Large and moderate deviations for the left random walk on GL d (R)

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Large and moderate deviations for the left random walk on GL d (R)

Christophe Cuny, Jérôme Dedecker, Florence Merlevède

To cite this version:

Christophe Cuny, Jérôme Dedecker, Florence Merlevède. Large and moderate deviations for the left random walk on GL d (R). 2016. �hal-01386519�

Large and moderate deviations for the left random walk on GL

( R )

Christophe Cuny^a, J´erˆome Dedecker^b and Florence Merlev`ede^c

Abstract

Using martingale methods, we obtain some upper bounds for large and moderate deviations of products of independent and identically distributed elements ofGL_d(R). We investigate all the possible moment conditions, from super-exponential moments to weak moments of orderp >1, to get a complete picture of the situation. We also prove a moderate deviation principle under an appropriate tail condition.

a Universit´e de la Nouvelle-Cal´edonie, Equipe ERIM, et CentraleSupelec, Laboratoire MAS.

Email: cuny@univ-nc.nc

b Universit´e Paris Descartes, Sorbonne Paris Cit´e, Laboratoire MAP5 (UMR 8145).

Email: jerome.dedecker@parisdescartes.fr

c Universit´e Paris-Est, LAMA (UMR 8050), UPEM, CNRS, UPEC.

Email: florence.merlevede@u-pem.fr

1 Introduction

Let (Ω,F,P) be a probability space and (Y_n)_n≥1 be independent and identically distributed random variables on (Ω,F,P) taking values inG:=GL_d(R),d≥2 (the group of invertibled-dimensional real matrices), with common distributionµ. Denote by Γ_µthe closed semi-group generated by the support ofµ. Letk · k be the euclidean norm on R^d, and for every g∈GL_d(R), let kgk:= sup_kxk=1kgxk.

In all the paper, we assume thatµisstrongly irreducible, i.e. that no proper finite union of subspaces of R^d are invariant by Γ_µ and that it is proximal, i.e. that there exists a matrix in Γ_µ admitting a unique (with multiplicity one) eigenvalue with maximum modulus.

For such a measureµ, it is known that there exists a unique invariant measure ν on the projective spaceX:=P_d−1(R) (see for instance Theorem 3.1 of [5]) in the following sense: for any bounded Borel functionh fromX to R

Z

X

h(u)ν(du) = Z

G

Z

X

h(g·u)µ(dg)ν(du). (1.1)

Moreover, if

Z

G

logN(g)µ(dg)<∞, whereN(g) := max(kgk,kg⁻¹k), (1.2) then (see for instance Corollary 3.4 page 54 of [5] or Theorem 3.28 of [3]), for everyx∈S^d−1,

logkY_n· · ·Y₁xk

n _n−→

→+∞λ_µ= Z

G

Z

X

σ(g, u)µ(dg)ν(du) almost surely, (1.3) where

σ(g,x) = log¯

kg·xk kxk

for g∈GL_d(R) and x∈R^d− {0}, ¯x denoting the class ofx inX . Note that the functionσ defined above is a cocycle, in the following sense:

σ(gg^′, u) =σ(g, g^′·u) +σ(g^′, u) for any g, g^′ ∈Gand u∈X . (1.4) Let A_k=Y_k· · ·Y₁ fork≥1 andA₀= Id. In this paper we wish to study the asymptotic behavior of

sup

kxk=1

P

1max≤k≤n|logkA_kxk −kλ_µ|> n^αy

, (1.5)

when α∈(1/2,1], under stronger moment conditions on logN(Y₁) than (1.2). This is a way to study rates of convergence in the strong law (1.3). In the probabilistic terminology, the case α ∈ (1/2,1) corresponds to themoderate deviation regime, andα = 1 to thelarge deviation regime.

The case α= 1/2 corresponds to the normalization of the central limit theorem. In that case, the asymptotic behavior of (1.5) is due to Benoist and Quint [4] as soon as logN(Y₁) has a moment of order 2 (note that Benoist and Quint do not deal with the maximum in (1.5), but their method also applies in that case, see also Theorem 1(ii) in [6]). A previous result is due to Jan [18] under a moment of order 2 +ǫ,ǫ >0.

In this paper, we shall give precise informations on the rate of convergence to 0 (asn→ ∞) of (1.5) when α ∈(1/2,1], under various moment conditions on the random variable logN(Y1): sub or super- exponential moments in Section 2, weak moments of order p >1 in Section 3, and strong moments of order p≥1 in Section 4.

In Section 2.2 we shall give a moderate deviation principle for the process n^−α(logkA_[nt]xk −[nt]λµ), t∈[0,1]

when logN(Y₁) satisfies Arcones’s tail condition [1] (which is true under an appropriate sub-exponential moment condition). In Section 4 we obtain some results in the spirit of Baum and Katz [2] which complement the results on complete convergence obtained in [4] in the case α = 1. When logN(Y₁) has a strong moment of order p∈(1,2) and α= 1/p, this gives the rate n^(p−1)/p in the strong law of large numbers, which was proved by another method in [6], Theorem 1(i).

All along the paper, the following notations will be used: letF0 ={∅,Ω} and Fk =σ(Y₁, . . . , Y_k), for any k ≥ 1. For any x ∈ S^d−1, define X_0,x = x and X_n,x = σ(Y_n, A_n−1x) for n ≥ 1. With these notations, for anyx∈S^d−1 and any positive integer k,

logkA_kxk=

k

X

i=1

X_i,x. (1.6)

The equality (1.6) follows easily from the fact that σ is a cocycle (i.e. (1.4) holds). In Section 6 we shall present some extensions of our results to general cocycles, in the spirit of Benoist and Quint [4].

2 The case of (sub/super) exponential moments

2.1 Upper bounds for large deviations

Letr >0. In this subsection, we assume that Z

G

e^δ(logN(g))rµ(dg)<∞, for someδ >0. (2.1) We first consider the case r≥1. In that case, using the spectral gap property, Le Page [19] proved the following large deviation principle: there exists a positive constant Asuch that, for anyy ∈(0, A),

nlim→∞

1

nlogP(|logkA_nxk −nλ_µ|> ny) =φ(y). (2.2) Of course, this is the best possible result for y∈(0, A). However, it does not give any information for large values ofy, and the rate function φ is not explicit (in particular, one cannot easily describe the behavior ofφwhen r varies in [1,∞)).

The following result, which is obtained via a completely different method, can be seen as a com- plementary result of (2.2). It gives an explicit (up to a constant) upper bound for φwhen y ∈(0, A), and this upper bound is valid for any y > 0. In particular, we can see the qualitative change in the behavior of large deviations for largey when r varies in [1,∞).

Theorem 2.1. Assume that (2.1) holds for some r≥1. Then there exists a positive constant C such that, for any y >0,

lim sup

n→∞

1

nlog sup

kxk=1

P

1max≤k≤n|logkA_kxk −kλ_µ|> ny

≤ −C y²1_y∈(0,1)+y^r1_y≥1

. (2.3) Forr ∈(0,1), there is no such result as (2.2). Instead, one can prove:

Theorem 2.2. Assume that (2.1) holds for some r ∈ (0,1). Then there exists a positive constant C such that, for any y >0,

lim sup

n→∞

1

n^rlog sup

kxk=1

P

1max≤k≤n|logkA_kxk −kλ_µ|> ny

≤ −Cy^r. (2.4)

Proofs of Theorems 2.1 and 2.2. SinceR

(logN(g))²µ(dg)<∞, we infer from the equality (3.9) in [4] that, for anyx∈S^d−1,

X_k,x−λ_µ=D_k,x+ψ(A_k−1x)−ψ(A_kx), (2.5) where ψ is a bounded function and D_k,x is Fk-measurable and such that E(D_k,x|Fk−1) = 0. The decomposition (2.5) is called amartingale-coboundary decomposition. Such a decomposition has been used for the first time in the paper [13] by Gordin (see also [14]).

Starting from (1.6) and (2.5), for any x∈S^d−1 and any positive integer k,

logkA_kxk −kλ_µ=M_k,x+ψ(x)−ψ(A_kx), (2.6) where M_k(x) =D1,x+· · ·+D_k,x is a martingale adapted to the filtration Fk. Clearly, since|ψ(x)− ψ(A_kx)| ≤2kψk∞, it is equivalent to prove (2.3) and (2.4) forM_k,x instead of (logkA_kxk −kλ_µ).

To do this, we first note that if (2.1) holds, then

E

e^δ|Xk,x|r Fk−1

∞=

Z

e^{δ|σ(g,Ak−1x)|r}µ(dg) ∞

≤ Z

G

e^δ(logN(g))rµ(dg)<∞. (2.7) Using again thatψ is bounded we infer from (2.5) and (2.7) that there exists a constantK such that

sup

kxk=1

E

e^δ|Dk,x|r Fk−1

∞< K , (2.8)

for any positive integerk.

Starting from (2.8), it remains to apply known results to the martingaleM_k(x).

To prove (2.3) (caser≥1), we apply Theorem 1.1 of [20], which implies that there exists a positive constant csuch that, for any y >0,

sup

kxk=1

P

1max≤k≤n|M_k,x|> ny

≤2 exp −nc y²1_y∈(0,1)+y^r1_y≥1

. (2.9)

The upper bound (2.3) follows directly from (2.9). Note that a direct application of Theorem 1.1 of [20] gives us (2.9) without the maximum over k. However, a careful reading of the proof reveals that one can take the maximum over k. The only argument that should be added to the proof is Doob’s maximal inequality for non-negative submartingales, which implies that

E

e^{λmax1≤k≤nMk,x}

≤E

e^λMn,x

, for any λ >0.

To prove (2.4) (case r ∈ (0,1)), we apply Theorem 2.1 of [12] (see also the proof of Proposition 3.5 of [7]) and more precisely the upper bound (13) in [12], which implies that there exist two positive constantsc₁ and c₂ such that

sup

kxk=1

P

1max≤k≤n|M_k,x|> ny

≤4 exp (−c₁(ny)^r), for any y > c₂n^{−(1−r)/(2−r)}. (2.10) The upper bound (2.4) follows directly from (2.10). ♦

2.2 A moderate deviation principle

Let (b_n)_n≥0 be a sequence of positive numbers satisfying the following regularity conditions:

The functionsf(n) =n²

b²_n andg(n) =b²_n are strictly increasing to infinity, and lim

n→∞

n

b²_n = 0. (2.11) Forx∈S^d−1, let

Zn,x =

logkA_[nt]xk −[nt]λ_µ

b_n , t∈[0,1]

.

The processZn,x takes values in the spaceD([0,1]) equipped with the usual Skorokhod topology. The following functional moderate deviation principle holds:

Theorem 2.3. Let (b_n)_n≥0 be a sequence of positive numbers satisfying the condition (2.11). Assume R(logN(g))²µ(dg)<∞ and

lim sup

n→∞

n

b²_nlognµ{logN > b_n}=−∞. (2.12) Then, for any x ∈ S^d−1, n⁻¹E((logkA_nxk −nλ_µ)²) → V as n→ ∞, where V does not depend on x.

Moreover, for any Borel setΓ⊂D([0,1]),

− inf

ϕ∈Γ^◦I_V(ϕ)≤lim inf

n→∞

n

b²_nlog inf

kxk=1

P(Z_n,x ∈Γ)≤lim sup

n→∞

n

b²_nlog sup

kxk=1

P(Z_n,x ∈Γ)≤ −inf

ϕ∈Γ¯I_V(ϕ), (2.13) where

IV(h) = 1 2V

Z 1 0

h^′(u)2

du

if simultaneously V >0, h(0) = 0and h is absolutely continuous, and I_V(h) = +∞ otherwise.

Remark 2.1. Let (b_n)_n≥0 be a sequence of positive numbers satisfying (2.11). If (X_i)_i≥1 is a sequence of independent and identically distributed random variables, Arcones [1] proved that the functional moderate deviation principle holds providedE(X₁²)<∞ and

lim sup

n→∞

n

b²_nlognP(|X₁|> b_n) =−∞. (2.14) Moreover, he showed that condition (2.14) is also necessary for the moderate deviation principle. Note that our condition (2.12) is exactly Arcones’s tail condition for the random variable logN(Y1). When b_n=n^α with α∈(1/2,1), the tail condition (2.12) is true if

µ{logN > x} ≤e^−xβa(x),

for β = 2−(1/α) and a function a such that a(x) → ∞ as x → ∞ (note that β ∈ (0,1), so only a sub-exponential moment is needed for logN(Y₁)).

Remark 2.2. Applying the contraction principle, Theorem 2.3 implies in particular that, for any Borel set Γ⊂R⁺,

−inf y²

2V, y∈Γ^◦

≤lim inf

n→∞

n

b²_nlog inf

kxk=1

P

max_1≤k≤n|logkA_kxk −kλµ|

b_n ∈Γ

≤lim sup

n→∞

n

b²_nlog sup

kxk=1

P

max_1≤k≤n|logkA_kxk −kλµ|

b_n ∈Γ

≤ −inf y²

2V, y∈Γ¯

. Note that a partial result in this direction has been obtained in [3], Proposition 11.12. In that Propo- sition, the authors proved a moderate deviation principle for (logkAnxk −nλµ) and the collection of open intervals, under an exponential moment for logN(Y₁). However, their result is stated in a more general framework than ours (see Section 6 of the present paper for an extension of Theorem 2.3 to general cocycles).

Proof of Theorem 2.3. Since R

(logN(g))²µ(dg) < ∞, the decomposition (2.6) holds, and it is equivalent to prove (2.13) for the process

Z˜_n,x =

M_[nt],x

b_n , t∈[0,1]

instead of Z_n,x. Now, by a standard argument, to get the result uniformly with respect to x ∈ S^d−1 in (2.13), it suffices to prove the functional moderate deviation principle for the process ˜Z_n,xn, where (x_n)_n≥1 is any sequence of points in S^d−1.

The result will follow from the next proposition, which is a triangular version of Theorem 1 in [11]. This proposition is in fact a corollary of a more general result for triangular arrays of martingale differences which can be deduced from Puhalskii’s results and Worms’s paper (see [22] and [23]). We refer to Theorem 5.1 of the Appendix for a complete statement and some elements of proof.

Before giving the statement of this proposition, we need more notations. Assuming (2.11), we can construct the strictly increasing continuous function f(x) that is formed by the line segments from (n, f(n)) to (n+ 1, f(n+ 1)). Similarly we defineg(x) and denote by

c(x) =f⁻¹(g(x)). (2.15)

Proposition 2.4. Let d_i,n)_1≤i≤n be a triangular array of real-valued square-integrable martingale differences, adapted to a triangular array of filtrations(Fi,n)_0≤i≤n. Let(b_n)_n≥0 be a sequence of positive numbers satisfying (2.11), and let

Z¯n=

d_1,n+· · ·+d_[nt],n

b_n , t∈[0,1]

. Assume that the three following conditions holds

1. There exists a positive number V such that, for any δ >0 and any t∈[0,1], lim sup

n→∞

n b²_nlogP







 1 n

[nt]

X

i=1

E(d²_i,n|Fi−1,n)



−V t

> δ



=−∞. (2.16)

2. For any ε >0 and δ >0, lim sup

n→∞

n

b²_n logP 1 n

n

X

i=1

E

d²_i,n1_|d

i,n|>εnb⁻¹n

Fi−1,n

> δ

!

=−∞. (2.17)

3.

n

b²_nlog sup

n≤m≤c(n+1)

sup

1≤k≤m

n

P |d_k,m|> b_n|Fk−1,m

∞

!

→ −∞ as n→ ∞, (2.18) where c(n) is defined in (2.15).

Then, for any Borel set Γ⊂D([0,1]),

− inf

ϕ∈Γ^◦I_V(ϕ)≤lim inf

n→∞

n

b²_nlogP Z¯_n∈Γ

≤lim sup

n→∞

n

b²_nlogP Z¯_n∈Γ

≤ −inf

ϕ∈Γ¯I_V(ϕ), (2.19) where I_V is defined as in Theorem 2.3.

Let us conclude the proof of Theorem 2.3. Let (xn)n≥1be any sequence of points inS^d−1. We apply Proposition 2.4 to the martingale differencesd_i,n=D_i,xn (recall that D_i,x is the martingale difference of the decomposition (2.5)). Condition (2.17) is clearly satisfied thanks to (2.5) and the fact that

E

X_k,x² _n1_|X

k,xn|>εnb⁻¹n

Fk−1

∞=

Z

(σ(g, A_k−1x_n))²1_|σ(g,A

k−1xn)|>εnb⁻¹n µ(dg) ∞

≤ Z

G

(logN(g))²1_logN(g)>εnb−1 n µ(dg). To check Condition (2.16), we apply Proposition 3.1 in [4], which implies that, for any δ > 0 and any t∈[0,1], there exist A >0 and α >0 such that, for the variance V defined in Theorem 2.3,

P







 1 n

[nt]

X

i=1

E(D²_i,xn|Fi−1)



−V t

> δ



≤Ae^−αn. Condition (2.16) follows then easily, since n²b⁻_n²→ ∞ asn→ ∞.

It remains to check Condition (2.18). By (2.5) again, it is equivalent to prove the condition for X_k,xm instead ofD_k,xm. Now

E

1_|X

k,xm|≥bn

Fk−1

∞=

Z

1_{σ(g,Ak−1xm)≥bn}µ(dg) ∞

≤µ{logN ≥bn} , and the result follows by (2.12). ♦

3 The case of weak moment of order p > 1

In this section, we study the asymptotic behavior of (1.5) when logN(Y₁) has only a weak moment of order p >1.

Theorem 3.1. Let p >1 and and assume that sup

t>0

t^pµ{logN > t}<∞. (3.1)

Let α∈(1/2,1] and α≥1/p. Then there exists a positive constant C such that, for any y >0, lim sup

n→∞ n^αp−1 sup

kxk=1

P

1max≤k≤n|logkA_kxk −kλ_µ|> n^αy

≤ C

y^p . (3.2)

Proof of Theorem 3.1.

The case p >2. In that case the decomposition (2.6) holds, and it is equivalent to prove (3.2) for M_k,xinstead of (logkA_kxk −kλ_µ). To do this, we shall apply the following inequality due to Haeusler [16]: for all γ, u, v >0,

P

1max≤k≤n|M_k,x| ≥γ

≤

n

X

i=1

P(|D_i,x| ≥u) + 2P

n

X

i=1

E(D²_i,x|Fi−1)≥v

!

+ 2 exp γu⁻¹ 1−log γuv⁻¹

. (3.3) Note that if (3.1) holds forp >2, then

E X_k,x² |Fk−1

∞=

Z

(σ(g, A_k−1x))²µ(dg) ∞

≤ Z

G

(logN(g))²µ(dg)<∞, (3.4) and there exists a positive constant C such that

E

1_|Xk,x|≥u Fk−1

∞=

Z

1_{σ(g,Ak−1x)≥u}µ(dg) ∞

≤µ{logN ≥u} ≤ C

u^p , (3.5) for anyu >0 and any positive integer k. Using again thatψ is bounded we infer from (2.5) and (3.4) that there exist two positive constants c₁, c₂ such that

sup

kxk=1

E D²_k,x|Fk−1

∞≤c₁, (3.6)

sup

u>0

u^p sup

kxk=1

P(|D_k,x| ≥u)≤sup

u>0

u^p sup

kxk=1

E

1_|Dk,x|≥u|Fk−1

∞≤c2, (3.7) for any positive integerk. Taking γ =n^αy,u=n^αy/r with r∈(0,∞), and v= 2nc₁ in (3.3), we get

P

1max≤k≤n|M_k,x| ≥n^αy

≤c₃ 1

y^pn^αp−1 + 1 y^2rn^(2α−1)r

, (3.8)

for some positive constantc₃. Selecting r >(αp−1)/(2α−1), the upper bound (3.2) follows directly from (3.8).

The case p∈(1,2). Let x∈S^d−1. We have logkA_nxk −nλ_µ=

n

X

k=1

(X_k,x−λ_µ) =

n

X

k=1

(D_k,x+R_k,x) :=M_n,x+U_n,x,

where

D_n,x =σ(Y_n, A_n−1x)− Z

G

σ(g, A_n−1x)µ(dg), and R_n,x = Z

G

σ(g, A_n−1x)µ(dg)−λ_µ. Notice that E(D_k,x|Fk−1) = 0. We use now the basic inequality

P

1max≤k≤n|logkA_kxk −kλ_µ| ≥n^αy

≤P

1max≤k≤n|M_k,x| ≥n^αy/2

+P

1max≤k≤n|U_k,x| ≥n^αy/2

. (3.9) We first deal with the second term on the right-hand side of (3.9). We shall need the following extension of Theorem 3 in [24]. The proof is given in Appendix.

Theorem 3.2. Let p ∈]1,2[ and (X_k)_k∈Z be a sequence of real-valued random variables in L^p and adapted to a non-decreasing filtration (Fk)_k∈^Z. Let Si=X1+· · ·+Xi and S_n^∗ = max1≤i≤n|Si|. Then, for any n≥1,

kS^∗_nkp ≤(2c_p+ 1)





n

X

j=1

kX_jk^pp





1/p

+ 2^(p−1)/p(2c_p+ 1)

r−1

X

j=0





2^r−j

X

k=1

kE(S_k2^j −S_(k−1)2^j|F(k−1)2^j)k^pp





1/p

, (3.10) where cp = 2^{1/p p}_p−1 and r is the unique positive integer such that 2^r−1 ≤n <2^r.

For k ≤ 0, set R_k,x =R_0,x and Fk =F0. Observe that |R_k,x| ≤ R

GlogN(g)µ(dg) < ∞ for every k≥0. Hence we may apply Theorem 3.2 withX_k:=R_k,x. With that choice, we have

S_k2j−S_(k−1)2j =

2^j

X

ℓ=1

R_(k−1)2j+ℓ,x,

and, using independence (twice), E(R_(k−1)2^j_+ℓ,x|F(k−1)2^j)

≤ sup

y∈S^d−1

Z

G

(E(σ(g, A_ℓ−1y))−λ_µ)µ(dg)

= sup

y∈S^d−1|E(X_ℓ,y)−λ_µ|. Letn≥1 andr ≥1 be such that 2^r−1 ≤n <2^r. We infer that there existsC_p>0, such that

1max≤k≤n|U_k,x| p

≤C_pn^1/p+C_p

r−1

X

j=0

2^(r−j)/p

2^j

X

ℓ=1

sup

kyk=1|E(X_ℓ,y)−λ_µ|

≤C_pn^1/p+ Cp2^1/p

2^1/p−1n^1/pX

ℓ≥1

sup_kyk=1|E(X_ℓ,y)−λ_µ|

ℓ^1/p . (3.11)

Recall that (3.1) holds forp∈(1,2). Hence, for any r < p, by (6) of [6], X

n≥1

n^r−2 sup

kyk=1|E(X_n,y)−λ_µ|<∞. (3.12)