. If d=1 assume furthermore thatµis non-arithmetic. Then, for any v ∈V , U˘ (v,·)is a Radon measure onV and we have the vague convergence˘

tlim→0₊U(t v,·)= 1 Lµ

ν⊗ℓ,

whereνis the uniqueµ-stationary measure onP^d⁻¹. Theorem. B^α. Assume T satisfies condition i-p,R

logγ(g)dµ(g)< ∞, L_µ<0, s_∞>0and there existsα>0with k(α)=1,R

|g|^αlogγ(g)dµ(g)< ∞. If d =1assume furthermore thatµis non-arithmetic. Then for any v∈P^d⁻¹we have the vague convergence onV˘

tlim→0₊t⁻^αU(t v,·)=e^α(v)

L_µ(α)ν^α⊗ℓ^α.

Up to normalization the Radon measureν^α⊗ℓ^αis the uniqueα-homogeneous measure which satisfies the harmonicity equationµ∗(ν^α⊗ℓ^α)=ν^α⊗ℓ^α.

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Forv∈P^d⁻¹, we consider the random variableM(v)=sup{|S_nv|; n∈N}. Then we have the following matricial version of Cramér’s estimate in collective risk theory (see [15]).

Corollary. With the notations of Theorem B^α, for any u∈P^d⁻¹, we have the convergence

tlim→∞t^αP{M(u)>t}=Ae^α(u)>0.

Theorems B, B^α are consequences of the arguments used in the proof of Theorem A and of a renewal theorem for a class of Markov walks onR(see [36]). An essential role is played by the law of large numbers for log|Snv|underQ^s(s=0,α); the comparison of

|S_n|and|S_nv|follows from the finiteness of the limit of (log|S_n| −log|S_nv|). This is the essential property used in [36], in a more general framework. For the sake of brevity, we have formulated these theorems in the context of ˘V instead ofV. Corresponding statements whereP^d⁻¹is replaced by the unit sphereS^d⁻¹are given in section 4. Also the above weak convergence can be extended to a larger class of functions.

In [35], renewal theorems as above were obtained for non negative matrices, the exten- sion of these results to the general case was an open problem and a partial solution was given in [39]. Theorems B and B^αextend these results to a wider setting. In view of the interpretation ofU(v,·) as a mean number of visits, Theorem B is a strong reinforcement of the law of large numbers forS_n(ω)v, hence it can be used in some problems of dy- namics for group actions onT-spaces. In this respect we observe that a specific version of the asymptotic homogeneity ofU stated in Theorem B has been of essential use in [30] for the description of theT-minimal subsets of the action of a large subsemigroup T⊂SL(d,Z) of automorphisms of the torusT^d.

On the other hand Theorem B^αgives a description of the fluctuations of a linear random walk on ˘V withP-a.e. exponential convergence to zero, under condition i-p and the existence inT of a matrix with spectral radius greater than one. These fluctuation properties are responsible for the homogeneity at infinity of stationary measures for affine random walks onV, that we discuss now.

Letλbe a probability measure on the affine groupH ofV,µits projection onGandT, Σas above. We assume thatT satisfies condition i-p, and suppλhas no fixed point inV. Ifd=1 we assume thatT is non-arithmetic. We consider the affine stochastic recursion onV

X_n₊₁=A_n₊₁X_n+B_n₊₁, (R)

where (An,B_n) areλ-distributed i.i.d random variables. From a heuristic point of view, the corresponding affine random walk can be considered as a superposition of an additive random walk onV governed byB_nand a multiplicative random walk onV\ {0}

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governed by A_n. Here, as it appears below in Theorem C, the non trivial multiplicative part A_n plays a dominant role, while the additive part B_n has a stabilizing effect sinceλhas a unique stationary probabilityρonV andρis not supported on a point. If E(log|A_n|)+E(log|B_n|)< ∞and the dominant Lyapunov exponentL_µ for the product A₁···A_n is negative, then R_n =Σⁿ₀⁻¹A₁···A_kB_k₊₁ converges λ^⊗^N−a.e. to R, the law ρ ofR is the uniqueλ-stationary measure onV,(V,ρ) is aλ-boundary (see [17])and suppρ=Λa(Σ) is the uniqueΣ-minimal subset ofV. IfT contains at least one matrix with an expanding direction, thenΛa(Σ) is unbounded and if I_µ =[0,∞[ there exists α>0 withk(α)=1, hence we can inquire about the “shape at infinity” ofρ. According to a conjecture of F. Spitzer the measureρshould belong to the domain of attraction of a stable law with indexαifα∈[0,2[ or a Gaussian law ifα≥2. Here we prove a multidimensional precise form of this conjecture, and more generally theα-homogeneity at infinity ofρ, where we assume thatµsatisfies the conditions of Theorem B^α,λsatisfies moment conditions and suppλhas no fixed point inV. We denote byΛ(Te ) the inverse image of the limit setΛ(T) inS^d⁻¹and byνe^αthe symmetric lifting ofν^αtoS^d⁻¹. Then our main result implies the following

Theorem. C. With the above notation we assume that T satisfies condition i-p, suppλ has no fixed point in V , s_∞>0, L_µ <0and α∈]0,s_∞[ satisfies k(α)=1. If d =1we assume alsoµis non-arithmetic. Then, ifE(|B|^α⁺^τ)< ∞andE(|A|^αγ^τ(A))< ∞for some τ>0, the uniqueλ-stationary measureρon V satisfies the following vague convergence on V\ {0}

tlim→0₊t⁻^α(t·ρ)=Cσ^α⊗ℓ^α,

where C >0,σ^αis a probability measure onΛ(Te )andσ^α⊗ℓ^αis aµ-harmonic Radon measure supported onR^∗Λ(Te ). If T has no proper convex invariant cone in V , we have σ^α=eν^α. The above convergence is also valid on any Borel function f such that the set of discontinuities of f isσ^α⊗ℓ^α-negligible and such that for some ε>0the function

|v|⁻^α|log|v||¹⁺^ε|f(v)|is bounded.

Briefly, we say thatρ satisfies Pareto’s asymptotics of indexα(see [47], page 74). The convergence in Theorem C can be considered as a Cramér type estimate for the random variable R and was stated in [26].This statement gives the homogeneity at infinity ofρ, hence the measureCσ^α⊗ℓ^αdefined by the theorem can be interpreted as the "tail measure" ofρ. In the context of extreme value theory for the process X_n, the convergence stated in the theorem implies thatρhas “multivariate regular variation” and this property plays an essential role in the theory (see [19], [47]). IfT has a proper convex invariant cone, thenCσ^αcan be decomposed as

Cσ^α=C₊ν^α₊+C₋ν^α₋,

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whereC₊, C₋≥0 andν^α₊⊗ℓ^α, ν^α₋⊗ℓ^α areµ-harmonic extremal measures onV \ {0}.

In section 5 the discussion of positivity forC,C₊,C₋in terms ofΛ(T) andΛa(Σ) lead us, via Radon transforms, to consider an associated linear random walk on the vector spaceV×R, which plays a dual role to the originalλ-random walkX_nonV. The proof of positivity forCdepends on the use of Kac’s recurrence theorem for this dual random walk. On the other hand, the proof ofα-homogeneity ofρat infinity follows from a Choquet-Deny type property for the linearµ-random walk onV \ {0}. The spectral gap property, stated in theorem A, plays an essential role in this study.

Ford =1, positivity of C =C₊+C₋ was proved in [20] using Levy’s symmetrisation argument, positivity ofC₊ andC₋ was tackled in [26] by a complex analytic method introduced in [11]. Ford=1 we haveν^α₊=δ₁,ν^α₋=δ₋₁ and the precise form of The- orem C gives that the conditionC₊=0 is equivalent to suppρ⊂]− ∞,c] withc∈R. In the Appendix we give an approach to part of Theorem C using tools familiar in Analytic Number Theory like Wiener-Ikehara’s theorem and a lemma of E. Landau but also results for Radon transforms of positive measures which are only valid forα∉N(see [5], [51]). However the discussion of positivity forC₊,C₋ seems to be not possible using only these analytical tools.

A natural question is the speed of convergence in Theorem C. Ford=1 see [20], if λ has a density. Ford >1 and under condition i-p, this question is connected with the possible uniform spectral gap for the operatorP^zof Theorem A,ifz=α+i t.

To go further we observe that Theorem C gives a natural construction for a large class of probability measures in the domain of attraction of a stable law. Using also spectral gaps and weak dependence properties of the processX_n, Theorem C allow us to prove convergence to stable laws for normalized Birkhoff sums along the affineλ-walk onV (see [18]) Furthermore ifd=1, and conditionally on regularity assumptions usual in extreme value theory, such convergences were shown in [10]; hence the results of [18] improve and extends the results of [10] in the case of GARCH processes (d≥1). If d>1 the above convergence is robust under perturbation ofλin the weak topology.

These convergences to stable laws are connected with the study of random walk in a random medium on the line or the strip (see [12]) ifα<2. On the other hand the study of the extremal value behaviour of the processX_n can be fully developed on the basis of Theorem C and on the above weak dependence properties ofX_n; in particular the asymptotics of the extremes of|X_n|are given by Fréchet type laws with indexα([31]), a result which extends the main result of [38] to the generic case. In a geometrical context, as observed in [44] for excursions of geodesic flow around the cusps of the modular surface, the famous Sullivan’s logarithm law is a simple consequence of Fréchet’s law for the continuous fraction expansion of a real number uniformly distributed in [0,1].

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Here also a logarithm law is valid for the random walkX_n (see [31]). The arguments developed in the proof of homogeneity at infinity forρcan also be used in the study of certain quasi-linear equations which occur in various domains related to branching random walks in particular (see [20]), [42]). For example the description of the shape at infinity of the fixed points of the multidimensional version of the “smoothing trans- formation” considered in [13] in the context of infinite particle systems in interaction depends on such arguments (see [9]). In an econometrical context, the stochastic recursion (R) can be interpreted as a mechanism which, in the long run, produces debt or wealth accumulation with a specific homogeneous structure at large values; hence,in the natural setting of affine stochastic recursions, this mechanism “explains” the re- markable power law asymptotic shape of wealth distribution empirically discovered by the economist V. Pareto ([43]).

For information on the role of spectral gap properties in limit theorems for Probability theory and Ergodic theory we refer to [1], [4], [6], [18], [21], [27], [28], [40]. For information on products of random matrices we refer to [4], [6], [16], [24]. Theorem A (resp B, B^αand C) is proved in sections 2,3 (resp 4 and 5).

We thank Ch.M. Goldie, I. Melbourne and D. Petritis for useful informations on stochastic recursions and extreme value theory.We thanks also the referees for careful reading and very useful sugestions.

2 Ergodic properties of transfer operators on projective spaces

In this section we study the qualitative properties of transfer operators onP^d⁻¹orS^d⁻¹. As mentioned in the introduction one can find in [4] a detailed study of a general class of transfer operators on flag manifolds with equicontinuity properties. Here, our transfer operators depend on a complex parameterz which is typically large with Rez>0.

Hence, for self-containment reasons in particular, we develop from scratch our study onP^d⁻¹ orS^d⁻¹ for Rez=s ≥0. A first step is to reduce these transfer operators to Markov operators (Theorems 2.6,2.16) with equicontinuity properties.

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2.1 Notation and preliminary results

LetV =R^dbe the Euclidean space endowed with the scalar product〈x,y〉 =Pd

1x_iy_i and the norm|x| =¡P_d

1|xi|²¢1/2

, ˘V the factor space ofV \ {0} by the finite group {±I d}. We denote byP^d⁻¹(respS^d⁻¹), the projective space (resp unit sphere) ofV and by ¯v (resp

˜

v) the projection ofv∈V onP^d⁻¹(respS^d⁻¹). The linear groupG=GL(V) acts onV, V˘ by (g,v)→g v. Ifv ∈V\ {0}, we writeg·v =_|^{g v}_{g v}_| and we observe thatGacts onS^d⁻¹ by (g,x)→g·x. We will also write the action ofg ∈G onx∈P^d⁻¹byg·x; we define

|g x|as|gx˜|ifx∈P^d⁻¹and ˜x∈S^d⁻¹has projectionx∈P^d⁻¹. Also, ifx,y∈P^d⁻¹,|〈x,y〉|

is defined as|〈x,e ye〉|wherex,e ye∈S^d⁻¹have projectionsx,y. Corresponding notations will be taken when convenient. For a subsetA⊂S^d⁻¹the convex envelope Co(A) ofA is defined as the intersection withS^d⁻¹of the closed convex cone generated byAinV. We denote byO(V) the orthogonal group ofV and bymtheO(V)-invariant measure on P^d⁻¹. A positive measureηonP^d⁻¹will be said to be proper ifη(U)=0 for every proper projective subspaceU6=P^d⁻¹.We denote by EndV the space of endomorphisms of the vector spaceV.

Let P be a positive kernel on a Polish space E and let e be a positive function on E which satisfiesPe=kefor somek>0. Then we can define a Markov kernelQ_e onEby Doob’s relativisation procedure:Q_eϕ=_ke¹P(ϕe). This procedure will be used frequently here. For a PolishG-spaceE we denote byM¹(E) the space of probability measures on E. If ν∈ M¹(E), and P is as above,νwill be said to beP-stationary if Pν=ν, i.e for any Borel functionϕ, ν(Pϕ)=ν(ϕ). We will writeC(E) (respC_b(E) for the space of continuous (resp bounded continuous) functions onE. If E is a locally compact G- space,µ∈M¹(G), andρis a Radon measure onE, we recall that the convolutionµ∗ρis defined as a Radon measure byµ∗ρ=R

δ_{g x}dµ(g)dρ(x), whereδ_yis the Dirac measure aty∈E. Aµ-stationary measure onEwill be a probability measureρ∈M¹(E) such that µ∗ρ=ρ. In particular, ifE=V or ˘V andµ∈M¹(G) we will consider the Markov kernel P onV (resp ˘P on ˘V) defined byP(v,·)=µ∗δ_v, (resp ˘P(v,·)=µ∗δ_v). OnP^d⁻¹(resp S^d⁻¹) we will write ¯P(x,·)=µ∗δ_x(resp ˜P(x,·)=µ∗δ_x).

If u is an endomorphism ofV, we denoteu^∗ its adjoint map, i.e.〈u^∗x,y〉 = 〈x,u y〉if x,y∈V. Ifµ∈M¹(G) we will writeµ^∗for its push forward by the mapg →g^∗and we define the kernel^∗P onV by^∗P(v,.)=µ^∗∗δ_v. Fors ≥0 we denoteℓ^s (resph^s) the s-homogeneous measure (resp function) onR^∗₊={t∈R; t>0} given by

ℓ^s(d t)= d t

t^s⁺¹ (resph^s(t)=t^s).

Fors=0 we writeℓ(d t)=^{d t}_t .

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Using the polar decompositionV\ {0}=S^d⁻¹×R^∗₊ and the corresponding functional decompositions onV\{0}, everys-homogeneous measureη(resp functionψ) onV\{0}

can be written as

η=π⊗ℓ^s (resp.ψ=ϕ⊗h^s),

whereπ(resp. ϕ) is a measure (resp. function) onS^d⁻¹. Similar decompositions are valid on ˘V =P^d⁻¹×R^∗₊. Ifg ∈G, andη=π⊗ℓ^s (respψ=ϕ⊗h^s) the directional component ofgη(resp.ψ◦g) is given by

ρ^s(g)(η)= Z

|g x|^sδ_g_·_xdη(x), ρ_s(g)(ψ)(x)= |g x|^sψ(g·x).

The representationsρ^sandρ_s extend to measures onGby the formulae ρ^s(µ)(η)=

Z

|g x|^sδ_g_·_xdµ(g)dπ(x), ρ_s(µ)(ψ)(x)= Z

|g x|^sψ(g·x)dµ(g).

We will write, forϕ∈C(P^d⁻¹) (resp.ψ∈C(S^d⁻¹))

P^sϕ=ρs(µ)(ϕ), (resp. ˜P^sψ=ρs(µ)(ψ)), ^∗P^sϕ=ρs(µ^∗)(ϕ), (resp.^∗P˜^sψ=ρs(µ^∗)(ψ)).

We endowS^d⁻¹(resp.P^d⁻¹) with the distance ˜δ(resp.δ) defined by δ(x,˜ y)= |x−y|(resp. δ( ¯x, ¯y)=inf{|x−y|;|x| = |y| =1}).

Forε>0,ϕ∈C(P^d⁻¹) (resp.ψ∈C(S^d⁻¹)), we denote [ϕ]ε=sup

x6=y

|ϕ(x)−ϕ(y)|

δ^ε(x,y) (resp. [ψ]ε=sup

x6=y

|ψ(x)−ψ(y)| δ˜^ε(x,y) ),

|ϕ| =sup{|ϕ(x)|;x∈P^d⁻¹} (resp.|ψ| =sup{|ψ(x)|;x∈S^d⁻¹}), and we write

H_ε(P^d⁻¹)={ϕ∈C(P^d⁻¹);[ϕ]ε< ∞},(resp.H_ε(S^d⁻¹)={ψ∈C(S^d⁻¹);[ψ]ε< ∞}.

The set of positive integers will be denoted byN. We denote byµⁿthen^th convolution power ofµ, i.e forψ∈C_b(G),µⁿ(ψ)=R

ψ(gn···g₁)dµ^⊗ⁿ(g₁,···,gn).

Definition 2.1. Ifs∈[0,∞[ andµ∈M¹(G), we denote k(s)=kµ(s)= lim

n→∞

µZ

|g|^sdµⁿ(g)

¶1/n

, Iµ={s≥0;kµ(s)< +∞}.

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We observe that the above limit exists, since by subadditivity ofg →log|g|, the quant- ityu_n(s)=R

|g|^sdµⁿ(g) satisfiesu_m₊_n(s)≤u_m(s)u_n(s). Also k_µ(s)=inf_n_∈N(u_n(s))^1/n, which impliesI_µ={s ≥0;R

|g|^sdµ(g)< +∞}. Furthermore, Hölder inequality implies thatI_µ is an interval of the form [0,s_∞[ or [0,s_∞], and logkµ(s) is convex on I_µ. Also k_µ∗ =k_µ since |g| = |g^∗|. If µand µ^′ commute and c∈[0,1], µ^′′ =µ+(1−c) µ^′ then k_µ′′(s)=ckµ(s)+(1−c)k_µ′(s), ifs∈Iµ∩I_µ′.

Definition 2.2. 1. An elementg ∈EndV is said to be proximal ifg has a unique ei- genvalueλ_g ∈Rof maximum modulus andλ_g is simple.

2. A semigroupT ⊂Gis said to be strongly irreducible if no finite union of proper subspaces isT-invariant.

Proximality ofg means that we can writeV =Rv_g⊕V_g^<withg v_g=λ_gv_g, g V_g^<⊂V_g^<and the restriction ofgtoV_g^<has spectral radius less than|λ_g|. In this case lim

n→+∞gⁿ·x¯=v¯_gif x∉V_g^<and we say thatλ_g is the dominant eigenvalue ofg. IfE⊂Gwe denote byE^{pr ox} the set of proximal elements ofE.The closed subsemigroup (resp. group) generated by Ewill be denoted [E] (resp.〈E〉). In particular we will consider below the caseE=suppµ where suppµis the support ofµ∈M¹(G).

Definition 2.3. A semigroupT ⊂Gis said to satisfy condition i-p ifT is strongly irreducible andT^prox6= ;.

As shown in ([22], [45]) this property ofT is satisfied if it is satisfied byZ c(T), the Zariski closure ofT. It can be proved that condition i-p is valid if and only if the connected component of the closed subgroupZ c(T) is locally the product of a similarity group and a semi-simple real Lie group without compact factor which acts proximally and irreducibly onP^d⁻¹. In this senseT is “large”. For example, ifT is a countable subgroup ofGwhich satisfies condition i-p thenT contains a free subgroup with two generators.

We recall that inCⁿ, the Zariski closure ofE⊂Cⁿis the set of zeros of the set of polyno- mials which vanish onE. The groupG=GL(V) can be considered as a Zariski-closed subset ofR^d²⁺¹. IfT is a semigroup, thenZ c(T) is a closed subgroup ofGwith a finite number of connected components. Ifd =1, condition i-p is always satisfied. Hence, when using condition i-p,d>1 will be understood.

Remark. The above definitions will be used below in the analysis of laws of large numbers and renewal theorems. A corresponding analysis has been developed in [35] for the case of non-negative matrices. We observe that proximality of an element inG is closely related to the Perron-Frobenius property for a positive matrix. IfT is not irreducible, we can consider the subspaceV⁺(T) generated by the dominant eigenvectors of

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the elements ofT^prox; thenV⁺(T) isT-invariant (see [21] p 120) and, ifT⁺is the restriction ofT toV⁺(T), thenT⁺satisfies condition i-p. In that way our results below could be used in reducible situations (see for example [9]).

Definition 2.4. AssumeT is a subsemigroup ofG which satisfies condition i-p. Then the closure of the set { ¯v_g;g ∈T^prox} will be called the limit set ofT and will be denoted Λ(T).

We recall that for a semigroupT acting on a topological space E, a subset X ⊂E is said to beT-minimal if any orbitT y(y∈X) is contained inX and dense inX. For the minimality ofΛ(T)⊂P^d⁻¹, see [2], [23].

With these definitions we have the

Proposition 2.5. Assume T ⊂G is a subsemigroup which satisfies condition i-p and S⊂T generates T . Then TΛ(T)=Λ(T) and Λ(T) is the unique T -minimal subset of P^d⁻¹. Ifµ∈M¹(G)is such that T=[suppµ]satisfies i-p, there exists a uniqueµ-stationary measure ν on P^d⁻¹. Also suppν=Λ(T) and ν is proper. Furthermore, if d >1, the subgroup of R^∗₊ generated by the set {|λ_g|;g ∈T^prox} is dense in R^∗₊. In particular, if ϕ∈C(Λ(T))satisfies for some t ∈R, |e^iθ| =1: ϕ(g·x)|g x|^{i t} =eⁱ^θϕ(x) for any g ∈S, x∈Λ(T)then t=0, e^iθ=1,ϕ=constant.

Remark. The first part of the above statement is essentially due to H. Furstenberg ([17], Propositions4.11, 7.4) . The second part is proved in ([28]. Proposition 3). For another proof and extensions of this property see [2], [25]. This property plays an essential role in the renewal theorems of section 4 as well as in section 5 ford >1. In the context of non negative matrices a modified form is also valid (see [9]); in ([35]), Theorem A) under weaker conditions onT, its conclusion is assumed as an hypothesis. Ifd =1, we will need to assume it, i.e we will assume thatT is non-arithmetic ; ifT =[suppµ] satisfies this condition, we say thatµis non-arithmetic.

2.2 Uniqueness of eigenfunctions and eigenmeasures on P

^d−1

Here we considers∈I_µand the operatorP^s(resp^∗P^s) onC(P^d⁻¹) defined by P^sϕ(x)=

Z

|g x|^sϕ(g·x)dµ(g), (resp.^∗P^sϕ(x)= Z

|g x|^sϕ(g·x)dµ^∗(g)).

For a measureνonP^d⁻¹,P^sνis defined by duality againstC(P^d⁻¹). Forz=s+i t∈Cwe will also writeP^zϕ(x)=R

|g x|^zϕ(g·x)dµ(g). Below we study existence and uniqueness

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for eigenfunctions or eigenmeasures ofP^s. We show equicontinuity properties of the normalized iterates ofP^sand ˜P^s.

Theorem 2.6. Assumeµ∈M¹(G)is such that the semigroup[suppµ]satisfies (i-p) and let s∈I_µ. If d=1we assume thatµis non arithmetic. Then the equation P^sϕ=k(s)ϕ has a unique continuous solutionϕ=e^s, up to normalization. The function e^sis positive ands -Hölder with¯ s¯=inf(1,s).

Furthermore there exists a uniqueν^s ∈M¹(P^d⁻¹)such that P^sν^s is proportional to ν^s. One has P^sν^s=k(s)ν^s and suppν^s=Λ([suppµ]). If^∗ν^s ∈M¹(P^d⁻¹)satisfies^∗P^s(^∗ν^s)= k(s)^∗ν^s, and e^sis normalized byν^s(e^s)=1, one has p(s)e^s(x)=R

|〈x,y〉|^sd^∗ν^s(y)where p(s)=R

|〈x,y〉|^sdν^s(x)d^∗ν^s(y).

The map s→ν^s(resp s→e^s)is continuous in the weak topology (resp uniform topology) and the function s→logk(s)is strictly convex.

The Markov operator Q^sonP^d⁻¹defined by Q^sϕ=_k(s)e¹ sP^s(ϕe^s)has a unique stationary measureπ^sgiven byπ^s=e^sν^sand we have for anyϕ∈C(P^d⁻¹)the uniform convergence of(Q^s)ⁿϕtowardsπ^s(ϕ). If z=s+i t , t∈Rand Q^zis defined by Q^zϕ=_k(s)e¹ sP^z(e^sϕ),then the equation Q^zϕ=e^iθϕwithϕ∈C(P^d⁻¹),ϕ6=0implies e^iθ=1, t=0,ϕ=constant.

Remark. 1. Ifs=0, thene^s=1, andν^s=νis the uniqueµ-stationary measure [17].

The fact thatνis proper is of essential use in [40], [6] and [28], for the study of limit theorems.

2. In section 3 we will also construct a suitable kernel-valued martingale which allows to prove thatν^sis proper (see Theorem 3.2), ifs∈Iµ. We note that analyticity ofk(s) is proved in Corollary 3.20 below. Continuity of the derivative ofkwill be essential in sections 3,4 and is proved in Theorem 3.10. Ifs=0 the corresponding martingale construction was done in [17].

3. By definition ofP^sand ˘P, the function (resp. measure)e^s⊗h^s(resp.ν^s⊗ℓ^s) satisfies the equation

P(e˘ ^s⊗h^s)=k(s)(e^s⊗h^s),(resp. ˘P(ν^s⊗ℓ^s)=k(s)ν^s⊗ℓ^s)

The proof of the theorem depends of a proposition and the following lemmas improving corresponding results for positive matrices in [35].

Lemma 2.7. Assumeσ∈M¹(P^d⁻¹)is not supported by a hyperplane. Then, there exists a constant c_s(σ)>0such that, for any u in EndV