Random fractals generated by a local gaussian process indexed by a class of functions

DOI:10.1051/ps/2010003 www.esaim-ps.org

RANDOM FRACTALS GENERATED BY A LOCAL GAUSSIAN PROCESS INDEXED BY A CLASS OF FUNCTIONS

Claire Coiffard

Abstract. In this paper, we extend the results of Orey and Taylor [S. Orey and S.J. Taylor, How often on a Brownian path does the law of the iterated logarithm fail? Proc. London Math. Soc. 28(1974) 174–192] relative to random fractals generated by oscillations of Wiener processes to a multivariate framework. We consider a setup where Gaussian processes are indexed by classes of functions.

Mathematics Subject Classification. 60J65, 28A80.

Received November 25, 2008. Revised July 24, 2009.

1. Introduction

Let{W(t), t∈[0,1]}denote a Wiener process. L´evy [8] studied the modulus of continuity ofW, and obtained the following limiting law. We have

h→0lim sup

0≤t≤1−h

W(t+h)−W(t)

2hlog(1/h) = 1, p.s. (1.1)

This result shows that some points of a Brownian path don’t follow the usual law of iterated logarithm. Ac- cording to this law, for each fixedt₀∈[0,1],

lim sup

h→0

W(t₀+h)−W(t₀)

√2hlog logh = 1, p.s. (1.2)

Orey and Taylor [11] introduced the random sets defined, for Λ∈[0,1], by E_Λ=

t∈[0,1] : lim sup

h↓0 (2hlog(1/h))^−1/2(W(t+h)−W(t))≥Λ

.

For each Λ>0 E_Λ collects the exceptional points in [0,1] where the law of the iterated logarithm (1.2) fails.

Orey and Taylor [11] showed that, with probability 1,E_Λis a random fractal with Hausdorff dimension, given by

dimE_Λ= 1−Λ². (1.3)

Keywords and phrases. Random fractals, Hausdorff dimension, Wiener process.

1LSTA, Universit´e Pierre et Marie Curie, 175 rue du Chevaleret, 75013 Paris, France;[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2011

Recall (see,e.g., Falconer [7]) that the Hausdorff dimension ofE⊂[0,1] is defined by dimE= inf{c >0 :s^c−mes(E) = 0}= sup{c >0 :s^c−mes(E) =∞}, wheres^c−mes(E) denotes the Hausdorff measure of E, given by

s^c−mes(E) = lim

δ→0inf

i≥1

|Ui|^c :E⊆

i≥1

U_i,|Ui| ≤δ

. (1.4)

We denote by |U_i|the diameter ofU_i, namely, the supremum of the Euclidean distance between two elements ofU_i. The infimum in (1.4) is taken over all collections{U_i:i≥1}of subsets with diameter|U_i|< δfor alli≥1 and such that E ⊆ _i≥1U_i. The identity (1.3) was extended in various directions. In particular, Deheuvels and Mason [4] and Deheuvels and Lifshits [3] established functional versions of (1.3). Dindar [6] extended this result to Wiener processes onR². Our work will rely in part on the approach of Mason [10] where processes are indexed by class of functions.

The aim of this paper is to provide a largely extended version of (1.1) and (1.3) in the framework of multipa- rameter Gaussian processes, indexed by classes of functions. We start by giving some notation which is needed for the statement of our results.

We consider a classF of bounded functions onI^d= [0,1]^d. We denote by f the sup-norm of the function f ∈ F. Let| · |₂ denote the usual Euclidean norm onR^d. Assume that the classF satisfies :

F.i. lim_|w|2→0sup_f∈F

R^d(f(x)−f(x+w))²dx= 0, F.ii. lim_λ→1sup_f∈F

R^d(f(x)−f(λx))²dx= 0, F.iii. for eachλ≥1,z∈R^d andf ∈ F,f(z−λ·)∈ F, F.iv. F is aVC-subgraph class,

F.v. F is pointwise measurable class. That is, there exists a countable subclassF₀ of F such that for each functionf ∈ F , there exists a sequence of functions{fm} ∈ F₀ such thatf_m(z)→f(z), z∈R^d . A collection of measurable functions F on a sample space X is called a VC-subgraph class if the collection of all subgraphs of the functions in F forms a VC-class of sets in X ×R. For a definition of a VC-class or Vapnik- ˘Cernonenkis class, we refer to p. 141 in Van der Vaart and Wellner [14]. A VC-class satisfies an entropy condition, of the following type. For eachε >0, the covering number N(ε,F, · ) is the minimal number of balls{g: g−f < ε}of radiusεneeded to cover the setF. For someA >0, the covering numberN(ε,F, · ) of a VC-classF grows polynomially inA/εasε↓0.

The multivariate local Gaussian process atz∈R^d, indexed byf ∈ F, is defined by W(h,z;f) =

R^df z−u

h^1/d

dW(u), (1.5)

whereW(z),z∈R^d denotes a standard Wiener process withdparameters inR^d. We set

Θ_h,z(f) = W(h,z;f) 2hlog(1/h)=

R^df(h^−1/d(z−u))dW(u)

2hlog(1/h) · (1.6)

We define an inner product off₁, f₂∈ F by setting f₁, f₂L2 :=

I^d

f₁(u)f₂(u)du.

LetG₂(F) be the Hilbert subspace ofL₂(F) spanned byF. For eachϕ∈G₂(F), we denote by Θϕthe functional Θ_ϕ(f) =

I^df(u)ϕ(u)du. For each Λ∈[0,1], we set H_Λ =

Θ_ϕ(f), f ∈ F, ϕ∈G₂(F) with

I^d

ϕ²(u)du≤Λ²

. (1.7)

Let_∞(F) denote the class of bounded functions onF, endowed with the sup-norm · F. For anyϑ∈ H1and ε >0, set

B_ε(ϑ) ={ψ∈_∞(F) : ψ−ϑ _F < ε}. (1.8) Finally, for anyε >0, we set

H₁^ε={ψ∈_∞(F) : inf

ϑ∈H1 ψ−ϑ F< ε}.

The following result gives a uniform functional law of the logarithm for local Gaussian processes indexed by a class of functions.

Theorem A. Let J be a compact subset of R^d with nonempty interior. Suppose that (F.i-v) hold, and set for γ >0,

h⁻¹_k =(1 +γ)^k. Then, with probability 1,

(a) for each ε >0, there exists aδsuch that, for all 0< h≤δ,{Θh,z:z∈J} ⊂ H^ε₁,

(b) for each ϑ∈ H1 and ε >0, there exists a k(ϑ, ε)such that for all k ≥k(ϑ, ε), there exists a zk ∈J such thatΘ_hk,zk ∈B_ε(ϑ).

Proof. The proof of this theorem is given in Mason [9].

Now for eachϕ∈G₂(F), following the lines of Orey and Taylor [11], we consider the set of points defined by L(Θ_ϕ) =

z∈I^d: lim inf

h↓0 Θ_h,z−Θ_{ϕ F} = 0

. (1.9)

This set collects the points of I^d where Θ_hd,z is infinitely often in a neighborhood of the function Θ_ϕ. Set further, for Λ≥0

L_Λ=

L(Θ_ϕ),Θ_ϕ∈ H₁, ϕ∈G₂(F),

I^d

ϕ²(u)du≥Λ²

. (1.10)

Our main result, stated in the following theorem, evaluates the Hausdorff dimensions ofL(Θ_ϕ) andL_Λ. Theorem 1.1. Assume that F fulfills F.i-v, and let ϕ∈G₂(F) be such that

I^d

ϕ²(u)du≤1.

Then for each Λ∈[0,1], the setsL(Θ_ϕ) andL_Λ are almost surely everywhere dense inI^d and satisfy dimL(Θ_ϕ) =d

1−

I^d

ϕ²(u)du

and dimL_Λ=d(1−Λ²). (1.11) The proof of Theorem1.1is based on an adaptation of the arguments of Deheuvels and Mason [5] and Deheuvels and Lifshits [3] and is given in Section2.

2. Proof of Theorem 1.1

Let the assumptions of Theorem1.1be in force. The proof of (1.11) reduces to show that with probability 1, for all 0≤Λ≤1,

dimL_Λ≤d(1−Λ²), (2.1)

and, for all Θ_ϕ∈ H1 andϕ∈G₂(F),

dimL(Θ_ϕ)≥d

1−

I^d

ϕ²(u)du

. (2.2)

To establish (2.1) and (2.2) we will need the preliminary facts given in the next section. In Section2.2we will establish (2.1) and in Section2.3we will provide a proof for (2.2).

2.1. Preliminary facts

Fact 2.1 below is a generalization of the well-known result of Schilder [12] relative to large deviations. For anyψ∈_∞(F), we set

I(ψ) = ₁

2

I^dξ²(u)du whenψ(f) =f, ξL2 for some ξ∈G₂(F),

∞ otherwise.

Also, for any subsetB⊂_∞(F), we set

I(B) = inf{I(ψ) :ψ∈B}.

Fact 2.1. Let F be a class of functions fulfilling Fi-Fii, and let {k:k≥1} be a sequence of constants such that _k →0 whenk→ ∞. Setε_k = (2 log(1/_k))⁻¹, fork= 0,1, . . .. Then,

(i) for each closed subsetF of _∞(F) lim sup

k→∞ ε_klogP{Θk,z∈F} ≤ −I(F), (2.3)

(ii) for each open subset Gof _∞(F) lim inf

k→∞ ε_klogP{Θk,z∈G} ≥ −I(G). (2.4)

Proof. It follows readily from Theorem 5.2 in Arcones [1]. (This same method is used for the proof of Prop. 1

in Mason [10].)

The next fact will be instrumental in the proof of (2.2).

Fact 2.2. Let A⊆I^d be such thatA =_∞

m=1E_m, where E₁⊇E₂ ⊇. . . and E_m = ^M_k=1^mJ_m,k, with{J_m,k : 1≤k≤M_m}being for eachm≥1, a collection of disjoint hypercubes ofR^d, such thatmax_1≤k≤Mm|Jm,k| →0 and M_m → ∞ as m → ∞. Then, whenever there exist two constants Δ >0 andδ > 0 such that, for every hypercube J⊆I^d with |J| ≤Δ, there is a constant m(J) such that for allm≥m(J)

M_m(J) := #{J_m,k⊆J : 1≤k≤M_m} ≤δ|J|^cM_m, (2.5) we have s^c−mes(A)>0.

Proof. See,e.g., Lemma 2.2 of Orey and Taylor [11].

The next fact gives a useful property of the binomial distribution.

Fact 2.3. Let S_N follow a binomial distribution with parametersN andp. Then, for allr∈[1,∞],

P(SN ≥N rp)≤exp(−N ph(r)), (2.6)

and for all r∈[0,1],

P(SN ≤N rp)≤exp(−N ph(r)), (2.7)

wherehis the Chernoff function associated with the standard Poisson process and defined by

h(r) =

⎧⎪

⎨

⎪⎩

rlogr−r+ 1 for r >0,

1 for r= 0,

∞ otherwise.

Proof. The proof of (2.6) and (2.7) is based on Markov inequalities (See,e.g., Chernoff [2]). This result is in

Lemma 3.8 of Deheuvels and Mason [5]).

2.2. Upper bounds

In this part, we establish (2.1). Actually, it is sufficient to show the result for 0<Λ<1. Indeed, if Λ₁≤Λ₂, L_Λ2 ⊆ L_Λ1 and by the properties of the Hausdorff dimension, dimL_Λ2 ≤ dimL_Λ1. So, for all 0 < Λ < 1, dimL₁≤d(1−Λ²). Since Λ∈(0,1) is arbitrary, dimL₁= 0. The inequality dimL₀≤dis trivial, and so, it is enough to prove (2.1) for 0<Λ<1.

In order to establish (2.1), we must first fix some notation. Set L_Λ+ =

L(Θ_ϕ),Θ_ϕ∈ H1, ϕ∈G₂(F),

I^d

ϕ²(u)du>Λ²

. (2.8)

Remember that for every setG, the neighborhood ofGin _∞(F) is defined by G^ε={φ∈_∞(F) : inf

ϑ∈G φ−ϑ F < ε}.

We set

L(h, ε,Λ) ={z∈I^d: Θ_h,z∈ H/ ^ε_Λ}, (2.9) and

L(ε,Λ) ={z∈I^d: Θ_h,z∈ H/ _Λ^ε i.o}. (2.10) We can see that for all Λ>0 and for all integerm₀≥1, we have

L_Λ+ ⊆ ^∞

m=m0

L(1/m,Λ).

Therefore, in order to establish (2.1), it is enough to show that for all 0<Λ<1 andε >0,

s^d(1−Λ2)−mes(L(ε,Λ))<∞. (2.11)

The following lemma will be crucial to control the oscillations between two points. For f ∈ F, consider the process

Y(f) =

R^df(u)dW(u).

Notice that forf₁, f₂∈ F, the usual pseudo metric betweenf₁ andf₂ is defined by

ρ(f₁, f₂) =

E(Y(f₁)−Y(f₂))²=

R^d(f₁−f₂)²(u)du.

Lemma 2.4. There exists a function ψ(δ)of δ >0, fulfillingψ(δ)→0 asδ↓0, and such that lim sup

h↓0 sup

ρ(f1,f2)≤√ δh

|Y(f₁)−Y(f₂)|

2hlog(1/h) =ψ(δ) a.s.

Proof. See,e.g., Mason [9].

Remember that| · |2 stands for the Euclidean norm. By F.i-ii, there exists a functionA(δ),δ >0, such that

δ→0limA(δ) = 0, (2.12)

f∈Fsup sup

|w|2≤δ

R^d(f(x)−f(x+w))²dx≤A(δ). (2.13) sup

f∈F sup

1−δ≤λ≤1+δ

R^d(f(x)−f(λx))²dx≤A(δ). (2.14) The following lemma will be needed to apply Lemma2.4in our proofs.

Lemma 2.5. For all 0< δ <1, andt,t ∈I^d andh, h >0,

f∈Fsup sup

|t−t|2≤δh^1/dE(W(h,t;f)−W(h,t;f))²≤A(δ)h. (2.15) and

f∈Fsup sup

(1−δ)^d≤h/h≤(1+δ)^dE(W(h,t;f)−W(h,t;f))²≤A(δ)h. (2.16) Proof. We start with (2.15). Observe that

E

W(h,t;f)−W(h,t;f) ₂

=E

R^df t−u

h^1/d −f

t −u h^1/d

dW(u)

₂

=

R^d

f

t−u h^1/d

−f

t −u h^1/d

₂ du

=h

R^d

f(x)−f

x+t −t h^1/d

₂ dx.

Since|h^−1/d(t −t)|2≤δ, we infer (2.15) from (2.13). The proof of (2.16) is very similar, since E

W(h,t;f)−W(h,t;f)₂

=

R^d

f

t−u h^1/d

−f t−u

h ^1/d ₂

du

=h

R^d

f(x)−f((h/h)^1/dx)²

dx.

Given this relation, we infer (2.16) from (2.14) in order to obtain (2.16).

Denote byx ≤x <x+ 1 the integer part of x. Forγ >0, set

ν_k =(1 +γ)^k/d, k= 1,2, . . . (2.17)

and for some integerK≥1 and for everyk≥1, set τ_k(i_r, K) = i_r

Kν_k for 0≤i_r≤Kν_k and 1≤r≤d.

In the multivariate framework, we use the notationi= (i₁, . . . , i_d) and we set τ_k(i, K) =

τ_k(i₁, K), . . . , τ_k(i_d, K) .

Finally, we also set

½i1,...,id,k(ε,Λ) =^½{Θ_1/ν^d

k,τk(i,K)∈ H/ ^ε/2_Λ }, (2.18)

I_i1,...,id,k(ε) =

dr=1

τk(ir√−1,K)

d ,^τk(ir√^+1,K) d

if^½_i1,...,id,k(ε,Λ) = 1,

∅ otherwise. (2.19)

Lemma 2.6. For all K≥1 large enough, there exists almost surely an N <∞ such that for allk≥N, V_k(ε,Λ) :=

1/ν^d_k≤h<1/ν_k−1^d

L(h, ε,Λ)⊆

0≤i1≤Kνk

. . .

0≤id≤Kνk

I_i1,...,id,k(ε), (2.20)

Proof. Lett∈ _1/ν^d

k≤h<1/ν_k−1^d L(h, ε,Λ). Then, Θ_h,t∈ H/ ^ε_Λ for someh∈[1/ν_k^d,1/ν_k−1^d [, which means that for any Θ_ϕ∈ HΛ,

Θh,t−Θ_{ϕ F}> εfor someh∈[1/ν^d_k,1/ν_k−1^d [.

We choosei= (i₁, . . . , i_d) such thatt∈_d

r=1

τk(ir√−1,K)

d ,^{τk(ir√+1,K)}

d

. Therefore, by triangular inequality, Θ_1/νd

k,τk(i,K)−Θ_{ϕ F} ≥ Θ_h,t−Θ_{ϕ F}− Θ_h,t−Θ_{h,τk(i,K) F}− Θ_h,τk(i,K)−Θ_1/νd

k,τk(i,K) F. (2.21) We first bound the term Θ_h,t−Θ_{h,τk(i,K) F}. Recalling (2.19), we see that

|t−τ_k(i, K)|₂≤ 2 Kν_k.

By applying Lemma2.5toδ= 2/K, we get sup

f∈FE(W(h,t;f)−W(h, τ_k(i, K);f))²≤A 2

K

1/ν_k^d. Notice that the functionb(h) =

2hlog(1/h) is increasing for 0< h <e⁻¹. We infer from Lemma2.4that for some 1/ν_k^d≤h <1/ν_k−1^d , there exists a functionψ(δ) ofδ >0, such thatψ(δ)→0 asδ↓0, and

lim sup

k→∞ sup

f∈F

|W(h,t;f)−W(h, τ_k(i, K);f)|

2hlog(1/h) = lim sup

k→∞ sup

ρ(f1,f2)≤√

A(^2d_K)ν_k^−d

|Y(f₁)−Y(f₂)|

2hlog(1/h)

≤lim sup

k→∞ sup

ρ(f1,f2)≤√

A(_K²)ν^−d_k

|Y(f₁)−Y(f₂)|

2ν_k^−dlog(ν_k^d)

≤ψ

A 2

K ·

Then, for allK≥1 large enough, there exists almost surely anN <∞such that for allk≥N,

Θ_h,t−Θ_{h,τk(i,K) F} ≤ε/4. (2.22)

We next provide an upper bound for Θ_h,τk(i,K)−Θ_1/νd

k,τk(i,K) F.

Recall that 1/ν^d_k ≤h≤1/ν_k−1^d . Thus, using (2.16) in Lemmas 2.5, and 2.4, we find that for K ≥1 large enough, there exists almost surely anN <∞such that for allk≥N,

Θh,τk(i,K)−Θ_1/νd

k,τk(i,K) F≤ε/4. (2.23)

Next, we infer from (2.21), (2.22) and (2.23) that Θ_1/νd

k,τk(i,K)−Θ_{ϕ F}≥ε/2. (2.24)

Finally, (2.18) and (2.24) jointly imply (2.20).

We now turn to the proof of (2.1). For all Λ∈(0,1) we have s^d(1−Λ2)−mes(V_k(ε,Λ))≤

Kνk

i1=0

. . .

Kνk

id=0

( 2

Kν_k)^d(1−Λ2)½_i1,...,id,k(ε,Λ), and so

s^d(1−Λ2)−mes(L_Λ+)≤

k≥1

s^d(1−Λ2)−mes(V_k(ε,Λ))

≤

k≥1 Kνk

i1=0

. . .

Kνk

id=0

( 2

Kν_k)^d(1−Λ2)½_i1,...,id,k(ε,Λ).

We setS_k=_Kνk

i1=0. . ._Kνk

id=0^½i1,...,id,k(ε,Λ). To establish (2.1), it is enough to show that ∞

k=1

E _Kν

k

i1=0

. . .

Kνk

id=0

2 Kν_k

_d(1−Λ²₎

½i1,...,id,k(ε,Λ)

≤^∞

k=1

2 Kν_k

_d(1−Λ²₎

ESk<∞. (2.25)

Observe that

ES_k =

Kνk

i1=0

. . .

Kνk

id=0

P(Θ_(1/νk)d,τk(i,K)∈ H/ _Λ^ε/2)

= (Kν_k+ 1)^dP(Θ_(1/νk)d,0∈ H/ ^ε/2_Λ ),

with 0= (0, . . . ,0). We now use Fact2.1, with_k = (1/ν_k)^d and F =_∞(F)− H^ε/2_Λ . Obviously, forρ >0, 2I(F)≥Λ²+ρ. We therefore obtain

ES_k ≤(Kν_k+ 1)^dexp{−2dlogν_k(Λ²+ρ)/2}

≤(Kν_k+ 1)^dν_k^−d(Λ2+ρ). Thus,

2 Kν_k

_d(1−Λ²₎

ESk= (1 +o(1))ν_k^−dρ= (1 +o(1))(1 +γ)^−kρ. We have shown that (2.25), and hence (2.1), holds.

2.3. Lower bounds

In this section, we will establish that for every Θ_ϕ ∈ H1 associated with a function ϕ∈ G₂(F) such that 0< λ²=

I^dϕ²(u)du<1,

dimL(Θ_ϕ)≥d

1−

I^d

ϕ²(u)du

. (2.26)

First we will discuss some consequences of this property. Let Λ∈(0,1). By choosingλ= Λ∈(0,1) and taking ϕ(u) = Λ foru∈I^d, the obvious inclusionL(Θ_ϕ)⊆L_Λ implies that

dimL_Λ≥d(1−Λ²) for each Λ∈(0,1). (2.27) We next suppose that (2.27) holds. It is obvious that dimL₁ ≥ 0 and according to the properties of the Hausdorff dimension,

dimL₀= dim

m≥1

L_1/m= sup

m≥1dimL_1/m= sup

m≥1(1−m⁻²) = 1.

Thus, to prove that (1.11) holds for each Λ∈[0,1], it is enough to efstablish (2.26), for all 0< λ <1.

For this purpose, we will apply Fact2.2, taking for Aa suitable subset ofL(Θ_ϕ) and c=d(1−λ²−η) with η > 0 small enough. In the following, our attention will be devoted to the construction ofA. We will require some additional notation. Let {hk:k≥1} be a sequence of constants verifying

H1) h_k→0, kh^d_k → ∞and 0< h_k<1, H2) for all 0< c <1,_∞

k=1h⁻¹_k exp{−h^−c_k /2}is a convergent series.

For allk≥1, we setm_k=(h^1/d_k )⁻¹andt_k(i) = (t_k(i₁), . . . , t_k(i_d)) = (i₁h^1/d_k , . . . , i_dh^1/d_k ). We define for all ε >0,ϕ∈G₂(F) andk≥1,

W_k,ϕ(ε) =

t_k(i₁), . . . ,t_k(i_d)

: 1≤i₁, . . . , i_d≤m_k, Θ_hk,tk(i)−Θ_{ϕ F}< ε

, (2.28)

and

U_k,ϕ(ε) =

t∈[0,1/2]^d: Θ_hk,t−Θ_{ϕ F} < ε

. (2.29)

Theε-neighborhood of a measurable setV ⊆I^d, forε >0, is defined by N(ε, V) =

x=(x1,...,xd)∈V d r=1

(x_r−ε, x_r+ε). (2.30)

For anyt∈I^d, we can chooseiin order to minimize |t_k(i)−t|. Before the construction of A, we show a few lemmas.

Lemma 2.7. For any ε∈ (0,1) and θ =θ(ε) <1 satisfying ψ(dθ)≤ε, there exists almost surely a k₀(ε, θ) such that, for allk≥k₀(ε, θ),

N

θh_k, W_k,ϕ(ε)

⊆U_k,ϕ(2ε). (2.31)

Proof. Lett_k(i) = (t_k(i₁), . . . , t_k(i_d))∈W_k,ϕ(ε) andt= (t₁, . . . t_d)∈[0,1/2]^d be such that for each 1≤r≤d, we have|t_k(i_r)−t_r|< θh^1/d_k . The triangle inequality entails that

Θ_hk,t−Θ_{ϕ F}≤ Θ_hk,t−Θ_{hk,tk(i) F}+ Θ_hk,tk(i)−Θ_{ϕ F}.

Using Lemmas2.4and 2.5, we conclude that for anyε >0, there existsk₀(ε) such that for everyk≥k₀(ε) Θ_hk,t−Θ_{hk,tk(i) F} < ψ(dθ).

So,

Θhk,t−Θ_{ϕ F} ≤2ε.

The conclusion of the lemma is therefore immediate.

For every measurable setE⊆I^d, we set m_k(E) = #

(i₁, . . . , i_d) : 0≤i₁, . . . ,i_d≤m_k,

d r=1

[t_k(i_r), t_k(i_r+ 1)]⊆E

, (2.32)

and, remembering (1.6)

N_k,ϕ(ε) = #

(i₁, . . . , i_d) : 1≤i1, . . . , i_d≤m_k, Θ_hk,tk(i)−Θ_{ϕ F}≤ε

. (2.33)

For every measurable setE⊆I^dwe define N_k,ϕ(ε, E) to be N_k,ϕ(ε;E) = #

(i₁, . . . , i_d) : 1≤i₁, . . . , i_d≤m_k;

(t_k(i₁), . . . , t_k(i_d))∈E, Θ_hk,tk(i)−Θ_{ϕ F}< ε

. (2.34)

Finally, for every 0≤i₁, . . . , i_d≤m_k, we set

X_i1,...,id=^½{ Θ_hk,tk(i)−Θ_{ϕ F}< ε}. (2.35) Observe that these variables are independent and identically distributed Bernouilli random variables with prob- ability of success

p_k(ε) :=P(X₀= 1) =P( Θhk,0−Θ_{ϕ F}< ε). (2.36) The following lemma provides an evaluation of this probability.

Lemma 2.8. For all δ∈(0, λ²), there exist0 < δ < δ, ε₀ =ε₀(δ)>0 and a k₁(ε, δ)≥1 such that for each ε∈(0, ε₀] andk≥k₁(ε, δ), we have

p_k(ε)≥h^λ_k^2−δ ≥h^λ_k². (2.37)

Proof. We set

N_ε(Θ_ϕ) ={Ξ∈_∞(F) : Ξ−Θ_{ϕ F} < ε}. (2.38) Recall the definition (1.6) of Θ_hk,tk(i) and (2.36). Observe that

p_k(ε) =P

Θ_hk,0∈N_ε(Θ_ϕ) .

Setting G=N_ε(Θ_ϕ), which is an open set of_∞(F), we apply Fact2.1 (ii), with_k=h_k. We obtain then for allklarge enough,

p_k(ε)≥exp{−ε⁻¹_k I(N_ε(Θ_ϕ))} ≥exp{−2 log(h⁻¹_k )I(N_ε(Θ_ϕ))}.

But, I(N_ε(Θ_ϕ)) < I(Θ_ϕ) = λ²/2. Thus, for any fixed δ > 0, there exists an ε₀(δ) > 0 such that for each ε∈(0, ε₀(δ)],λ²−δ <2I(N_ε(Θ_ϕ))< λ². It follows that (2.37) holds choosingδ ∈(0, λ²−2I(N_ε(Θ_ϕ))).

LetE be an union of disjoint hypercubes of Lebesgue measure vol(E) greater than S≥3^dh_k. Then we have vol(E)

3^dh_k ≤

1−2h^1/d_k S^1/d

_d vol(E)

h_k ≤m_k(E)≤ vol(E)

h_k · (2.39)

Lemma 2.9. Let ε∈(0, ε₀) whereε₀ is as in Lemma2.8 and let δ∈(0,1−λ²) be fixed. For any σ∈(0,1) there exists almost surely ak₂(ε, σ, δ)≥1such that, for all k≥k₂(ε, σ, δ), we have

|N_k,ϕ(ε, E)−m_k(E)p_k(ε)|< σm_k(E)p_k(ε), (2.40) whereE is an union of disjoint hypercubes of Lebesgue measurevol(E)greater than h^1−λ_k ^2−δ

Proof. Since N_k,ϕ(ε, .) and m_k(.) are additive set functions, it suffices to prove (2.40) when E is a hypercube ofI^d with Lebesgue measure greater thanh^1−λ_k ^2−δ.

Fix σ and δ such that 0< σ < σ and 0 < δ < δ. We next prove that it is enough to prove the lemma whenE=J is a hypercube of the form

d r=1

[t_k(i_r), t_k(i_r+l(k))),

for some 0≤i₁, . . . , i_d ≤2m_k, and withl(k) :=h^−(λ_k ^2+δ)/d. To see this, assume that (2.40) is satisfied for E =J, σ =σ and δ = δ. Let K(k) := h^δ_k^−δ. From (H1), we can choose k(δ, σ)< ∞ such that, for all k≥k(δ, σ), we have

3≤K(k), (2.41)

3^dh_k < h^1−λ_k ^2−δ, (2.42)

(1 +σ)

1 + 4/(K(k)−2) 1−2h^(λ_k ^2+δ)/d

_d

≤(1 +σ), (2.43)

and

1−σ≤(1−σ)

1−2h^(λ_k ^2+δ)/d 1 + 4/(K(k)−2)

d

. (2.44)

For allk≥1, we have

K:=K(E, k) =

! vol(E)^1/d h^(1−λ_k ^2−δ)/d

"

.

Observe thatK≥K(k)→ ∞. Fork≥k(δ, σ), there exist (K+ 2)^d disjoint hypercubesJ₁, . . . , J_(K+2)d of the form_d

r=1[t_k(i_r), t_k(i_r+l(k))] such that

(K−2) ^d

=1

J⊆E⊆

(K+2) ^d

=1

J.

Given the form of the hypercubesJ, we see that vol(J) = (1 +o(1))h^1−λ_k ^2−δ. Moreover, we have the following inequalities.

(K−2)^dvol(J₁)≤vol(E)≤(K+ 2)^dvol(J₁).

Hence, applying (2.39) and Lemma2.9forE=J, it follows that, for all klarge enough,

N_k,ϕ(ε, E)≤

(K+2)^d

=1

N_k,ϕ(ε, J)≤(1 +σ)

^(K+2)d

=1

m_k(J)

p_k(ε)

≤(K+ 2)^d(1 +σ)vol(J₁)h⁻¹_k p_k(ε)

≤ (K+ 2)^d

(K−2)^d(1 +σ)vol(E)h⁻¹_k p_k(ε)

≤(1 +σ)

1 +_K−2⁴ 1−2h^(λ_k^2+δ)/d

_d

m_k(E)p_k(ε)

≤(1 +σ)m_k(E)p_k(ε). (2.45)

A similar argument shows that

N_k,ϕ(ε, E)≥

(K−2)^d

=1

N_k,ϕ(ε, J)≥(K−2)^dN_k(J₁)

≥(K−2)^d(1−σ)m_k(J₁)p_k(ε)

≥(K−2)^d(1−σ)vol(J₁)

h_k (1−2h^(Λ_k ^21+δ)/d)^dp_k(ε)

≥(K−2)^d

(K+ 2)^d(1−σ)vol(E)

h_k (1−2h^(Λ_k ^21+δ)/d)^dp_k(ε)

≥(1−σ)

1−2h^(λ_k^2+δ)/d 1 +_K−2⁴

d

m_k(E)p_k(ε)

≥(1−σ)m_k(E)p_k(ε). (2.46)

We obtain that it suffices to show the lemma whenE is a hypercubeJ of the form J =

d r=1

[t_k(i_r), t_k(i_r+(k))] with 0≤i_r≤2m_k and 1≤r≤d. (2.47) Note here that the total number of hypercubes of this form is bounded above by h⁻¹_k . For all k ≥1, we set Q_k = P(N_k,ϕ(ε, J) > r_k,1) wherer_k,1 = (1 +σ)m_k(J)p_k(ε). We now apply Fact 2.3 for S_N = N_k,ϕ(ε, J), N =m_k(J),p=p_k(ε) andr= (1 +σ). We obtain forksufficiently large,

Q_k≤exp{−mk(J)p_k(ε)h(1 +σ)}.

Moreover, forklarge enough, we have m_k(J)≥

#

vol(J)^1/d h^1/d_k −2

$_d

≥ 1

2^dh^−λ_k ^2−δ. Thus, applying Lemma 2.8, we see that forklarge enough,

h⁻¹_k Q_k≤h⁻¹_k exp

− 1

2^dh^−δ_k h(1 +σ)

.

We use (H2) to obtain_∞

k=1h⁻¹_k Q_k<∞. Therefore, the Borel-Cantelli lemma implies that with probability 1, for allksufficiently large, we have

N_k,ϕ(ε, J)≤(1 +σ)m_k(J)p_k(ε).

Define likewiseR_k =P(N_k,ϕ(ε, J)< r_k,2) wherer_k,2 = (1−σ)m_k(J)p_k(ε). By a similar argument, we show that with probability 1, for allk sufficiently large,

N_k,ϕ(ε, J)≥(1−σ)m_k(J)p_k(ε).

Finally, (2.40) is proved for any hypercube of the form (2.47). This concludes the proof of Lemma2.9.

We shall now prove the existence of a sequence of setsE₁, E₂, . . .fulfilling the assumptions of Fact2.2and such that A =_∞

m=1E_m ⊆L(Θ_ϕ). Let see the steps needed for the construction of these sets. In a first step, we establish the existence of this sequence via an induction argument. In a second step, we show that{Em:m≥1}

satisfies (2.5). Finally, in a last step, we apply the Fact2.2to establish (2.2).

Step 1: Existence ofE_m. We choose two sequences of nonnegative constants{σ_m:m≥1}and{δ_m:m≥1} such that

(1i) 0< σ_m<1/2, for anym≥1;

(1ii) _m

i=1(1 +σ_i)²/(1−σ_i)²≤2;

(1iii) δ₀= 0, δ_m≥0, for anym≥1;

(1iv) _∞

m=1δ_m≤η/3<¹₆min(λ²,1−λ²).

We select two decreasing sequences of positive constants{ε_m:m≥1} and{θ_m:m≥1}such that (2i) ε₁<1,ε_m↓0,ε_m< δ_m;

(2ii) θ_m<(1/2)ε²_m;

(2iii) (1 + 2θ_m−1)^{d(1−λ2−2Δm−1)}≤ _1+1/2σ^1+σm

m; (2iv) 3^{d(1−λ2−2Δm−1)}≤ _1+1/2σ^1+σm_m.

For further use, we choose k₂(ε_m, σ_m, δ_m) ≥ k₁(ε_m, δ_m ≥ k₀(ε_m) as in Lemmas 2.7, 2.8 and 2.9. We set Δ_m=_m

k=1δ_k. Observe that for everym≥1, Δ_m< 1

6min(λ²,1−λ²),

and 0< δ_m<Δ_m<1. The construction of the setsE_mrelies on an inductive argument. That is to say that given E_m−1 and {M_m−1, k_m−1, E_m−1^∗ }, we evaluate E_m and{Mm, k_m, E_m^∗}. The constantsM_m−1, k_m−1 and the setsE_m−1^∗ are defined below.

We set

k₀= 1, L₀= 1, M₀= 1, (2.48)

and, for everym≥1,

L_m=θ_mh^1/d_k

m and L^∗_m−1=L_m−1−L_m. (2.49)

We choosek_ma positive integer such that the following conditions are verified.

(3i) k_m>max{k_m−1, k₂(ε_m, σ_m, δ_m)};

(3ii) 3^dL_m<3^dh^1/d_k

m < h^(1−λ_k ^2−δm)/d

m < L^∗_m−1< L_m−1; (3iii) L^d_m−1/(L^∗_m−1)^d≤1 + (1/2)σ_m;

(3iv) 1−σ_m≤(1−_L^2h∗^1/dkm m−1)^d; (3v) 2 ^hδmkm

h^λ2+_km−1^δm−1 ≤θ_m^d form≥2;