On tails of stationary measures on a class of solvable groups

On tails of stationary measures on a class of solvable groups

Dariusz Buraczewski

Institute of Mathematics, Wroclaw University, 50-384 Wroclaw, pl. Grunwaldzki 2/4, Poland Received 13 December 2005; received in revised form 30 May 2006; accepted 4 July 2006

Available online 14 December 2006

Abstract

LetGbe a subgroup of GL(R, d)and let (Q_n, M_n) be a sequence of i.i.d. random variables with values inR^dGand lawμ.

Under some natural conditions there exists a unique stationary measureν onR^d of the processXn=MnX_n−1+Qn. Its tail properties, i.e. behavior ofν{x: |x|> t}asttends to infinity, were described some over thirty years ago by H. Kesten, whose results were recently improved by B. de Saporta, Y. Guivarc’h and E. Le Page. In the present paper we study the tail ofνin the situation when the groupG₀is Abelian andR^dis replaced by a more general nilpotent Lie groupN. Thus the tail behavior ofνis described for a class of solvable groups of typeN A, i.e. being semi-direct extension of a simply connected nilpotent Lie groupN by an Abelian group isomorphic toR^d. Then, due to A. Raugi, (N, ν) can be interpreted as the Poisson boundary of (N A, μ).

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

SoitGun sous groupe de GL(R, d) et soit(Q_n, M_n)∈R^dGune suite de variables aléatoires indépendantes de loi μ.

Sous des hypothèses convenables il y a une unique mesure stationnaireνsurR^d pour le processus auto-régressif linéaireXn= M_nX_n−1+Q_n. Les propriétés asymptotiques de la queueν{x: |x|> t},t→ ∞, ont été étudiées par H. Kesten il y a 30 ans et plus récemment de nouveaux résultats ont été obtenus par B. de Saporta, Y. Guivarc’h and E. Le Page. Dans cet article on étudie le cas oùGest abélien etR^dest remplacé par un groupe de Lie nilpotentN. On obtient alors le comportement à l’infini de la queue deνpour une classe particulière de groupes de tyleN Aproduits semi-direct d’un groupeNsimplement connexeNavecG=R^d. Dans ce cas pariculier (N, ν) est un bord de Poisson au sens de A. Raugi.

©2006 Elsevier Masson SAS. All rights reserved.

Keywords:Solvable Lie groups; Stationary measure; Poisson kernel

1. Introduction

We study random recursions on solvable Lie groupsS, which satisfy the following assumptions

• Sis the semi-direct product of an Abelian groupA, isomorphic toR^d, acting on a simply connected nilpotent Lie groupN,

E-mail address:dbura@math.uni.wroc.pl (D. Buraczewski).

1 Research partially supported by KBN grant 1 P03A 018 26. The manuscript was prepared when the author was staying at Department of Mathematics, Université de Rennes and at Department of Mathematics, University Pierre & Marie Curie, Paris VI. The visits were financed by the European CommissionIHP Network 2002–2006Harmonic Analysis and Related Problems(Contract Number: HPRN-CT-2001-00273 – HARP) and European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability”, MTKD-CT-2004-013389. The author would like to express his gratitude to the hosts for hospitality.

0246-0203/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpb.2006.07.002

• there exists a contracting elementa0∈A, i.e. for everyx∈N: limk→∞δ_a^k

0(x)=0,whereδa₀stands for the action ofa₀onN, and 0 is the unit element inN.

Various classical objects like symmetric spaces, bounded homogeneous domains inCⁿ and manifolds of negative curvature admit simply transitive actions of such groups and therefore they are of considerable interest from many points of view [1,16,20].

Given a probability measureμonSwe define a random walk S_n=X_n· · ·X₁,

where{X_i}^∞_i=1is a sequence of independent identically distributed (i.i.d.) random variables with lawμ.

We write X1=QM,

withQ=πN(X1)∈N,M=πA(X1)∈A, where πN andπA denote canonical projections of S ontoN andA, respectively. We shall assume that

• μis mean-contracting, that is the element of the group corresponding to the vector

SlogMdμ(Q, M)is con- tracting;

•

S(|logM| +log⁺|Q|)dμ(Q, M) <∞, for convenient norms onAandNthat will be defined in Section 2.

Under these hypotheses the limitRofπ_N(S_n)exists in law (A. Raugi [21]) and gives rise to the measureνthat is the only stationary measure for the Markov chainπ_N(S_n)i.e.

μ∗ν=ν. (1.1)

This means that for every positive, Borel measurable functionf onN, we have μ∗ν(f )=

f

π_N(g·x)

μ(dg)ν(dx)=ν(f ).

Moreover, ifμis spread out (i.e. some power of μis nonsingular with respect to the Haar measure onS) and its support generates the groupS, A. Raugi [21] proved that (N, ν) is the Poisson boundary of this process, i.e. using the stationary measureνone can reconstruct boundedμ-harmonic functions onS, knowing their boundary value onN.

Our aim is to study behavior of ν

x: |x|> t

=P

|R|> t

ast tends to infinity, provided some further hypothesis onμ.

When the Abelian group is one dimensional, i.e.A=R⁺, the tail behavior is well understood. IfN=R, it was observed by H. Kesten [17] that the tail behavior ofνis strictly related to properties of the Laplace transform ofπA(μ) and that under natural conditions there existsα >0 such that

tlim→∞t^αP

|R|> t =C,

for some positive constantC. His proof was later essentially simplified by A.K. Grinceviˇcius [12] and Ch. Goldie [11].

The general situation of solvable groups being extensions of nilpotent groups by one-dimensional Abelian group of automorphisms was studied in [2], where similar results were obtained. Much more can be said aboutν when the measureμcomes from a second-order, subelliptic, left-invariant differential operatorLonS, i.e. when instead ofμ we consider a semigroup of measuresμt, whose infinitesimal generator isL, and the measureνsatisfies

˘

μ_t∗ν=ν for everyt.

Then, the measure ν has a density and its behavior along some rays tending to infinity has been described in [6]

and [3].

The situation when the groupAacting onN is multidimensional is much more complicated. In the context of general solvable groups, the only results we know, concerning behavior at infinity of the stationary measure, were obtained in some particular cases when the measureμis connected with an subelliptic operator onS(compare above).

IfX=G/Kis a noncompact symmetric space,Sis the solvable part of the Iwasawa decomposition ofG=SKandL is the Laplace–Beltrami operator,νhas a smooth densitym, called Poisson kernel, which can be explicitly computed (see e.g. [8]). The formulas however are not very transparent as far as the pointwise decay at∞is concerned.

More general situation was studied by E. Damek and A. Hulanicki [4,5]. They considered on solvable groups S=N A, with diagonal action ofAonN, a large class of left-invariant second order, degenerate elliptic operators L and identified the Poisson boundary of (S,L) with (N1, ν), whereN₁ is some normal subgroup ofN. Then the stationary measureνonN₁has again smooth densitymand they proved, without knowing an explicit formula form that

N₁

τ_N1(x)^εm(x)dx <∞,

for some positiveε, whereτN₁is the Riemannian distance ofxfrom the identity, anddxis the Haar measure onN.

The case whenN is an Euclidean space, but the measure is general (not coming from a differential operator) was studied by many authors. AssumeN=R^mand there exists a group of matricesG(not necessarily Abelian) acting onR^m. Consider the stochastic recursion

R_n+1=M_n+1R_n+Q_n+1,

where (Qn, Mn) is a sequence of i.i.d.,R^m×Gvalued random variables distributed according to the given probability measureμ. Then under suitable assumptionsRnconverges to a random variableR, whose distributionνisμ-invariant.

Asymptotic properties ofR were studied by several authors [17–19,7,14]. Their main assumptions (except mean- contractivity and finiteness of some exponential moments) were proximality and (or) irreducibility. Let μ¯ be the canonical projection ofμontoG. Then proximality means that the semigroup generated by the support ofμ¯ contains a proximal element, i.e. a matrix having a unique real dominant eigenvalue (i.e. the corresponding eigenspace is one- dimensional). The action is called irreducible if there does not exist a finite union of proper subspaces ofR^m, which is invariant under the action of the support ofμ.¯

In this paper we study the reducible situation on general solvable groups. Our assumptions are natural generaliza- tion of one-dimensional situation, i.e. first of all we require finiteness of some exponential moments ofπ_A(μ). The main results of the paper are presented in Section 3.4 as Main Theorem A and Main Theorem B. In full generality we prove that there exists a constantχ₀such that for anyε >0

C1t^−χ0P

|R|> t Cεt^{−(χ0−ε)},

whereC₁andC_ε are positive constants, andC_ε depends onε. Notice that the result is new even in the case when an Abelian group of matricesA=G acts onN =R^m and the measureμdoes not satisfies to the assumptions of proximality and irreducibility required by the papers mentioned above.

We obtain more detailed description of the tail of the measure ν, when the action ofAis fully reducible, i.e.A acts diagonally on N. This corresponds to the classical situations of symmetric spaces and bounded homogeneous domains. Then we prove, without assuming proximality ofμ, the existence of positive constantsχ₀andC₂such that

C₁t^−χ0P

|R|> t C₂t^−χ0.

If we assume existence of a dominant root (see Section 3 for precise definitions), that in some sense substitutes the notion of proximality, we show

tlim→∞t^χ0P

|R|> t =C₃, for someC₃>0.

The outline of the paper is as follows. In Section 2 we introduce a class of solvable Lie groups for which our results holds and describe precisely their structure. In Section 3 we include a brief account of random walks on solvable groups: existence of an invariant measure and its properties in the case when the groupAis one-dimensional. Then we describe our assumptions and state the main results of the article. Their proofs are contained in Sections 4 and 5, respectively.

The author is grateful to the referee for helpful comments and corrections, improving the presentation of this paper and some arguments in the proof.

2. A class of solvable Lie groups

LetAbe an Abelian group isomorphic toR^d, acting on a nilpotent, connected and simply connected Lie groupN, i.e.

δ_a(xy)=δ_a(x)δ_a(y), a∈A, x, y∈N, (2.1)

whereδadenotes the action ofa∈AonN.

The semi-direct productNAis a solvable Lie group denoted byS. We shall denote by◦the action of the group SonN, i.e.

(x, a)◦y=x·δ_a(y), for(x, a)∈Sandy∈N.

Then the group multiplication inSis given by (x, a)·(y, b)=

(x, a)◦y, ab .

Lete(0,I respectively) be the neutral element ofS(N,Arespectively).

Our main assumption onSis that the action ofAonN is contractive i.e. that there exists an elementa∈Asuch that

klim→∞δ_a^k(x)=0, for everyx∈N. (2.2)

The Lie algebras of A, N, S are denoted by A,N andS respectively. Then S =N ⊕A and of course for every H∈A, adH preservesN. The exponential maps are global diffeomorphisms both betweenN andN, and between AandA. Their inverse will be denoted by log. Then for anyX∈N

δ_a exp(X)

=exp

e^ad(loga)X

. (2.3)

We shall denote the foregoing action of the groupAon the Lie algebraN, using the same symbolδ_a(X). LetN^C (N^C) be the complexification ofN (Nrespectively). For anyλin the set(A^∗)^Cof continuous homomorphisms from Ato(C,+)define

N^Cλ =

Z∈N^C: there existsksuch that(adH−λI )^kZ=0, for anyH∈A

. (2.4)

Then, it is known that forλ1, λ2∈(A^∗)^C

N^C_λ1,N^C_λ2 ⊂N^C_λ1+λ2. (2.5)

Moreover any spaceN^C_λ is preserved by the action of the groupA, i.e.

δ_a(Z)∈N^C_λ, forZ∈N^C_λ. (2.6)

We shall say thatλis a root if the appropriate spaceN^Cλ is nonempty. The set of all roots will be denoted byΔ. Then, of course, ifλ∈Δthen alsoλ¯∈Δand

N^C=

λ∈Δ

N^Cλ.

Leti_λ=dim_CN^C_λ. For anyλ choose a basis{Z_λ,1, . . . , Z_λ,iλ}ofN^C_λ, such that with respect to this basisAacts triangularly, i.e. for anyH∈A

adH (Z_λ,j)=λ(H )Z_λ,j+W_λ,j−1, (2.7)

for someW_λ,j−1∈span{Z_λ,1, . . . , Z_λ,j−1}. Theni_λ=i_λ¯, moreover may assume thatZ_λ,j =Z_λ,j¯ and ifλis real then all the vectorsZ_λ,j are real.

For a chosen basis{H₁, . . . , H_d}ofAintroduce coordinates inA: any elementH ofAcan be uniquely written as

H=

t_i(H )H_i. Notice that one can compute the action ofAonN^C, taking (2.3) and (2.7) into account, we obtain δ_expH(Z_λ,k)=e^{λ(H )}·

jk

P_λ,k,j(H )Z_λ,j, (2.8)

wherePλ,k,k=1, andPλ,k,j forj smaller thatkare some polynomials ofti(H ). One can easily see that the polyno- mials depend ont_i(H )only ifλ(H_i)=0.

Thus, the assumption that the action ofAis contractive implies that the negative Weyl chamber A⁻⁻=

H∈A: Rλ(H ) <0 for allλ∈Δ

(2.9) is not empty. LetA⁺⁺= −A⁻⁻be the positive Weyl chamber.

For anyz∈N^Cletz_λ,idenotes itsλ, icomponent, i.e.

z=exp z_λ,iZ_λ,i

.

A root λ₀ will be called simple if it cannot be written as a sum of other roots, i.e. for all possible choices of nonnegative integer numbers{c_λ}λ∈Δ, such that

c_λ>1, λ₀=

λ∈Δ

c_λλ.

The set of all simple roots will be denoted byΔ1.

For instance, letA=R², choose two vector fieldsH1,H2forming a basis ofA, and denote byλ1,λ2two function- als onAsuch thatλi(Hj)=δij. Then, ifΔ= {λ1, λ1/2, (λ1+λ2)/2, λ1+2λ2, λ2}, the set of simple roots consists of three elements:Δ1= {λ1/2, (λ1+λ2)/2, λ2}.

We have the following simple lemma

Lemma 2.10.Any rootλ₀can be written in the form λ0=

λ∈Δ1

cλλ, (2.11)

wherecλare nonnegative integer numbers.

Proof. SupposeH∈A⁺⁺and let us number all the rootsλ₁, λ₂, . . . , λ_kin the following way Rλ1(H )Rλ2(H )· · ·Rλk(H ).

We shall proceed by induction. Of course,λ₁is a simple root and (2.11) holds withc_λ1=1. Assume the lemma holds forλ₁, . . . , λ_i−1. If the rootλ_i is simple then it satisfies (2.11). Otherwise,λ_i can be written as

λ_i=

λ∈Δ

c_λλ,

wherec_λare positive integer and

c_λ>1. ThereforeRλ_i(H ) >Rλ(H )for anyλsuch thatc_λis nonzero. But this set contains either simple roots or other roots satisfying already (2.11). Therefore (2.11) also holds forλ_i. 2

The group multiplication inN is given by the Campbell–Hausdorf formula:

exp(X)·exp(Y )=exp

X+Y+ [X, Y]/2+ · · ·

, forX, Y∈N. (2.12)

Since the Lie algebraN is nilpotent, the sum above is finite. In particular if we fix a simple rootλ0, then in view of (2.5)

(x·y)_λ0,i=x_λ0,i+y_λ0,i, (2.13)

forx, y∈N andii_λ0.We shall describe the Campbell–Hausdorf formula more precisely later in Section 5.

2.1. Norms onNandA

Now we are going to construct a norm on N adapted to the action ofA. In the caseAis one-dimensional and diagonalizable W. Hebisch and A. Sikora [15] have built onN a smooth outside zero norm, homogeneous on the action of one-dimensional group of dilations, i.e. satisfying |δ_a(x)| =a|x|. Their ideas were used later in [2] to

construct a homogeneous norm with respect to general one-dimensional group of dilationsA. Here we shall adopt the construction for our purpose. Since we will need some further properties of the norm we give some details.

FixH0∈A⁺⁺such thatRλ(H0) >1 for all rootsλand letA0= {expt H0, t∈R}be an one parameter subgroup ofA. We change coordinates inA0, identifying

A0expt H0∼e^t∈R⁺. Forb∈R⁺andz∈N^Cdefine

σ_b(x)=δ_exp(logb)H0(z). (2.14)

Thenσ defines the action ofR⁺ onN^C, preservingN, and the semi-direct product N R⁺ is a solvable group, belonging to the class of solvable Lie groups studied in [2]. A key step of the construction is the following lemma:

Lemma 2.15.([15,2])There exists an open rectangle Ω=

Z=

λ,i

z_λ,iZ_λ,i∈N^C: |z_λ,i|< c_λ,i

, (2.16)

wherecλ,iare some positive constants, such that

if log(z),log(w)∈Ω,forz, w∈N^Cand0< b <1then log

σb(z)σ1−b(w)

∈Ω. (2.17)

We define the norm onN^C:

|z| =inf b: log

σ_b−1(z)

∈Ω

=inf e^t: log

δ_exp⁻¹_{t H}

0(z)

∈Ω .

One can easily check that this norm is continuous and satisfies to the following properties

• | · |is symmetric:|z⁻¹| = |z|;

• |z| =0 if and only ifz=0;

• | · |is subadditive, i.e.|z·w||z| + |w|;

• |σ_b(z)| =b|z|, for anyb∈R⁺. Finally, we define a norm onA:

a =max

|z|=1

δ_a(z).

Observe that

δ_a(z)a|z| and a₁a₂a₁a₂.

We shall often use the following constants being closely related to properties of the foregoing norms dλ=Rλ(H0), λ∈Δ,

and their simple property ifλ₀=

c_λλ,thend_λ0=

c_λd_λ. (2.18)

A crucial step in the proof of our main results will be the following lemma:

Lemma 2.19.There exist constantsCandDsuch that aCmax

λ∈Δ

e

Rλ(H )

dλ

· 1∨max

i

t_i(H )^D

for anya=expH∈A, wheret_i(H )denotesith coordinate ofH in the fixed basis ofA.

Proof. First, we shall prove that amax

λ∈Δ sup

{z_λ∈N_λ^C:|z_λ|=1}

δa(zλ). (2.20)

In fact, every space N^C_λ is invariant under the action of A (2.6), and writing any element of N^C as z= exp(

λ,iz_λ,iZ_λ,i)and using the fact that the action ofAonN^Cis linear we have a = sup

|z|=1

δ_a(z)= sup

|z|=1

inf

b: σ_b−1δ_a

λ,i

z_λ,iZ_λ,i

∈Ω

= sup

|z|=1

inf

b: σ_b−1δa

i

zλ,iZλ,i

∈Ωfor all rootsλ

max

λ∈Δ sup

{z_λ∈N_λ^C:|z_λ|=1}

δa(zλ), which proves desired inequality (2.20).

Define the functiong(H )=maxi|t_i(H )|. In view of (2.20) it is enough to justify that for any rootλthere exist constantsC_λ, D_λsuch that ifg(H ) > C_λ, then

σ_b⁻¹δ_expH(Z_λ)∈Ω

forb=exp{^{Rλ(H )}_dλ } ·(1∨g(H )^Dλ)and anyZλ=

izλ,iZλ,i∈ ¯Ω∩N^C_λ. In view of (2.8) δ_a(Z_λ)=e^{λ(H )}·

k

z_λ,k

jk

P_λ,k,j(H )Z_λ,j

, whereP_λ,k,j are some polynomials oft_j(H )andP_λ,k,k=1.

Next we have

σ_b⁻¹δ_a(Z_λ)=δ_{exp(−logb)H0}δ_a(Z_λ)

=e^{−logb·λ(H0)+λ(H )}·

k

zλ,k

j

P¯λ,k,j(H,logb)Zλ,j

,

whereP¯_λ,k,j are some polynomials oft_j(H )and logb. Substitutingbin the formula above we obtain σ_b⁻¹δ_a(Z_λ)=

1∨g(H )₋D_λRλ(H₀)

·e^−iIλ(H0)(

Rλ(H )

Rλ(H0)+Dλlog⁺g(H ))

·e^{iIλ(H )}

×

k

j

¯¯

P_λ,k,j

t_i(H ), D_λlog⁺g(H ),Rλ(H ) z_λ,j

Z_λ,k,

whereP¯¯_λ,k,j are polynomials coming from appropriately modified polynomialsP¯_λ,k,j and degrees of these polyno- mials depends only on the structure of the solvable groupS. Finally, choosingD_λlarge enough, there existsC_λsuch that ifg(H ) > C_λthen for allk

1∨g(H )₋D_λRλ(H₀)

j

P P¯¯ _λ,k,j

t_i(H ), D_λlog⁺g(H ),Rλ(H )|z_λ,j|c_λ,k, which proves the lemma. 2

3. Random walks onN Agroups and main theorems

3.1. Random walks

Given a probability measureμonSwe define a random walk:

S_n=(Q_n, M_n)· · ·(Q₁, M₁),

where(Qn, Mn)is a sequence of i.i.d.S-valued random variables with a distribution μ. The law ofSn is thenth- convolutionμ^∗nofμ.

Our aim is to study theN-component ofS_n, i.e. the Markov chain onNgenerated by the random walk onS:

Rn=πN(Sn)=(Qn, Mn)◦Rn−1,

R0=δ0, (3.1)

whereπ_N denotes the canonical projection π_N:S→S/A. Byπ_A we shall denote the analogous projection ofS ontoA.

It was proved by A. Raugi [21] that whenμis mean-contracting, i.e.

ElogM=

logMμ_A(dM)∈A⁻⁻, (3.2)

whereμ_A=π_A(μ), and under the following integrability condition ElogM +log⁺|Q|<∞

(the norms used by A. Raugi were different, but his proof gives the result also in our case)R_n converges in law to a random variableR, whose distribution will be denoted byν, andRdoes not depend on the choice ofR₀. Moreover, νis a unique stationary solution of the stochastic equation

ν=μ∗ν, where

μ∗ν(f )=

f (g◦x)μ(dg)ν(dx).

The above equation can be also written in the form R=d(Q, M)◦R,

whereRand(Q, M)are independent distributed according toνandμ, respectively.

The random variableRis constructed as a pointwise limit of the “backward” process:

R₀^∗=0, R_n^∗=π_N

(Q₁, M₁)· · ·(Q_n, M_n)

=Q₁·δ_Π1(Q₂)· · ·δ_Πn−1(Q_n), (3.3) whereΠn=M1· · ·Mn.

Our aim is to study, under some additional hypothesis, behavior of ν

x: |x|> t

=P

|R|> t ast tends to infinity.

3.2. Asymptotic behavior ofRwhendimA=1

When the Abelian groupAis one-dimensional, the behavior of the above sequence is well-known. The simplest example of a solvable group is the “ax+b” group, i.e. semi-direct product ofN=RandA=R⁺, with the group action

(x, a)·(y, b)=(x+ay, ab), x, y∈R, a, b∈R⁺.

Then, Kesten [17] proved (under some further assumptions) that there exist positive constantsαandCsuch that

tlim→∞t^αP

|R|> t =C.

His proof was later essentially simplified by Grinceviˇcius [12] and Goldie [11]. Their ideas were used in [2] to handle with general situation of homogeneous groups, when the groupSis a semi-direct product of a nilpotent groupN and of an one-dimensional group of dilationsA=R⁺. In this case the norm| · |is homogeneous for the action ofR⁺, i.e.

|δ_a(x)| =a|x|for everya∈R⁺,x∈N, and we have the following theorem:

Theorem 3.4.([2])LetS=N R⁺and assume that

• ElogM <0,

• there existsα >0, such thatEM^α=1,

• the law ofλlogMis non-arithmetic, i.e. there does not exista >0such thatλlogM∈aZ,

• EM^αlogM <∞,

• E|Q|^α<∞.

Then

tlim→∞t^αP

|R|> t =C. (3.5)

for some constantC. Moreover, if the action ofR⁺onN is diagonalizable then the constantCis nonzero if and only if for everyx∈N,

P

(Q, M)◦x=x <1.

If the action is not diagonalizable, the constant C is positive under the additional hypothesis that |Q|is bounded almost surely.

We shall often use description of asymptotic behavior of P

maxn {M1· · ·Mn}> t

,

whereMiare i.i.d. real valued random variables satisfying the assumptions of Theorem 3.4. It was observed by Kesten, that the sequence is strictly connected with asymptotic behavior ofR. Then it is well known that there exists a positive constantCsuch that

tlim→∞t^αP

maxn {M1· · ·M_n}> t

=C (3.6)

(see [9] for more details).

3.3. Laplace transform

In order to describe the tail ofRwe shall need some further assumptions onμ. Consider the Laplace transform of the measureμ_A=π_A(μ):

L(α)=

A

e^α(logM)μ_A(dM)=E e^α(logM) whereα∈A^∗. We assume that

for anyλ∈Δthere existsχλ>0 such thatL χ_λRλ

d_λ

=E e^χλ

Rλ(logM)

dλ =1. (3.7)

Then it is known that the Laplace transform is well defined for all functionals onAbelonging to the convex hull V of 0 andχλRλ/dλ for all rootsλ∈Δ. FurthermoreLis convex onV and because of (3.2) and (3.7) it is strictly smaller than 1 on the set

V₀=

α∈A^∗: α=

λ∈Δ

c_λ·χ_λRλ

d_λ , for nonnegative numbersc_λsatisfying 0<

c_λ<1

, i.e.

ifα∈V₀thenL(α) <1. (3.8)

Define

χ₀=min

λ∈Δ{χ_λ}, then the following holds

Lemma 3.9.Let λ0=

cλλfor some nonnegative numbers cλ. Assume that for some root λ1 the constantcλ₁ is nonzero andχ_λ1> χ₀. Thenχ_λ0> χ₀.

Proof. Let us write χ0Rλ0

d_λ0 =

λ

χ0dλcλ

d_λ0χ_λ ·χλRλ d_λ ,

and notice that because of our assumptions and (2.18) we have

λ

χ₀d_λc_λ d_λ0χ_λ < 1

d_λ0 ·

d_λc_λ=1.

Therefore by (3.8) L

χ₀Rλ₀ d_λ0

<1, which impliesχ0< χ_λ0. 2

Corollary 3.10.There exists a simple rootλ₀such thatχ_λ0 =χ₀.

We conclude that to computeχ₀it suffices to consider only simple roots:

χ0=min

λ∈Δ₁{χλ}. (3.11)

3.4. Main theorems

For any rootλandji_λ, letV_λ,j be the real subspace onN spanned byZ_λ,j ifλis real and byRZ_λ,jandIZ_λ,j, otherwise. Then forX∈N, byX|V_λ,j we shall denote the projection ofXonV_λ,j.

Now we can state the main results of the paper Main Theorem A.Assume

(A1) ElogM∈A⁻⁻;

(A2) for any rootλthere exists a positive numberχ_λsuch thatE[e^{χλRλ(logdλ M)}] =1;

(A3) the Laplace transform of the measure μ_A is finite in some neighborhood U of 0 in A^∗ i.e. if α∈U, then L(α) <∞;

(A4) E|Q|^χ0 <∞, forχ₀defined in(3.11).

Assume moreover that there exists a simple rootλ₀such thatχ_λ0=χ₀satisfying (A5) the law ofRλ0(logM)is non-arithmetic;

(A6) E[e

χλ0Rλ0(logM)

dλ0 |Rλ₀(logM)|]<∞;

(A7) for anyX∈V_λ0,iλ

0

P log

(Q, M)◦expX

Vλ0,iλ0

=X <1.

Then there exists a positive constantC₁and for anyε >0there existsC_εsuch that C₁t^−χ0P

|R|> t C_εt^{−(χ0−ε)}. A simple rootλ0is called dominant if

χ_λ0=χ₀

and ifχλ=χ0for some other rootλ, then there exists a constantcλlarger than 1 such thatλ=cλλ0.

Of course it may happen that dominant root does not exists, i.e. for two different simple rootsλ₁, λ₂, such that λ₁=cλ₂for any constantc, we haveχ₀=χ_λ1=χ_λ2.

Main Theorem B.Assume that the action ofAonNis diagonalizable and (B1) ElogM∈A⁻⁻;

(B2) for any rootλthere exists a positive numberχ_λsuch thatE[e^{χλλ(dλlogM)}] =1;

(B3) for any rootλ,E[e^χλλ(

logM)

dλ |λ(logM)|]<∞; (B4) E|Q|^χ0<∞;

whereχ₀was defined in(3.11). Assume moreover that there exists a simple rootλ₀such thatχ_λ0=χ₀satisfying (B5) the law ofλ₀(logM)is non-arithmetic;

(B6) there existsii_λ0 such that for everyX∈V_λ0,i,P[log((Q, M)◦expX)|Vλ0,i=X]<1.

Then there exists a positive numberC1such that 1

C₁t^−χ0P

|R|> t C1t^−χ0.

Moreover if there exists inΔ₁a dominant rootλ₀satisfying both(B5)and(B6), then

tlim→∞t^χ0P

|R|> t =C₂, for some positive numberC₂. 4. Proof of Main Theorem A

4.1. Upper estimates

In order to prove the upper bound of the tail ofR, we shall use Lemma 2.19 and prove existence ofχth moment of Rfor anyχsatisfying 0< χ < χ0and then the estimates follows immediately (Corollary 4.9).

Lemma 4.1.Under the hypothesis(A1)–(A4)the stationary measure ofRhas all moments smaller thanχ₀, i.e.

E|R|^χ<∞

for allχsatisfying0< χ < χ₀.

Proof. Fixχsuch thatχ < χ< χ0, then by definition ofχ0

L χRλ

d_λ

<1, λ∈Δ.

For any rootλlet us define a positive number aλ=

⎧⎨

⎩

χ , ifL χRλ

dλ

> L χRλ

dλ

, χ, otherwise.

Then, since the Laplace transform is convex L

βRλ dλ

< L a_λRλ

dλ

(4.2) for any rootλandβ∈(χ , χ). We may choose positiveδsatisfying

0< δ < 1

L(a_λRλ/d_λ)−1, for anyλ∈Δ. (4.3)

Consider the function f (s)=E

e^si|ti(logM)| .

For any sequenceσ of 0 and 1’s having the lengthddefine the element ofA^∗by the formula α_σ(H )=

d i=1

(−1)^{σ (i)}t_i(H ), H∈A, and notice thatf can be dominated by the sum

f (s)

σ∈{0,1}^d

L(sα_σ).

By (A3) for small values ofsthe Laplace transformL(sti)is well-defined, moreover it is continuous as a function of sand tends to 1 astgoes to 0. Therefore alsof is continuous and tends to 1. So, there existsθ, such that

f (s) <1+δ, forsθ. (4.4)

Next, choose a positive numberεsatisfying ε <min

θ (χ−χ ) χ χ , θ

2χ

. (4.5)

Finally, define q= θ

εχ, p= θ

θ−εχ. (4.6)

Then notice that ¹_p+¹_q=1, by (4.5)

q >2 and p <2 (4.7)

and moreover

χ < pχ < χ. (4.8)

Recall thatRwas constructed as the limit in distribution ofRn. Therefore it is enough to estimateχth moment of Rnindependently onn. We have

E|R_n|^χ1

χ =

EQ_n·δ_Mn(Q_n−1)· · ·δ_M1...Mn(Q₀)^{χ χ1}

E

_n−1

k=0

M_k+1· · ·M_n|Q_k| _χ¹

χ

+

E|Q_n|^χ1

χ

n−1

k=0

E

Mk+1· · ·Mn|Qk| ^χ1

χ +

E|Qn|^χ1

χ

E|Q|^χ_χ¹ 1+

∞ k=1

EM₁· · ·M_k^χ_χ¹ . Thus, we have to prove that the series

∞ k=1

EM₁· · ·M_k^χ_χ¹

is convergent.

For this purpose, observe that by Lemma 2.19, the Hölder inequality and (4.6)

EM1· · ·Mk^χCE maxλ

e

χRλ(logΠk )

dλ

· 1∨max

i

ti(logΠk)^{χ D}

C

E maxλ

e

pχRλ(logΠk )

dλ ¹p

· E

1∨max

i

t_i(logΠ_k)^{qχ D1}_q

C

E

λ

e

pχRλ(logΠk ) dλ

¹

p·

E

i

e^εqχ

jk|t_i(logM_j)|¹

q

C

λ

E e

pχRλ(logM) dλ

k p·

E

e^{εqχi|ti(logM)|}

k q. Therefore, applying (4.6), (4.8) and (4.2) we obtain

∞ k=1

EM1· · ·M_k^χ1

χ C

∞ k=1

λ

L

pχRλ d_λ

^k

p·f (εqχ )^{kq 1}

χ

C

λ

∞ k=1

L

a_λRλ d_λ

¹

p ·(1+δ)^{1q k}

χ

C

λ

∞ k=1

L

a_λRλ d_λ

·(1+δ) _2χ¹ k

, where for the last inequality we used (4.7).

Finally by (4.3)

L a_λRλ

d_λ

·(1+δ) ¹

2χ

<1, therefore the series above converges. 2

Corollary 4.9.For anyεthere existsC_εsuch that P

|R|> t C_εt^{−(χ0−ε)} Proof. We have

t^χ0−εP

|R|> t

{x:|x|>t}

|x|^χ0−εν(dx)E|R|^χ0−ε

and by the lemma above the value is finite. 2 4.2. Lower estimates

To prove the lower estimate we choose a simple rootλ₀such thatχ_λ0=χ₀, satisfying (A5)–(A7), and then study projection of the random walk R_n on a suitable one or two dimensional linear subspace ofNλ0 or N^C_λ0 ⊕N^C_λ¯

0, respectively, depending whether λ₀ is real or complex. In both cases the projected random walk can be explicitly computed. Ifλ₀is real we obtain just a random walk onRgenerated by the action of “ax+b” group onR, described in Section 3.2, and we conclude the result from Theorem 3.4. The case whenλ₀is complex is more complicated. Then we obtain a random walk onR²generated by the action ofR⁺×O(2)onR², as studied in [2]. But our assumptions are different and we cannot apply the results proved there, so we shall give here a complete proof based on some ideas of A.K. Grinceviˇcius [13] and Ch. Goldie [11].

Fix a simple rootλ₀satisfying all the assumptions (A1)–(A7). We shall consider two cases.

Case I.λ₀is real.

Then we have the following lemma.