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TAIL-HOMOGENEITY OF STATIONARY MEASURES FOR SOME MULTIDIMENSIONAL STOCHASTIC RECURSIONS

DARIUSZ BURACZEWSKI, EWA DAMEK, YVES GUIVARC’H, ANDRZEJ HULANICKI AND ROMAN URBAN

Abstract. We consider a stochastic recursion Xn+1 = Mn+1Xn+ Qn+1, (n ∈ N), where (Qn, Mn) are i.i.d. random variables such that Qn are translations, Mnare similarities of the Euclidean space Rdand Xn∈ Rd. In the present paper we show that if the recursion has a unique stationary measure ν then the weak limit of properly dilated ν exists and defines a homogeneous tail measure Λ. The structure of Λ is studied and the supports of ν and Λ are compared. In particular, we obtain a product formula for Λ.

1. Introduction and the main result

Let V = Rd be the d-dimensional Euclidean space, endowed with the natural scalar product hx, yi =Pd

1xiyiand the corresponding norm: |x|2=Pd

1|xi|2. The norm of a linear transformation g of V is denoted |g|. We say that g ∈ Gl(V ) is a similarity, if

(1.1) |gx| = |g||x|, x ∈ V

and we will denote by G the group of all similarities.

We consider the group H = Rd⋊G of transformations V ∋ x → hx = gx + q ∈ V, where g ∈ G and q ∈ V . We study the stochastic recursion

(1.2) Xn+1= Mn+1Xn+ Qn+1, n ≥ 0,

where (Qn, Mn) is an H valued i.i.d. sequence with distribution µ, Qn ∈ V , Mn∈ G.

If E log |Mn| < 0 and E log+|Qn| < ∞ then 1.2 has a unique (in law) stationary solution given by Z0= Z = Q0+

X

k=1

M0· ... · Mk−1Qk

(see e.g. [DF], [Gre], [V]). We have

(1.3) Z0= M0Z1+ Q0,

where Z1= Q1+P

k=2M0· ... · Mk−1Qk and so, for the law ν of Z, we have ν(f ) = µ ∗ ν(f ) =

Z

H

hν(f ) dµ(h) = Z

V

Z

H

f (hx)dµ(h)dν(x).

“This research project has been partially supported by European Commission via IHP Network 2002-2006 Har- monic Analysis and Related Problems(contract Number: HPRN-CT-2001-00273 - HARP) and a Marie Curie Transfer of Knowledge Fellowship Harmonic Analysis, Nonlinear Analysis and Probability (contract number MTKD-CT-2004- 013389). D. Buraczewski, E. Damek, A. Hulanicki and R. Urban were also supported by KBN grants 1 P03A 018 26 and N201 012 31/1020.

1

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Here we are interested in the asymptotics as t → ∞ of ν({v ∈ V : |v| > t}). It is expected that under natural conditions (including E|M |χ = 1 for a χ > 0), this function decays as t−χ (see [K], [Gri], [Gol]) and indeed this is the case as, among other things, is shown in the Main Theorem 1.6.

In general, if Mn are only in GL(V ), (1.1) not being required, the remarkable fact that the tail of ν is homogeneous has been observed by Kesten ([K], see also [Gri], [Gol], [Gui2], [KP], [LeP]). In our setting, however we obtain a stronger result about the “asymptotic shape” of ν.

Let ¯µ be the projection of µ on G via the homomorphism h = (q, g) 7→ g and let Gµbe the closed subgroup of G generated by the support of ¯µ. Provided there is χ > 0 such thatR

G|g|χ d¯µ(g) = 1, we prove that the family of finite positive measures νg, g ∈ Gµ defined on V \ {0} by

νg(W ) = |g|−χP{gZ ∈ W } = |g|−χgν(W )

converge weakly to a nonzero Radon measure Λ on V \ {0} , as |g| → 0. The measure Λ, called the tail measure of ν, is “homogeneous” with respect to Gµ, i.e. gΛ = |g|χΛ, g ∈ Gµ. It turns out that Λ can be written in a product form (see Theorem 1.6).

Let Hµ be the subgroup generated by suppµ. The only restrictions on Hµ are: the action of Hµ

on V has no fixed point and Gµ is not compact. What follows in this paper applies to any such subgroup.

Our main result, in the case when V = R2 is identified with C and G with C, deals with Xn, Qn, Mn that are complex valued variables for which we write

Xn= |Xn|eiargXn, arg Xn ∈ [0, 2π).

Then, in this special case, under additional hypothesis Gµ= C, our main result yields

Theorem 1.4. Assume that the random variable (Q, M ) ∈ C × C is bounded, E log |M | < 0, 1−MQ is not a constant a.e. and there is χ > 0 with E(|M |χ) = 1. Then, for the Z that satisfies (1.3), there is C > 0 such that for any θ, θ ∈ [0, 2π)

t→+∞lim tχP{|Z| > t, θ < arg Z < θ} = C|θ − θ| 2π .

To formulate the main result of the paper in its proper generality we need some more notation.

The group G of similarities (1.1) is the direct product G = R+× O(V ) of dilations x 7→ ax and the orthogonal group O(V ), x 7→ kx. The map h = (q, g) 7→ |g| is a homomorphisms of H onto R+. Let µr be the corresponding projection of µ on R+ i.e. the law of |Mn| and let Rµ be the closed subgroup of R+ generated by suppµr. Since P{|Mn| = 1} < 1, we have

(1.5) Rµ= R+ or Rµ= hci = {cn: n ∈ Z} for a c > 1.

and so in the first case we can find Y in the Lie algebra G of Gµ such that |eY| 6= 1. Let Cµ = Gµ∩ O(V ). In both cases Gµ is a semi-direct product of Aµ and Cµ (see Appendix C), where

• Aµ= {e(log a)Y : a > 0} if Rµ = R+

• Aµ= {gn1 : n ∈ Z} if Rµ=< c >, |g1| = c.

Let ρ and l be Haar measures on Cµand Aµrespectively. We assume that ρ(Cµ) = 1 and we normal- ize l in the way that under above identification, l is either daa on R+ or the counting measure on hci multiplied by log c. Then for the Haar measure λ on Gµwe haveR f (g) dλ(g) = R f (ak) dl(a)dρ(k) i.e. λ = l ∗ ρ.

We are going to use convolution in a generalized sense i.e. whenever we have a group ˜H acting on a topological space ˜V and two measures θ on ˜H and β on ˜V then

hf, θ ∗ βi = Z

H× ˜˜ V

f (hv) dθ(h)dβ(v).

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Let λχ(dg) = |g|−χλ(dg) and lχ(da) = a−χl(da). Let Σ1= S1= {x ∈ V : |x| = 1} if Rµ= R+ or Σ1= {v ∈ V ; 1 ≤ |v| < c} if Rµ =< c >. Then Σ1is a compact or relatively compact fundamental domain for the action of Aµon V \ {0}. For any v ∈ V \ {0} there are unique a(v) ∈ Aµ and ¯v ∈ Σ1

such that v = a(v)¯v. Let r(v) = |a(v)|. Then we have well defined generalized polar coordinates r(v) ∈ Rµ and ¯v ∈ Σ1and a generalized projection pr(v) = ¯v. Clearly, if Rµ = R+ then r(v) = |v|

and if furthermore Y1= I, pr is the radial projection of V \ {0} on S1.

Let Tµ be the closed sub-semigroup of H generated by the support of µ. We observe that for every h ∈ H such that |g| < 1 there is a unique fixed point h+= (I − g)−1q ∈ V which is attractive.

The following fact has been proved, (see 2.4 2.7).

suppν is the closure of {h+: h ∈ Tµ, |g| < 1}

and it is unbounded if µr((0, 1]) < 1. The Mellin transform of µr is denoted by κ(s) = E|M |s, for 0 ≤ s < s,

where s= supp{s ∈ R+: κ(s) < ∞}. The main result of the paper is the following

Main Theorem 1.6. Assume that the action of suppµ on V has no fixed point, E log |M | < 0 and there is χ ∈ (0, s] such that κ(χ) = 1. Let mχ= E(|M |χlog |M |). If E|Q|χ < ∞ then there exists

|g|→0,g∈Glim µ

|g|−χ(gν) = Λ, in the weak sense. Λ = m1

χλχ∗ (ν − ¯µ ∗ ν) is a Radon measure on V \ {0} that satisfies gΛ = |g|χΛ, g ∈ Gµ.

In particular, Λ is Cµ - invariant. The convergence

|g|→0,g∈Glim µ|g|−χ(gν)(f ) = Λ(f ),

is valid for bounded continuous functions that vanish in a neighborhood of zero. Moreover, there is a finite Cµ - invariant measure σµ on Σ1 such that

Λ = χlχ∗ σµ. and

χmχσµ(W ) = E[rχ(Z0)1W( ¯Z0) − rχ(M0Z1)1W( ¯Z1)]

if W ⊂ Σ1 is Cµ - invariant.

The family of measures σt on ¯Σ1 defined by

σt(W ) = tχP{|Z| > t, ¯Z ∈ W }, for W ⊂ S1, converges weakly to a finite measure

σ= χlχ([1, ∞))σµ

and σ1) = limt→∞σt1). If Rµ= R+ then σ= σµ. If s= ∞ and lims→∞

E|Q|s

κ(s)

1s

< ∞ or if s< ∞, χ < s and lims→s

E|Q|s

κ(s) = 0, then Λ is non-zero. The function s → E|Z|s admits a meromorphic extension Lµ(z) to the domain ℜz < χ + ε for some ε > 0 with a simple pole at χ, where the residue of Lµ(z) is m1

χ

E(|Z − Q|χ− |Z|χ) < 0. Furthermore,

(1.7) suppσµ= \

t>0

suppσt.

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Remark 1.8. If Y1 = I and pr is the radial projection of V \ {0} on S1 we can compare suppσµ

and suppν. Let S the sphere at infinity of V and let ¯V = V ∪ S. We write ¯U for the closure of U ⊂ V in ¯V . pr gives a natural identification of S with the unit sphere S1 and 1.7 implies

suppσµ= pr(S∩ suppν) Remark 1.9. For d = 1, the theorem says

σµ = C+δ1+ Cδ−1

with C+ > 0 (respectively C > 0), if and only if, suppν ∩ R+ (respectively suppν ∩ R) is unbounded. The case d = 1, Rµ= R+ was investigated in [Gol] where the inequality C++ C > 0 was obtained. It may happen that only one of the constants C+or Cdoes not vanish. The question under which condition this is C+ or C, was not decided in [Gol]. Progress in this direction was made in [GL2]: C = 0 is equivalent to the existence of an interval [a, ∞) preserved under the action of suppµ. Then suppν = [b, ∞) with b > a. Otherwise suppν = R.

Here we propose such a condition formulated in terms of the support of ν that follows from a somewhat detailed study of Λ also for d > 1. The proof uses analytic tools like Dirichlet series to study the function C ∋ s 7→ E|Z|s in the manner known from analytic number theory. To wit, the implicit formulae for σµ and the residue of Lµ at χ that extend formulae of [Gol] and [DeH] are proved.

The theorem can be interpreted as a description of an “asymptotic shape” of ν and it is valid in a much more general setting i.e. when V is replaced by a connected and simply connected nilpotent Lie group N and G by a group of “similarities” of N defined in terms of a homogeneous norm (see Appendix D).

The main result of [Gui2] is formally similar to Theorem 1.6, but since Gµis “large”, the analogue of σµ that appears in [Gui2] is determined as a stationary measure on S1. In the general context, suppσµ may be much smaller then pr (S∩ suppν). This is not the case here and suppσµ is determined by the asymptotic behavior of suppν at ∞.

Let {µt} be the semi-group of measures on V ⋊R+generated by an elliptic left-invariant operator L or, more generally , by a subelliptic left-invariant operator on N ⋊ R+. If the law µ of (Q, M ) is equal to µt for some t, then ν ∈ C(N ) can be interpreted as the exit measure of the diffusion generated by L on N ⋊ R+. Then σµ has a continuous density c(u) [BDH] and

a→∞,a∈Rlim +

aδ+χν(au) = c(u),

where δ is the homogeneous dimension of N , (see Theorem 2.15 of [BDH]). c(u) is strictly positive provided L is elliptic. The same is true in the case when the invariant vector-fields that appear as squares in L generate the Lie algebra of N ⋊ R+ and then suppν = N = suppΛ. But for a general H¨ormander type operator L no conclusions about the support of ν has was in [BDH] and so the present result is new also for the “differential” case. For the latter see also [DH] and [DHZ].

For d > 1, it is interesting to look at the special case when the group Cµ acts transitively on S1

and Rµ = R+. Let u be a unit vector in V . Let µ = δu∗ ¯µ, and let sk ∈ G be a sequence of i.i.d random variables with the law ¯µ. Then, as in Theorem 1.4

Z = u +

X

n=0

s0· ... · snu and we have

t→+∞lim tχP{|Z| > t and pr(Z) ∈ W } = C|W |,

for every subset W of S1with negligible boundary, |W | being the d − 1 volume of W . In this case Λ = Cσ ⊗ l, where σ is the volume measure on S1 and C > 0.

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If d = 1 and ak were positive i.i.d random variables, the tail of Z = 1 +P

1 a1· · · ak played an important role in problems of random walk in random medium [KKS] Also, the law of Z appeared in some problems of mathematical physics [DeH], [DLNP] which in turn led to study of the dominant Lyapunov exponent γ(ε) of the 2 × 2 random matrices. It turns out that the asymptotic of γ(ε), when ε is small, is governed by the tail of Z. To deal with the corresponding problems in higher dimension requires an extension of the above to the situation when the vector space V is replaced by a nilpotent Lie group. We give a few indications in Appendix D.

The theorem implies that when Aµ = {aI : a > 0}, ν belongs to the domain of attraction of a stable law of index χ. A natural question is to describe the asymptotic behavior of the normalized sums of the non independent random variables Xk. In [GL1] and [BDG] convergence of these random variables towards stable laws is shown.

The authors are grateful to Waldemar Hebisch and Albert Raugi for fruitful discussions concerning several points of the paper. The proof of Theorem D.13 was suggested to us by Waldemar Hebisch.

2. Existence of the tail measure Λ

In this section we prove the first part of Theorem 1.6 i.e. existence of Λ and its radial decompo- sition (see Theorem 2.8).

We say that µ satisfies hypothesis (H) if no point of V is suppµ invariant and Elog |M | < 0,

(2.1)

there exists χ > 0, such that E|M |χ= 1, (2.2)

E|Q|χ < ∞.

(2.3)

Clearly 2.1 and 2.2 imply that µr((0, 1]) < 1.

Proposition 2.4. Assume that µ satisfies hypothesis (H). Then the series

X

k=0

M0· ... · Mk−1Qk

converges P a.e. to Z0 ∈ V . The law ν of Z0 is the unique µ - stationary measure ν, ν = µ ∗ ν and Z0= M0Z1+ Q0, where Z1=P

k=1M1· ... · Mk−1Qk. For every x ∈ V , µn∗ δx tends to ν as n → ∞. Moreover, the support of ν is unbounded and also for every β < χ

(2.5) E|Z0|β< ∞.

For any sphere U ⊂ V , ν(U ) = 0.

Remark 2.6. The first part of the above proposition is well known, but we include it here for the sake of completeness.

Proof. As in [V] we consider the seriesP

k=0M0· ... · Mk−1Qk and we observe that its general term converges exponentially fast to zero a.e.. Indeed, we have

log |M0· ... · Mk−1Qk| =

k

X

0

log |Mi| + log |Qk|.

The condition E log+|Q| < ∞ implies lim sup

k→∞

1

klog |Qk| ≤ 0, a.e.

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By the law of large numbers

k→∞lim 1 k

k

X

i=0

log |Mi| = E log |M | < 0, a.e.

Hence

Z0=

X

k=0

M0· ... · Mk−1Qk

is a.e. well defined and it satisfies

Z0= M0Z1+ Q0.

Since (Q0, M0) and Z1 are independent, for the law ν of Z0we have µ ∗ ν = ν.

It follows that suppν is invariant under Tµ. Moreover, h0· · · hnx = M0· · · Mnx +

n

X

k=0

M0· · · Mk−1Qk, for x ∈ V . Therefore, for any f continuous and bounded

n→∞lim µn∗ δx(f ) = lim

n→∞

Ef (h0· · · hnx) = Ef (Z0) = ν(f ) and so for any µ-stationary measure ρ

ρ = lim

n→∞µn∗ ρ = lim

n→∞

Z

V

µn∗ δxdρ(x) = ν.

Moreover, if |g| > 1, then the modulae of eigenvalues of g are strictly bigger then 1 and h = (q, g) ∈ Tµ has a unique fixed point h

h= (I − g)−1(q).

By (2.1) and (2.2), P{|M | > 1} > 0 and so such h exists in the support of µ. For any x 6= h we have

n→∞lim |hnx| = lim

n→∞|gn(x − h) + h| = +∞

Since suppµ has no fixed point, suppν has more than one element and so we can take x 6= h, x ∈ suppν. Then hnx ∈ suppν and so suppν is unbounded. The proof of (2.5) is contained in Appendix C.

Finally, we observe that any intersection of spheres is a finite union of irreducible algebraic submanifolds of V . Let U be the collection of such irreducible submanifolds. If U, U ∈ U, U ⊂ U and dimU = dimU then U = U. Moreover, any U ∈ U carries a well defined finite volume measure and its barycenter is defined by

c(U ) = 1 σ(U )

Z

xdσ(x).

For any h ∈ H and U ∈ U, h(U ) ∈ U and h(c(U )) = c(h(U )). Assume that there are U ∈ U such that ν(U ) 6= 0 and let U0 be the set of such U of minimal dimension. Then for U, U ∈ U0

such that U 6= U we have ν(U ∩ U) = 0. Since ν(V ) = 1, it follows that for any ε > 0, the set {U ∈ U0, ν(V ) ≥ ε} is finite. Therefore, the function U → ν(U ) from U0 to R+ attains its maximum at a finite number of points U1, ..., Ur. Moreover, ν(Uj) =R ν(g−1Uj) dµ(g) and so Ujs are permuted by the action of (suppµ)−1. The same is true for their barycenters. It follows that the barycenter of the set c(U1), ..., c(Ur) is invariant under the action of H, which is impossible. Hence

for any sphere U , ν(U ) = 0. 

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For h = (q, g), |g| < 1 let h+= (I − g)−1(q) be the fixed point of h. We have the following lemma which shows that the support of ν depends only of Tµ, not of µ itself.

Lemma 2.7. Under the hypotheses of Proposition 2.4, the support of ν is S ={h+: h ∈ Tµ, |g| < 1}

Furthermore, S is the unique Tµ-minimal subset of V .

Proof. First we prove that S ⊂ suppν. Let |g| < 1. Notice that hn(h+) = h+ and hn(x) = gn(x − h+) + h+, hence limn→∞hn(x) = h+ and limn→∞hnν = δh+. On the other hand for every h ∈ Tµ,

supphν = h(suppν) ⊂ suppν.

Hence h+∈ suppν and so S ⊂ suppν.

Now we show that S is Tµ invariant. Let h, h ∈ Tµ, h = (q, g), h = (q, g) and |g| < 1. Then limn→∞|ggn| = 0. Therefore,

n→∞lim(hhn)+= h(h+).

Hence

h(h+) ∈ S

If ρ is any probability measure on S, we have, by Proposition 2.4 ν(S) ≥ lim sup

n→∞ µn∗ ρ(S) and since S is Tµ invariant,

µn∗ ρ(S) = 1.

This shows that ν(S) = 1, suppν ⊂ S and the first conclusion follows. For the second one let S be a closed Tµ-invariant subset of V . For any h ∈ Tµ with |g| < 1 and any v ∈ S, hnv ∈ S and

h+= limn→∞hnv ∈ S. Hence S ⊂ S. 

Given a function f let Dis(f ) be the set of all discontinuities of f .

Theorem 2.8. Assume that µ satisfies hypothesis (H) and mχ = E|M |χ| log |M || < ∞. Then the formula mχΛ = λχ∗ (ν − ¯µ ∗ ν) defines a Radon measure Λ on V \ {0} such that

(2.9) lim

|g|→0,g∈Gµ

|g|−χEf (gZ) = lim

|g|→0,g∈Gµ

|g|−χgν(f ) = Z

V

f (x) dΛ(x), for every function f such that Λ(Dis(f )) = 0 and for some ε > 0:

sup

x

|x|−χ| log |x||1+ε|f (x)| < ∞.

For every g ∈ Gµ

(2.10) δg∗ Λ = |g|χΛ.

In particular, Λ is Cµ - invariant.

Moreover, there is a finite Cµ - invariant measure σµ on Σ1 such that Λ = χlχ∗ σµ.

and

χmχσµ(W ) = E[rχ(Z0)1W( ¯Z0) − rχ(M0Z1)1W( ¯Z1)]

if W ⊂ Σ1 is Cµ - invariant.

The family of measures σt on ¯Σ1 defined by

σt(W ) = tχP{|Z| > t, ¯Z ∈ W },

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for W ⊂ S1, converges weakly to a finite measure

(2.11) σ= χlχ([1, ∞))σµ.

If Rµ= R+ then σ= σµ.

Theorem 2.8 will follow from Lemmas 2.12, 2.19, 2.25, 2.27, 2.31. The idea is to transfer the problem to Gµ and to study the function ¯f (g) = Ef (gZ) = gν(f ), g ∈ Gµ. The renewal theorem on Gµ is applied to obtain the asymptotic of |g|−χf (g) and so of ν. Two properties of ν − ¯¯ µ ∗ ν are used: it has mass zero and it is small at infinity.

Define convolution of a function φ with a measure η on G as φ ∗ η(g) =

Z

G

φ(gh)dη(h).

The following simple lemma gives a handy formula for the “Poisson integral” ¯f (g) = gν(f ) as the ¯µ - potential on G of ψf = f − f ∗ ¯µ.

Lemma 2.12. Let f be a bounded continuous function and let f (g) = Ef (gZ) = (gν)(f ).¯

Then both ¯f and ψf = ¯f − ¯f ∗ ¯µ are bounded and continuous on G. If f (0) = 0, we have

(2.13) lim

m→∞

f ∗ ¯¯ µm= 0, and

(2.14) f =¯

X

n=0

ψf∗ ¯µn, the convergence being uniform on compact sets in G. Moreover,

(2.15) ν =

X

0

¯

µk∗ (ν − ¯µ ∗ ν).

Proof. Continuity and boundedness of f and ψf follows immediately from regularity of f and dom- inated convergence. For the second assertion we apply the law of large numbers for the random walk a1· .... · an generated by µron R+. Since E log |M | < 0, we have limn→∞a1· .... · an = 0 P a.e and so

n→∞lim |(g1· .... · gn)x| = lim

n→∞|g1| · ... · |gn||x| = 0 P a.e hence

(2.16) lim

n→∞δg∗ ¯µn( ¯f ) = lim

n→∞

Z

f ((gg1· ...gn)x))dP(ω)dν(x) = 0.

We write

f =¯

m

X

n=0

ψf∗ ¯µn+ ¯f ∗ ¯µm+1.

and now (2.16) implies (2.14). Furthermore the above arguments show that the series in (2.14) is absolutely and uniformly convergent on compact subsets of G. Substituting ψf = ¯f − ¯f ∗ ¯µ into 2.14

we obtain 2.15. 

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To prove (2.9) we shall use an extended renewal theorem for subgroups of D = R+× K, where K is a compact metrisable group (see Appendix A) Such groups are unimodular as well as their closed subgroups.

Let ˜D be a subgroup of D and let ∆n = {g ∈ ˜D : n < log |g| ≤ n + 1}, n ∈ Z. We say that a bounded Borel function φ is d-R-i (direct Riemann integrable) on ˜D if

– the set of discontinuities of φ is negligible with respect to the Haar measure of ˜D, – P

nsupg∈∆n|φ(g)| < ∞.

(2.17)

A continuous function ψ on D satisfying the second condition on D is d-R-i on any subgroup of D.

The renewal theorem A.1 will be later on applied to D = G and ˜D = Gµ.

Let Hεbe the set of bounded functions on V that vanish in a neighborhood of 0 and that satisfy the ε-H¨older condition:

(2.18) |f (x + y) − f (x)| ≤ Cf|y|ε.

Hε+will denote the set of positive elements in Hε. For x ∈ V , η > 0 let Bη(x) = {y ∈ V, |y − x| < η}.

The following lemma is of fundamental importance for the rest of the paper.

Lemma 2.19. Assume that 0 < ε < s ≤ s, η > 0, f ∈ Hε and suppf ∩ Bη(0) = ∅, for η > 0. Let X be a Gµ-invariant subset of V that contains the support of f . Then, if E(|M |s+ |Q|s+ε) < ∞ and E(|Z|s−ε2s 1X(Z)) < ∞, the function

(2.20) ψs,f(g) = |g|−sψf(g)

is d-R-i on Gµ and

X

n

sup

g∈∆n

s,f(g)| ≤ CsCfηε−s,

where Cs does not depend of f, η. If X is G - invariant, then ψs,f is d-R-i on G. If s = χ, the assumption E(|M |χ+ |Q|χ) < ∞ is sufficient.

Remark 2.21. The argument takes into account that ν − ¯µ ∗ ν is smaller at infinity than ν.

Proof. Since ψχ,f is continuous, we have to check only the second condition in (2.17). We have ψf(g) = Ef (gZ0) − f (gM0Z1).

If en < |g| ≤ en+1 the expectation above is restricted to the non vanishing set of the integrand i.e.

to

Pn = {(|Q0| + |M0Z1| ≥ ηe−n−1, Z0∈ X or Z1∈ X}

Here, and later on, we use Z0= M0Z1+ Q0. We define the random variable n0= −1 + log η − log(|Q0| + |M0Z1|)

and then Pn= {n0≤ n, Z0∈ X or Z1∈ X}. Therefore by the H¨older property of f and (1.3),

|g|−s|f (gZ0) − f (gM0Z1)| ≤ Cf|g|ε−s|Q0|ε. Then for |g| ∈ (en, en+1]:

s,f(g)| ≤ CfEh

e−n(s−ε)|Q0|ε1Pni and

X

n

sup

g∈∆n

s,f(g)| ≤ CfEh

|Q0|ε X

n≥n0

e−n(s−ε)1Pni . But

X

n≥n0

e−n(s−ε)≤ e−n0(s−ε)

1 − e−(s−ε) = es−εηε−s

1 − e−(s−ε)(|Q0| + |M0Z1|)s−ε.

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and so it remains to estimate

(2.22) E|Q0|ε(|Q0| + |M0Z1|)s−ε(1X(Z0) + 1X(Z1)).

Then the inequality (a + b)λ≤ 2λ(aλ+ bλ), for a, b, λ ≥ 0 gives

|Q0|ε(|Q0| + |M0Z1|)s−ε≤ 2s−ε|Q0|s+ |Q0|ε|M0|s−ε|Z1|s−ε.

Since (M0, Q0) and Z1 are independent, |M0Z1| = |M0||Z1| and s − ε < s −εs2, we have E|Q0|ε|M0Z1|s−ε1X(Z1) = E|M0|s−ε|Q0|εE|Z1|s−ε1X(Z1)

≤ E|Q0|sεs E|M0|s1−εs E|Z1|s−ε1X(Z1)

≤ E|Q0|sεs E|M0|s1−εs E|Z1|s−ε2s1X(Z1) < ∞, (2.23)

On the other hand

|M0Z1|s−ε≤ 2s |M0Z1+ Q0|s−ε+ |Q0|s−ε = 2s |Z0|s−ε+ |Q0|s−ε.

Hence by H¨older inequality

E|Q0|ε|M0Z1|s−ε1X(Z0) ≤ 2sE|Q0|s+ 2sE[|Q0|ε|Z0|s−ε1X(Z0)

≤ 2sE|Q0|s+ 2s E|Q0|s+εs+εε E|Z0|s−ε2s1X(Z0)s+εs

< ∞.

(2.24)

Combining (2.23) and (2.24) we obtain the required estimate of (2.22). For s = χ notice that the above calculation gives

Eh

|Q0|ε(|Q0| + |M0Z1|)χ−ε(1X(Z0) + 1X(Z1))i

≤ 2χ+1E|Q0|χ+ E|Q0|χχε E|M0|χ1−χε

E|Z1|χ−ε

which is finite. 

Now we are able to express |g|−χf (g) as a potential.¯

Lemma 2.25. Let ¯µχ = |g|χµ. Then ¯¯ µχ is a probability measure with finite and strictly positive mean mχ. Let Uχ =P

0 µ¯kχ be the potential of ¯µχ. If f ∈ Hεfor some ε, then ψχ,f is Uχ integrable and for every g ∈ Gµ

|g|−χf (g) = (δ¯ g∗ Uχ)(ψχ,f).

Proof. In view of our assumptions mχ =R

G(log |g|)|g|χd¯µ(g) ∈ (0, ∞). By the previous lemma with s = χ, ψχ,f is d-R-i, ψχ,f∗ ¯µkχ, (δg∗ Uχ)(ψχ,f) are well defined (see [Fe]) and

|g|−χf∗ ¯µk)(g) = ψχ,f∗ ¯µkχ(g).

Furthermore, if f is non-negative, we have |g|−χ f ∗ ¯¯ µk(g) = | · |−χf ) ∗ ¯¯ µkχ(g), hence the last convolution is well defined. The same is true for any bounded f , since we can write f as the difference of two nonnegative bounded functions. Therefore, for any n ≥ 0

n

X

k=0

ψχ,f ∗ ¯µkχ(g) = |g|−χf (g) − |g|¯ −χ( ¯f ∗ ¯µn+1)(g).

By Lemma 2.12, limn→∞f ∗ ¯¯ µn+1= 0 and so for g ∈ G

n→∞lim

n

X

k=0

ψχ,f∗ ¯µkχ(g) = (δg∗ Uχ)(ψχ,f) = |g|−χf (g).¯



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Remark 2.26. From Lemma 2.25 it follows that |g|−χf (g) is bounded when g varies. Indeed, ψ¯ s,f

is d-R-i and, on the real line, potentials of d-R-i functions are bounded. Clearly, this remains valid on G itself by the natural projection from G to R+.

Lemma 2.27. The formula Λ(f ) = 1

mχ

Z

G

g(ν − ¯µ ∗ ν)(f )|g|−χ dλ(g) = 1 mχ

λχ∗ (ν − ¯µ ∗ ν)(f ), defines a nonnegative Radon measure on V \ {0} with the property

δg∗ Λ = |g|χΛ, for g ∈ Gµ.

In particular, Λ is Cµ invariant. Moreover, any f ∈ Hεfor some ε > 0 is Λ integrable and

|g|→0,g∈Glim µ

|g|−χ(gν)(f ) = lim

|g|→0,g∈Gµ

|g|−χEf (gZ) = Λ(f ).

Proof. By Theorem A.1 applied to δg∗ Uχ we have

|g|→0,g∈Glim µ

|g|−χf (g) =¯ lim

|g|→0,g∈Gµ

δg∗ Uχ

ψχ,f = 1 mχ

Z

Gµ

ψχ,f(g) dλ(g).

Hence the map (2.28) f 7→ 1

mχ

Z

Gµ

ψχ,f(g)dλ(g) = 1 mχ

Z

Gµ

g(ν − ¯µ ∗ ν)(f )|g|−χ dλ(g) = 1 mχ

χ∗ (ν − ¯µ ∗ ν) defines a nonnegative Radon measure Λ on V \ {0}. The relation

δg∗ Λ = |g|χΛ, g ∈ Gµ

follows from mχΛ = λχ∗ (ν − ¯µ ∗ ν). 

Notice that

(2.29) sup

t>0tχν|x| > t < ∞.

and

(2.30) sup

t>0

tχΛ|x| > t < ∞.

Indeed, we take a function f0∈ Hεsuch that f0(x) ≥ 1B1(0)c(x) for any x ∈ V . Since

|g|→0,g∈Glim µ

|g|−χEf0(gZ) = Λ(f0) is finite, (2.29) follows. δg∗ Λ = |g|χΛ implies (2.30).

Lemma 2.31. For any sphere U ⊂ V

Λ(U ) = 0.

Proof. Let θ = ν − ¯µ ∗ ν. Then Λ(U ) = R θ(g−1U ) dλχ(g). If Λ(U ) > 0, we have θ(g−1U ) > 0 for some g ∈ Gµ. Hence ν(g−1U ) = θ(g−1U ) + (¯µ ∗ ν)(g−1U ) > 0, which is impossible by the last

assertion of Proposition 2.4. 

Now we are able to complete the proof of Theorem 2.8.

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Proof of Theorem 2.8. For (2.9) we start with continuous functions. For M > η, let f ∈ Cc BM(0)\

Bη(0) and let fn∈ Hε be a sequence of functions satisfying kfn− f kL < 1

n suppfn ⊂ Bcη

2(0).

Then

|g|−χE|f − fn|(gZ) ≤ 1

n|g|−χP|Z| > η/(2|g|) ≤ C1 n and

Z

V

|f − fn|(x) dΛ(x) ≤ C n.

Therefore, in view of Lemma 2.27 the statement is proved for f ∈ Cc(V \ {0}).

To prove (2.9) in its full generality, we use continuous functions φη, φM, φ such that suppφη ⊂ B(0), φη(x) = 1, x ∈ Bη(0)

suppφM ⊂ BM

2(0)c, φM(x) = 1, x ∈ BM(0)c φηM + φ = 1

and we let η → 0, M → ∞. Then, let

fη= φηf, fM = φMf and fη,M = φf.

We then have

f = fη+ fη,M + fM, and Λ(Dis(fη,M)) ≤ Λ(Dis(f )) = 0. Therefore,

|g|→0,g∈Glim µ

|g|−χgν(fη,M) = Z

V

fη,M(x)dΛ(x), by the Portmanteau theorem. For the other integrals we prove

|g|−χgν(fη) ≤ Cηε, Λ(fη) ≤ Cηε,

|g|−χgν(fM) ≤ CM−ε, Λ(fM) ≤ CM−ε. for some C > 0. Indeed, by the hypothesis on f and (2.29)

|g|−χ|gν(fM)| ≤ C|g|−χ Z

V

|g(x)|χ1{|g(x)|≥M}| log |gx||−1−ε dν(x)

≤ C X

n≥⌊log2M

|g|−χ Z

2n≤|g(x)|≤2n+1

|g(x)|χ| log |gx||−1−ε dν(x)

≤ C X

n≥⌊log2M

|g|−χ2(n+1)χn−1−εν{|gx| ≥ 2n}

≤ C X

n≥⌊log2M

n−1−ε≤ CM−ε

that is arbitrarily small, by the appropriate choice of M and so the first conclusion follows.

Now we calculate mχΛ = λχ∗ (ν − ¯µ ∗ ν) using the generalized polar decomposition of V \ {0}.

For v ∈ V \ {0} we write

v = a(v)¯v, ¯v ∈ Σ1, a(v) ∈ Aµ, |a(v)| = r(v)

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and we desintegrate θ = ν − ¯µ ∗ ν along the fibres Aµ¯v:

hf, ν − ¯µ ∗ νi = Z

f (r, ¯v) dθv¯(r)d¯θ(¯v),

where θv¯and supp¯θ are finite measures supported by Aµv and Σ¯ 1respectively. It follows hf, mχΛi =

Z

f (gx) dλχ(g)dθ(x)

= Z

f (bk(r, ¯v)) dlχ(b)dρ(k)dθ¯v(r)d¯θ(¯v)

= Z

f (ba(v)k¯v) dlχ(b)dρ(k)dθ¯v(r)d¯θ(¯v)

= Z

f (bk¯v) rχdlχ(b)dρ(k)dθ¯v(r)d¯θ(¯v)

= Z

f (bk¯v) dlχ(b)dρ(k)θ¯v(rχ)d¯θ(¯v) Let σµ be a Radon measure on Σ1 defined by

hh, σµi = 1 χmχ

Z

h(k¯v) dρ(k)θv¯(rχ)d¯θ(¯v).

Then σµ is Cµ invariant and

(2.32) hf, Λi = χ

Z

f (b¯v) dlχ(b)dσµ(¯v).

Since Λ is a Radon measure and Σ1 is relatively compact, σµ is finite. Furthermore, if W ⊂ Σ1 is Cµ invariant then

χmχσµ(W ) = Z

1W(¯v)θ¯v(rχ) d¯θ(¯v)

= Z

r(v)χ1W(pr(v)) dθ(v)

= Z

r(v)χ1W(pr(v)) d(ν − ¯µ ∗ ν)(v)

= E[rχ(Z0)1W( ¯Z0) − rχ(M0Z1)1W( ¯Z1)].

If W = Σ1 we substitute M0Z1= Z0− Q0and we have

χmχσµ1) = E[rχ(Z) − rχ(Z − Q)].

For W ⊂ Σ1with σµ(∂W ) = 0 we consider ˜W =S

a∈Aµ,|a|≥1aW and we notice that Λ(Dis(fW˜) = 0.

Indeed, if Rµ = R+ this follows immediately from 2.32. If Rµ = hci, then we apply Lemma 2.31.

Therefore,

|a|→0,a∈Alim µ

σ|a|(W ) = lim

|a|→0,a∈Aµ

|a|−χν(a−1W ) = Λ( ˜˜ W ).

On the other hand by 2.32

Λ( ˜W ) = χσµ(W )lχ([1, ∞)).

The required convergence of σt to σ and 2.11 follow. If Rµ = R+ then χlχ([1, ∞)) = 1 and σ= σµ.



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3. Support of the tail measure

In this section we show the second part of the Main Theorem 1.6, i.e. we show that Λ does not vanish and we describe the support of ν. More precisely, we prove

Theorem 3.1. Assume that µ satisfies hypothesis (H), χ < s and E|Q|s< ∞ for every s < s. If one of the following conditions is fulfilled

(3.2) s= ∞ and sup

s<∞

E|Q|s κ(s)

1s

= D < ∞

(3.3) s< ∞ and lim

s→s

E|Q|s κ(s) = 0,

then Λ is non-zero. The function s → E|Z|s admits a meromorphic extension Lµ(z) to the domain ℜz < χ + ε for some ε > 0. Lµ(z) has a simple pole at χ with the residue m1χE(|Z − Q|χ− |Z|χ) < 0.

Furthermore,

(3.4) suppσµ= \

t>0

suppσt. If Aµ= {aI : a > 0} then 3.4 implies

suppσµ= pr(S∩ suppν).

To prove Theorem 3.1, we need a number of lemmas and propositions. The essential conclusions are summarized in Propositions 3.9, 3.12 and Corollary 3.14. The idea is as follows. Assume Λ = 0.

Then E[|Z|χ− |Z − Q|χ] = 0, hence E|Z|sextends holomorphically to 0 < ℜz < s. Then using a functional equation for E[|Z|s] we get contradiction with the fact that |Z| is unbounded. Moreover, we extend this argument to “directions” of V in order to calculate suppΛ in terms of the asymptotics of suppν at ∞.

For f ∈ Hε+, x 6= 0 and a complex number z such that ℜz > 0 let fˆz(x) = |x|−z

Z

Gµ

|g|−zf (gx) dλ(g).

f (x) is well defined because by definition of Hˆ ε+, suppf ⊂ {x ∈ V : |x| > δ} for a positive δ. Hence

| ˆfz(x)| ≤ |x|−ℜz|f |

Z

|g|>η|x|−1

|g|−ℜzf (gx) dλ(g),

which is finite. Moreover, ˆfzis continuous outside 0 and Gµ- homogeneous of degree 0 i.e. ˆfz(gx) = fˆz(x), g ∈ Gµ, hence bounded. Finally, supp ˆfz= Gµsuppf .

For any positive function f belonging to Hε, ¯f as in section 2 and for any complex number z define

λz( ¯f ) = Z

Gµ

|g|−zf (g)dλ(g) provided the integral is absolutely convergent.

Notice that

λz( ¯f ) = E|Z|zz(Z) and |λz( ¯f )| ≤ λℜz(| ¯f |)

therefore, by Proposition (2.4), λz( ¯f ) is well defined for z satisfying 0 < ℜz < χ. First we show (Proposition 3.7) that χ in the last inequality may be replaced by χ + η < s for some η > 0 provided lim|g|→0|g|−χf (g) = 0.¯

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Lemma 3.5. Assume that f ∈ Hε+ satisfies λs( ¯f ) = R

Gµ|g|−sf (g) dλ(g) < +∞ for some s ∈¯ [0, s). Then for ℜz < s the functions |g|−z( ¯f ∗ ¯µ)(g) and ψz,f(g) = |g|−z( ¯f − ¯f ∗ ¯µ)(g) are λ integrable. The functions of z

λz( ¯f ∗ ¯µ) = Z

Gµ

|g|−z( ¯f ∗ ¯µ)(g) dλ(g)

Λz(f ) = Z

Gµ

ψz,f(g) dλ(g) = (λz∗ (ν − ¯µ ∗ ν))(f ) are holomorphic and

(3.6) Λz(f ) = (1 − κ(z))λz( ¯f ).

In particular, (3.6) is valid for ℜz < χ.

Proof. We have

f ∗ ¯¯ µ(g) = E[f (gM0Z1)]

and so

λz( ¯f ∗ ¯µ) = Z

Gµ

|g|−zE[f (gM0Z1)] dλ(g).

Since λz( ¯f ) < ∞, the change of variables g → gM0yields λz( ¯f ∗ ¯µ) = E|M0|z

Z

Gµ

|g|−zf (g) dλ = κ(z)λ¯ z( ¯f ) Hence for Re z < s, λz( ¯f ) and λz( ¯f ∗ ¯µ) are well defined, holomorphic and

Λz(f ) = λz( ¯f ) − λz( ¯f ∗ ¯µ) = (1 − κ(z))λz( ¯f ).

 Lemma 3.7. There exists η > 0 such that for every f ∈ Hε+ for ε < mins−χ

2 , χ, 1 satisfying Λχ(f ) = 0, the function g 7→ |g|−zf (g) is λ-integrable if 0 < ℜz < χ + η. Furthermore, λ¯ z( ¯f ) and Λz(f ) are holomorphic for 0 < ℜz < χ + η.

Proof. By Proposition 2.4, E|Z|s< ∞ for s < χ. Define η1= χ+1ε2 . Then χ + η1+ ε < χ + 2ε < s

and χ + η1χ+ηε2

1 < χ + η1χ+1ε2 = χ. Moreover, if s < χ + η1then also s −εs2 < χ. Hence, in view of Lemma 2.19, applied to s ≤ χ + η1, ψs,f is d-R-i for s < χ + η1 and so Lemma 3.5 implies that λz( ¯f ) and Λz(f ) are holomorphic functions of z in the domain 0 < ℜz < χ+η1. Since κ has a simple zero at χ and Λχ(f ) = 0, 1−κ(z)Λz(f ) defines a holomorphic function in Bη(χ) for some η independent of f . For ℜz < χ, Λz(f ) = (1 − κ(z))λz( ¯f ). Hence, 1−κ(z)Λz(f ) is the holomorphic extension of λz( ¯f ) to Bη(χ). On the other hand,

λz( ¯f ) = Z

|g|>1

|g|−zf (g) dλ(g) + σ¯ f(z), whereR

|g|>1|g|−zf (g) dλ(g) is holomorphic for 0 < ℜz and σ¯ f(z) =R

|g|≤1|g|−zf (g) dλ(g). Indeed,¯ if suppλ = R+, then

Z

|g|>1

|g|−s dλ(g) = Z

1

a−sda a and if suppλ = hci then

Z

|g|>1

|g|−s dλ(g) = log cX

n≥1

c−ns.

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In any case,

Z

|g|>1

|g|−sf (g) dλ(g)¯ is convergent for s > 0. Therefore,

Λz(f ) 1 − κ(z)−

Z

|g|>1

|g|−zf (g) dλ(g)¯

is the holomorphic extension of σf(z) to the ball Bη(χ). Then from Landau lemma (see Appendix B), we infer that the abscissa of convergence of σf is larger than χ+η and the conclusion follows. 

For f ∈ Hε+ let

Cf = Cc({f > 0}) ∩ Hε+.

Now we are going to show in Proposition 3.9 that λs(¯h) is well defined for s < sprovided Λχ(f ) = 0 and h ∈ Cf. Before, we need a lemma.

Lemma 3.8. Assume that f ∈ Hε+ and h ∈ Cf. Then there is C > 0 such that ˆhs≤ C ˆfsfor every s > 0.

Proof. Notice that both ˆfsand ˆhsare continuous, Gµinvariant and supp(ˆhs) = Gµ(supph) therefore it is enough to prove that ˆfsis positive on the support of h. Let x ∈ supph, then x 6= 0 and f (x) > 0.

Then

|x|ss(x) = Z

Gµ

|g|−sf (gx) dλ(g) > 0.

and so ˆhs(x) ≤ ˆfs(x) for x ∈ supph. Finally,

ˆhs(gx) = ˆhs(x) ≤ C ˆfs(x) = C ˆfs(gx)

the conclusion follows. 

Proposition 3.9. Assume that f ∈ Hε+ satisfies Λχ(f ) = mχΛ(f ) = 0 and let h ∈ Cf. Then for every s ∈ [0, s)

λs(¯h) = E|Z|sˆhs(Z) < ∞.

Proof. Let

s0= sup{s < s: for every h ∈ Cf, λs(¯h) < ∞}.

By Lemmas 3.8 and 3.7,

CE|Z|sˆhs(Z) ≤ E|Z|ss(Z) = λs( ¯f ) < ∞

for s < χ + η i.e. s0 ≥ χ + η. If s0< s, we can find s1 < s and ε > 0 with s1εs2

1 < s0 < s1

and so for every h ∈ Cf

E[|Z|s1ε2s1ˆhs(Z)] < ∞.

Applying Lemmas 2.19 and 3.8 to X = suppˆhs, h ∈ Ch, we see that Λs1(h) is well defined. Now we argue as in the proof of the previous proposition, replacing χ by s1εs2

1 and χ + η by s1. Therefore λs1(¯h) < ∞.

This contradicts s0< sand so the proposition follows.  We start to describe the support of Λ. First we notice that outside suppΛ all the moments of Z smaller than s exist. This will lead to conclusions about the support of ν in Proposition 3.12.

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Lemma 3.10. Let W0 be an open subset of Σ1 such that pr−1(W0) ∩ suppΛ = ∅. Then for every s < s and ¯W ⊂ W0

E|Z|s1pr−1( ¯W) < ∞,

In particular, for every continuous φ, Aµ - homogeneous of degree 1 and supported in pr−1( ¯W ) Eφ(|Z|)s < ∞.

Proof. We have

E[|Z|s1pr−1( ¯W)(Z)] ≤ 1 + E[|Z|s1pr−1( ¯W)(Z)1|x|≥1(Z)].

Let h ∈ Hεbe a nonnegative function compactly supported in pr−1(W0) such that h = 1 if x ∈ ¯W . First we observe that for a positive D we have

(3.11) 1pr−1( ¯W)(x) ≤ D

Z

|a|≤1

h(ax) dl(a) 3.11 is clear for x ∈ ¯W becauseR

|a|≤1h(ax) dl(a) > 0 for x ∈ ¯W . If x /∈ Σ1, pr(x) ∈ ¯W then x = b¯x with |b| > 1, ¯x ∈ ¯W . Then

Z

|a|≤1

h(ax) dl(a) = Z

|a|≤|b|

h(a¯x) dl(a) ≥ Z

|a|≤1

h(a¯x) dl(a) > 0.

Let f, f ∈ Hε nonnegative compactly supported in pr−1(W0) such that h ∈ Cf and f ∈ Cf. Λ(f) = 0 and so by Proposition 3.9 λs( ¯f ) < ∞. As before we prove that ˆfs is positive on the support of h and so for a constant C

h ≤ Cs. Therefore,

E|Z|sh(aZ) ≤ CE|Z|ss(aZ)

= C Z

Gµ

|g|−sEf (gaZ) dλ(g)

= C|a|sλ¯s( ¯f ) and

E|Z|s1pr−1( ¯W)(Z)1|x|≥1(Z) ≤ D Z

|a|≤1

E|Z|sh(aZ) dl(a)

≤ DCλs(f ) Z

|a|≤1

|a|sdl(a) < ∞

 Proposition 3.12. Assume that the hypotheses of Theorem 3.1 are satisfied. Then for any open set W ⊂ Σ1, such that suppΛ ∩ pr−1( ¯W ) = ∅, the set suppν ∩ pr−1( ¯W ) is bounded. If s < ∞ then suppν ∩ pr−1( ¯W ) = ∅.

Proof. Since suppΛ is Cµ - invariant, we have also that suppΛ ∩ pr−1(CµW ) = ∅. Hence we¯ can assume that W is Cµ-invariant. Therefore, it suffices to show that for any non-negative φ ∈ C1(V \ suppΛ), Gµ - homogeneous of degree 1, φ(Z) is bounded. Let φ be such a function. In view of Lemma 3.10, E[φ(Z)s] < ∞ if s < s. Also since φ is homogeneous of degree 1, supx(x)| < ∞.

It follows |φ(Z − Q) − φ(Z)| ≤ C|Q|. We have

φ(Z0− Q0) = φ(M0Z1) = |M0|φ(Z1) and so

Eφ(Z0− Q0)s = κ(s)Eφ(Z1)s.

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