A strictly stationary $\beta$-mixing process satisfying the central limit theorem but not the weak invariance principle

HAL Id: hal-00911758

https://hal.archives-ouvertes.fr/hal-00911758v2

Submitted on 14 Oct 2014

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A strictly stationary β-mixing process satisfying the central limit theorem but not the weak invariance

principle

Davide Giraudo, Dalibor Volný

To cite this version:

Davide Giraudo, Dalibor Volný. A strictly stationary β-mixing process satisfying the central limit theorem but not the weak invariance principle. Stochastic Processes and their Applications, Elsevier, 2014, 124, pp.3769-3781. �10.1016/j.spa.2014.06.008�. �hal-00911758v2�

A STRICTLY STATIONARY β-MIXING PROCESS SATISFYING THE CENTRAL LIMIT THEOREM BUT NOT THE WEAK

INVARIANCE PRINCIPLE

DAVIDE GIRAUDO AND DALIBOR VOLNÝ

Abstract. In 1983, N. Herrndorf proved that for a φ-mixing sequence satis- fying the central limit theorem andlim inf_n→∞σn²/n >0, the weak invariance principle takes place. The question whether for strictly stationary sequences with finite second moments and a weaker type (α, β, ρ) of mixing the central limit theorem implies the weak invariance principle remained open.

We construct a strictly stationaryβ-mixing sequence with finite moments of any order and linear variance for which the central limit theorem takes place but not the weak invariance principle.

1. Introduction and notations

Let (Ω,F, µ) be a probability space. If T: Ω → Ωis one-to-one, bi-measurable and measure preserving (in sense that µ(T⁻¹(A)) =µ(A) for all A∈ F), then the sequence f◦T^k

k∈Z is strictly stationary for any measurable f: Ω → R. Con- versely, each strictly stationary sequence can be represented in this way.

For a zero mean square integrable f: Ω → R, we define S_n(f) :=

n−1

X

j=0

f ◦T^j, σ_n²(f) :=E(S_n(f)²)andS_n^∗(f, t) :=S_⌊nt⌋(f) + (nt− ⌊nt⌋)f◦T^⌊nt⌋, where⌊x⌋is the greatest integer which is less than or equal to x.

We say that(f◦T^j)_j>1 satisfies thecentral limit theorem with normalization a_n if the sequence(a⁻¹_n S_n(f))_n>1converges weakly to a strandard normal distribution.

Let C[0,1] denote the space of continuous functions on the unit interval endowed with the norm kgk∞:= sup

t∈[0,1]|g(t)|.

Let D[0,1] be the space of real valued functions which have left limits and are continuous-from-the-right at each point of[0,1). We endow it with Skorohod metric (cf. [2]). We define S_n^∗∗(f, t) :=S_⌊nt⌋(f), which gives a random element ofD[0,1].

We shall say that the strictly stationary sequence f◦T^j

j>0 satisfies the weak invariance principle in C[0,1] with normalization a_n (respectively in D[0,1]) if the sequence of C[0,1](of D[0,1]) valued random variables a⁻¹_n S_n^∗(f,·)

n>1 (resp.

a⁻¹_n S_n^∗∗(f,·)

n>1) weakly converges to a Brownian motion process in the corre- sponding space.

LetA andB be two sub-σ-algebras ofF, where (Ω,F, µ)is a probability space.

We define theα-mixing coefficients as introduced by Rosenblatt in [14]:

α(A,B) := sup{|µ(A∩B)−µ(A)µ(B)|, A∈ A, B∈ B}.

Date: September 2, 2014.

2000Mathematics Subject Classification. 60F05; 60F17.

Key words and phrases. Central limit theorem, invariance principle, mixing conditions, strictly stationary process.

1

Define theβ-mixing coefficients by β(A,B) := 1

2sup

I

X

i=1 J

X

j=1

|µ(A_i∩B_j)−µ(A_i)µ(B_j)|,

where the supremum is taken over the finite partitions{Ai,16i6I}and{Bj,16j6J} ofΩ of elements ofA(respectively ofB). They were introduced by Volkonskii and Rozanov [16].

Theρ-mixing coefficients were introduced by Hirschfeld [8] and are defined by ρ(A,B) := sup

|Corr(f, g)|, f ∈L²(A), g∈L²(B) , whereCorr(f, g) := [E(f g)−E(f)E(g)] [kf −E(f)k_L²kg−E(g)k_L²]⁻¹.

Ibragimov [9] introduced for the first timeφ-mixing coefficients , which are given by the formula

φ(A,B) := sup{|µ(B|A)−µ(B)|, A∈ A, B∈ B, µ(A)>0}. The coefficients are related by the inequalities

(1) 2α(A,B)6β(A,B)6φ(A,B), α(A,B)6ρ(A,B)62p

φ(A,B).

For a strictly stationary sequence(X_k)_k∈Z andn>0we define αX(n) =α(n) = α(F−∞⁰ ,Fn^∞) where Fu^v is the σ-algebra generated by X_k with u 6 k 6 v (if u = −∞ or v = ∞, the corresponding inequality is strict). In the same way we define coefficientsβ_X(n),ρ_X(n),φ_X(n).

We say that the sequence(X_k)_k∈Zisα-mixing if lim

n→+∞αX(n) = 0, and similarily we defineβ,ρ and φ-mixing sequences.

α,β andφ-mixing sequences were considered in the mentioned references, while ρ-mixing sequences first appeared in [12].

Inequalities (1) give a hierarchy between theses classes of mixing sequences.

If (a_N)_N>1 and (b_N)_N>1 are two sequences of positive real numbers, we write a_N ≍b_N if there exists a positive constantC such that for each N,C⁻¹a_N 6b_N 6 Ca_N.

The main results are

Theorem A. Let δ be a positive real number. There exists a strictly stationary real valued processY = (Y_k)_k>0= f◦T^k

k>0 satisfying the following conditions:

a) the central limit theorem with normalization √n takes place;

b) the weak invariance principle with normalization √

n does not hold;

c) we have σ_N(f)²≍N;

d) we have for some positive C,β_Y(N)6C/N^1/2−δ; e) Y₀ ∈L^p for any p >0.

Alternatively, we can construct the process in order to have a control of the mixing coefficients on a subsequence.

Theorem A’. Let (c_j)_j>0 be a decreasing sequence of positive numbers. Then there exists a strictly stationary real valued processY = (Y_k)_k>0 = f◦T^k

k>0 satisfying conditions a), b), c), e) in Theorem A, and:

d’) there is an increasing sequence (m_k)_k>1 of integers such that for each k, β_Y(m_k)6c_mk.

2

Remark 1. Herrndorf proved ([7], Theorem 2.13) that if(ξ_n)is a strictly stationary φ-mixing sequence for which σ_n → ∞,S_n/σ_n converges in distribution to a stan- dard normal distribution and σ_n⁻¹max_16i6n|ξ_i| → 0 in probability, then the weak invariance principle takes place. So Herrndorf’s result does not extend toβ-mixing sequences.

Remark 2. Rio et al. proved in [13] that the condition R1

0 α⁻¹(u)Q²(u)du < ∞ implies the weak invariance principle, where α⁻¹(u) := inf{k, α(k)6u} and Q is the right-continuous inverse of the quantile function t 7→ µ{X₀ > t}. If the process is strictly stationary, with finite moments of order 2 +r, r >0, the latter condition is satisfied whenever P_∞

n=1(n+ 1)^2/rα(n) < ∞ (Ibragimov [10] found the condition P_∞

n=1α(n)^1−2/r < ∞). Since Y₀ ∈ L^p for all p < ∞, we have that P

Nα(N)^r = +∞for anyr <1, hence in Theorem A’ we can thus hardly get such a bound as in d’) for the whole sequence.

Remark 3. Ibragimov proved that for a strictly stationaryρ-mixing sequence with finite moments of order2+δfor some positiveδ, the weak invariance principle holds, cf. [11]. In particular, this proves that our construction does not give a ρ-mixing process. Shao also showed in [15] that the condition P

nρ(2ⁿ)<∞ is sufficient in order to guarantee the weak invariance principle inD[0,1]for stationary sequences having order two moments. So a potentialρ-mixing counter-example has to adhere to restrictions on the moments as well as on the mixing rates.

About the method of proof

In proving the result we will use properties of coboundaries h =g−g◦T (g is called a transfer function). For a positive integerN and a measurable functionv, we denoteS_N(v) :=

N−1

X

j=0

U^jv(Here and below,U^jv:=v◦T^j.). BecauseS_n(g−g◦T) = g−g◦Tⁿ, for any sequencea_n→ ∞we have(a_n)⁻¹S_n(g−g◦T)→0in probability hence adding a coboundary does not change validity of the central limit theorem.

If, moreover, g ∈ L² then n^−1/2kS_n^∗(g−g◦T)k∞ → 0 a.s. hence adding of such coboundary does not change validity of the invariance principle (if norming by√

nor byσ_nwithlim inf_nσ²_n/n >0), cf. [6], pages 140-141. On the other hand, ifg6∈L², adding a coboundary can spoil tightness even if g−g◦T is square integrable, cf.

[17]. A similar idea was used in [5]. In the proof of Theorem A and A’ we will find a coboundary g−g◦T which is β-mixing and spoils tightness. The coboundary has all finite moments but the transfer function is not integrable. We then add an m such that(m◦Tⁱ)_i∈Zand (h◦Tⁱ)_i∈Zare independent (enlarging the probability space), andm◦Tⁱ is i.i.d. with moments of any order (in particular, it satisfies the weak invariance principle).

The proof uses the fact that |µ(A∩B)−µ(A)µ(B)|6µ(A). The method does not seem to apply to processes which are ρ-mixing and for this kind of processes the problem remains open.

2. Proof

2.1. Construction ofh. Let us consider an increasing sequence of positive integers (n_k)_k>1 such that

n₁ >2 and

∞

X

k=1

1 n_k <∞,

3

and for each integerk>1, letA⁻_k, A⁺_k be disjoint measurable sets such thatµ(A⁻_k) = 1/(2n²_k) =µ(A⁺_k).

Let the random variablese_k be defined by

(2) e_k(ω) :=







1 if ω∈A⁺_k,

−1 if ω∈A⁻_k, 0 otherwise.

We can choose the dynamical system (Ω,F, µ, T) and the sets A⁺_k, A⁻_k in such a way that the family(e_k◦Tⁱ)_k>1,i∈Z is independent. We defineA_k :=A⁺_k ∪A⁻_k and

(3) h_k :=

nk−1

X

i=0

U⁻ⁱe_k−U^−nk

nk−1

X

i=0

U⁻ⁱe_k, h:=

+∞

X

k=1

h_k.

Since µ{h_k6= 0} 6 2/n_k, the function h is almost everywhere well-defined (by the Borel-Cantelli lemma).

It will be useful to express, forN >nk, the sum SN(hk) as a linear combination of U^pe_k. Denote s_k:=

nk−1

X

j=0

U^−je_k. AsN >n_k and h_k=s_k−U^−nks_k, we have

S_N(h_k) =

N−1

X

j=0

(U^js_k−U^j−nks_k)

=

N−1

X

j=0

U^js_k−

N−nk−1

X

j=−nk

U^js_k

=−

−1

X

j=−nk

U^js_k+U^N

−1

X

j=−nk

U^js_k.

We also have

−1

X

j=−nk

U^js_k=

nk

X

j=1

U^−js_k

=U⁻¹

nk−1

X

i=0 nk−1

X

j=0

U^−(i+j)e_k

−1

X

j=−nk

U^js_k=U^−2nk





nk

X

j=1

jU^je_k+

nk−1

X

j=1

(n_k−j)U^nk+je_k



. (4)

The previous equations yield (5) S_N(h_k) =

nk

X

j=1

jU^j+N−2nke_k+

nk−1

X

j=1

(n_k−j)U^j+N−nke_k

−

nk

X

j=1

jU^j−2nke_k−

nk−1

X

j=1

(n_k−j)U^j−nke_k.

4

Each h_k is a coboundary, as if we define v_k :=

nk−1

X

j=0

U^−js_k, then v_k −U⁻¹v_k = s_k−U^−nks_k=h_k (so in this case the transfer function is −U⁻¹v_k).

Since µ{v_k 6= 0} 6 2/n_k, Borel-Cantelli’s lemma shows that the function g :=

−

+∞

X

k=1

U⁻¹v_kis almost everywhere well defined under our assumption thatP

k1/n_k is convergent. Becauseh=g−U g,h is a coboundary.

2.2. Mixing rates. We show that the process(Uⁱf)_i∈Z is β-mixing. In doing so we use the following proposition (cf. [4], Theorem 6.2).

Proposition 4. Let(X_k,i)_i,k= 1,2, . . . be mutually independent strictly stationary processes with respective mixing coefficients β_k(n), let X_i = P_∞

k=1X_k,i converge.

The process(X_i)_i is strictly stationary with mixing coefficientsβ(n)6P_∞

k=1β_k(n).

This reduces the proof of β-mixing of (Uⁱf)_i∈Z (in Theorems A and A’) to that of (Uⁱh)_i∈Z and thereby to that of(Uⁱh_k)_i∈Z fork>1.

In the following text we denote by β_k(n) the mixing coefficients of the process (h_k◦Tⁱ)i∈Z.

Lemma 5. For k>1, we have the estimate β_k(0)64/n_k.

Proof. Suppose k is a positive integer. For−∞6j6l6∞, letH^lj denote theσ- field generated by Uⁱh_k, j 6i6l, (i∈Z), and let Gj^l denote the σ-field generated by Uⁱe_k, j 6 i 6 l, (i ∈ Z). Define the σ-fields B1 := G−∞^−2nk, B2 := G−2n⁰ k+1

and B³ := G1^∞. Now H⁰−∞ ⊂ B¹ ∨ B² and H^∞0 ⊂ B² ∨ B³. Therefore β_k(0) 6 β(B1 ∨ B2,B2∨ B3). The σ-fields B1, B2, B3, are independent; hence the σ-fields B1∨ B3 and B2∨ B2 (with index2in both places) are independent; this implies by a result given e.g. in [4, Theorem 6.2],

β(B1∨ B2,B2∨ B3)6β(B1,B3) +β(B2,B2) = 0 +β(B2,B2).

Thusβ_k(0)6β(B²,B²). Also, theσ-fieldB²has an atomP0:=T0

i=−2nk+1

Uⁱe_k= 0 that satisfies µ(P₀)>1−2/n_k (sinceµ(Uⁱe_k 6= 0) = 1/n²_k for each i). By Lemma 2.2 of [3], ifBis aσ-field which has an atomD, thenβ(B,B)62[1−µ(D)]. Hence β(B2,B2)62[1−µ(P₀)]64/n_k. This finishes the proof of Lemma 5.

Denoting by β(N) the mixing coefficients of the sequence (h◦Tⁱ)_i∈Z, Proposi- tion 4, Lemma 5 and the fact thatβ_k(N) = 0 when N >2n_k yield

Corollary 6. For each integer k, we have

(6) β(N)6X

j>1

β_j(N)6 X

j:2nj>N

4 n_j.

Now we can prove d) and d’). Let i(N) denote the unique integer such that n_i(N)6N < n_i(N)+1. For sequences (u_N)_N>1,(v_N)_N>1 of positive numbers, u_N . vN means that there isC >0 such that for eachN,uN 6C·vN.

Proposition 7. Let δ >0. With the choice n_k:=⌊16^(2+δ)k⌋, we have d).

Proof. We deduce from (6) β(2n_k)6X

j>k

4

n_j .X

j>0

16^{−(2+δ)k(2+δ)j} .16^−(2+δ)k.

5

Consequently,

β(2N)6β(2n_i(N)). 4

n_i(N) . 1 n

1 2+δ

i(N)+1

. 1

N^2+δ1 .

Hence d) is fulfilled.

Proposition 8. Given (c_k)_k>1 as in Theorem A’, one can recursively choose a sequence (n_k)_k>1 growing to infinity arbitrarily fast, such that for the construction given above, one has that for each k>1, β(2n_k)6c_2nk.

Proof. Suppose that the sequence (n_k)_k>1 satisfies n_k+1> 8

c_2nk and n_k+1 >2n_k, k>1.

Then

β(2n_k)6

∞

X

j=k+1

β_j(2n_k)6

∞

X

j=k+1

β_j(0),

and, by Lemma 5 and the condition nj+l>2^lnj for j, l>1, we derive β(2n_k)6

∞

X

j=k+1

4

n_j 6 8 n_k+1.

The assumption n_k+1 >8/c2nk yieldsβ(2n_k)6c2nk for eachk>1.

This proves d’) with m_k:= 2n_k.

Remark 9. The sequence of integers (n_k)_k>1 can be chosen to meet all other con- ditions imposed in this paper.

2.3. Proof of non-tightness.

Lemma 10. There exists N₀ such that

(7) µ

( max

2nk6N6n²_k|S_N(h_k)|>n_k )

>1/4 whenever nk>N0.

Proof. For2n_k 6N 6n²_k, thanks to (5), we have {|S_N(h_k)|=n_k} ⊃

U^N−nke_k

= 1 ∩\

j∈I

U^je_k= 0 ∩ \

j∈JN

U^je_k = 0 ,

whereI = [1−2n_k,−1]∩ZandJ_N = ([N−2n_k+1, N−1−n_k]∪[N+1−n_k, N−1])∩Z. We define

B_N,k :=

U^N−nke_k

= 1 ∩

−1−nk

\

j=1−2nk

U^N+je_k= 0 ∩

−1

\

j=1−nk

U^j+Ne_k= 0

=

U^N−nke_k

= 1 ∩ \

j∈JN

U^je_k= 0 .

6

We have|S_N(h_k)|=n_konT

j∈I

U^je_k= 0 ∩B_N,k and the sets T

j∈I

U^je_k= 0 , Sn²_k

N=2nkB_N,k belong to independentσ-algebras. Therefore

(8) µ

( max

2nk6N6n²_k|S_N(h_k)|>n_k )

>

1− 1

n²_k 2nk

µ





n²_k

[

N=2nk

B_N,k



.

Recall Bonferroni’s inequality, which states that for any integer n and any events A_j, j∈ {1, . . . , n}, we have

(9) µ





n

[

j=1

A_j



>

n

X

j=1

µ(A_j)− X

16i<j6n

µ(A_i∩A_j).

It can be proved by induction. Notice that µ(B_N,k) = 1

n²_k

1− 1 n²_k

2nk−2

> 1 n²_k

1− 2

n_k

and for i6=j

µ(Bi+2nk−1,k∩Bj+2nk−1,k)6µ

U^i+nk−1ek

= 1 µ

U^j+nk−1ek

= 1 = 1 n⁴_k hence

µ





n²_k

[

N=2nk

B_N,k



=µ





(nk−1)²

[

N=1

B_N+2nk−1,k





>

(nk−1)²

X

N=1

1 n²_k

1− 2

n_k

− X

16i<j6(nk−1)²

µ(B_i+2nk−1,k∩B_j+2nk−1,k)

>

1− 2

n_k 3

− 1 2

which together with (8) and the inequality(1−1/n²_k)^2nk >1−2/n_k implies that

(10) µ

( max

2nk6N6n²_k|S_N(h_k)|>n_k )

> 1 4, whenever n_k >N₀, whereN₀ >3 is such that(1−2/n)

(1−2/n)³−1/2

>1/4

for n>N0.

Lemma 11. Assume that the sequence(n_k)_k>1 satisfies the following two conditions of lacunarity:

for eachk>K, 16

k

X

j=1

n²_j 6n_k+1; (11)

for eachk>K, n_k+1 >(k+ 1)²n_k, (12)

where 26K <∞. Then we have for k large enough

(13) µ

( 1 nk

max

2nk6N6n²_k|S_N(h)|>1/2 )

>1/8.

7

Proof. Fix an integer k>K. For 2n_k6N 6n²_k, we have (14)

1 n_k

k−1

X

j=1

SN(hj)

6 2

n_k

k−1

X

j=1

(nj + 1)² 6 1 2.

Indeed, using (5), for j < k, we can give an upper bound ofS_N(h_j)(asN >2n_k>

2n_j) as

|S_N(hj)|62

nj

X

l=1

l+ 2

nj−1

X

l=1

l= 2n²_j, and (14) holds by (11).

Now fixj > k. WritingS_N(h_j) =

N−1

X

i=0

Uⁱs_j−

N−nj−1

X

i=−nj

Uⁱs_j, wheres_j :=Pnj−1

i=0 U⁻ⁱe_j, we can see that

n²_k

[

N=2nk

{S_N(h_j)6= 0} ⊂

n²_k

[

i=−2nj+1

T⁻ⁱA_j,

hence using (12)

(15) µ





n²_k

[

N=2nk

{S_N(hj)6= 0}



6 n²_k+ 2n_j

n²_j 6 2n_k

n_j 62j⁻². LetE_k :=Sn²_k

N=2nk

S

j>k+1{S_N(h_j)6= 0}. By (15), we have

(16) µ(E_k)6 X

j>k+1

2j⁻² 62/k.

By (14), 1

n_k max

2nk6N6n²_k|SN(h)|> 1

n_k max

2nk6N6n²_k

SN



 X

j>k

hj





−1 2, hence

µ ( 1

n_k max

2nk6N6n²_k|S_N(h)|> 1 2

)

>µ





 1

n_k max

2nk6N6n²_k

S_N



 X

j>k

h_j





>1







>µ









 1

n_k max

2nk6N6n²_k

S_N



 X

j>k

h_j





>1







∩E_k^c





=µ ( 1

n_k max

2nk6N6n²_k|S_N(h_k)|>1 )

∩E_k^c

!

>µ ( 1

n_k max

2nk6N6n²_k|S_N(h_k)|>1 )!

−µ(E_k).

The result follows from Lemma 10 and (16).

The previous lemma yields together with Theorems 8.1 and 15.1 of [2] and the convergence to 0 of the finite dimensional distributions of (N^−1/2S_N^∗(h))_N>1 and (N^−1/2S_N^∗∗(h))_N>1 the following corollary.

8