self-normalized partial sums of linear processes

Functional central limit theorems for

self-normalized partial sums of linear processes

Alfredas Ra£kauskas and Charles Suquet July 10, 2010

Abstract

We prove the invariance principle under self-normalization by blocks for linear processes with summable lters and i.i.d. innovations in the domain of attraction of the normal distribution.

Keywords: Domain of attraction, invariance principle, linear processes, self-normalization, weak convergence.

1 Introduction and results

We consider linear processes

Xk =X

i∈Z

ai_k−i, k∈N, (1)

where(ai, i∈N)is a square summable (P

i∈Za²_i <∞) sequence of real numbers and(i, i∈Z)are i.i.d. centered random variables in the domain of attraction of the normal law (writen1∈DAN). This implies in particular thatE|i|^p<∞ for each0< p <2and, consequentlyXkis well dened (see, e.g., Brockwell and Davis [4]). Central limit theorem for partial sumsSn =X1+· · ·+Xn, n∈N, and functional limit theorems for processes build from partial sums(Sk, k∈N) has been extensively studied in the literature. We refer to the survey paper by Merlevède, Peligrad and Utev [15] for recent results on the central limit theorem and its weak invariance principle for stationary sequences under nite second moment assumption. In this paper we shall not assume that _i has nite variance. In such context self-normalization is an appropriate technique.

Usually this means the normalization byVn, where

V_n²=X₁²+· · ·+X_n². (2) A rich literature is devoted to the limit behavior of the sequenceV_n⁻¹S_n in the case of independent random variablesX₁, X₂, . . . Particularly, the central limit theorem is completely solved by Giné, Götze, and Mason [8] and invariance principles were proved in Ra£kauskas and Suquet [18, 19], Csörg®, Szyszkowicz and Wang [5]. For other important aspects of limit behaviour of self-normalized sums we refer to Logan et al. [14], Hahn and Zhang [9] and references therein.

For self-normalization techniques in time series analysis, we refer to Klüppelberg and Mikosch [12]. We also refer to a survey paper by de la Peña, Klass and Lai [6]

for recent results of the theory and applications of self-normalized processes in dependent variables.

We consider self-normalization using block-sums Bm,j of Xk's. Choosing N = [n/m], wherem=m(n), dene

U_n²=

N

X

j=1

B_m,j² , with B_m,j=

jm

X

i=(j−1)m+1

X_i. (3)

Let us remind that1∈DAN(domain of attraction of the normal distribution) means that there exists a sequencebn↑ ∞such that

b⁻¹_n

n

X

k=1

k−−−−→

n→∞ N(0,1), in distribution. (4) The following theorem is our contribution to the central limit theorems for linear processes.

Theorem 1. If P

i∈Z|ai|<∞,P

i∈Za_i6= 0 and₁∈DAN, then U_n⁻¹S_n −−−−→

n→∞ N(0,1), in distribution, (5) provided that m→ ∞andm/n→0 asn→ ∞.

Clearly the condition 1 ∈ DAN is necessary for the convergence (5) to hold on the whole class of lters considered in Theorem 1. Indeed looking at the special case where a0 = 1andai = 0fori6= 0, we have Xk =k and then the membership of1 in DAN is necessary by Giné, Götze, and Mason [8].

Earlier the convergence (5) was obtained by Juodis and Ra£kauskas [10]

under the stronger condition P

jj|aj|<∞. Theorem 1 is actually a corollary of functional limit theorems proved in this paper.

We dene the polygonal line process ξ_n(t) =

[nt]

X

i=1

X_i+ (nt−[nt])X_[nt]+1, t∈[0,1] (6) and view it as a random element in the Banach space C[0,1] of continuous functions on[0,1]equipped with the uniform norm

||x||= sup

t∈[0,1]

|x(t)|. (7)

We consider also the step partial sums process{ζ_n(t) :t∈[0,1]} dened by

ζn(t) =

[nt]

X

i=1

Xi (8)

as a random element of the Skorohod space D[0, 1] of all functions on [0, 1]

which have left-hand limits and are continuous from the right, equipped with the Skorohod topology (see, e.g. [2, Section 14].

LetW ={W(t) :t∈[0,1]} denote the standard Brownian motion on[0,1]. By −→^D we denote the convergence in distribution in the indicated space.

Our main results are the following two theorems.

Theorem 2. If P

i∈Z|a_i|<∞,P

i∈Za_i6= 0 and₁∈DAN, then

U_n⁻¹ξ_n −→^D W in the space C[0,1], (9) asn→ ∞, provided that m→ ∞ andm/n→0 asn→ ∞.

Theorem 3. If P

i∈Z|ai|<∞,P

i∈Zai6= 0 and1∈DAN, then

U_n⁻¹ζn−→D W in the space D[0,1], (10) asn→ ∞, provided that m→ ∞ andm/n→0 asn→ ∞.

Using the same assumption about the ai's, Kulik [13] obtained a strong approximation result for the process β⁻¹V_n⁻¹ζn with β =|P

iai|(P

ia²_i)^−1/2. In view of statistical applications, the interest of the self-normalization byUn

instead of Vn is that we do not need to know this coecientβ. Of course in practical situations, the choice ofm=m(n)is an important problem. Obtaining an optimal choice under so general assumptions as above seems out of reach.

2 Useful facts

We gather here some information about the DAN property, selfnormalization and linear processes, used in the proofs of our limit theorems. To avoid no- tational confusion, we denote by Vn(), Un(), ζn⁽⁾, ξ⁽⁾n the objects dened substitutingX byin (2), (3), (8), (6) respectively.

If₁∈DAN, then with the normalizing sequence (b_n)as in (4), one has for eachτ >0,

nP(|₁|> τ b_n)−−−−→

n→∞ 0, (11)

n

b²_nE²₁1_{{|1|≤τ bn}}−−−−→

n→∞ 1, (12)

see e.g. Araujo and Giné [1], Chap. 2, Cor. 4.8(a), Th. 6.17 (i) and Cor. 6.18 (b). Moreover

b⁻²_n

n

X

k=1

²_k−−−−→^Pr

n→∞ 1. (13)

Lemma 4. If 1∈DANwith normalizing sequence (bn)_n≥1 as in (4), then n

b_nE |1|1_{|1|>bn}

−−−−→

n→∞ 0. (14)

Proof. Integrating by part we obtain n

bn

E |1|1_{|1|>bn}

=nP(|1|> bn) + n bn

Z ∞ b_n

P(|1|> x) dx.

The rst term in the above right hand side tends to zero by (11). To prove the same convergence for the second term, it is convenient to expressP(|1|> x)in terms of the truncated second momentL(x) :=E²₁1_{|1|≤x}. Since ₁∈DAN,

it is well known (e.g. Feller [7]) that L is slowly varying and that x²P(|₁|>

x) =o(L(x))asx→ ∞. This may be rewritten as P(|1|> x) =L(x)g(x)

x² , x >0, (15)

with some non negative function g such that g(x) tends to 0 at innity. In particular, g is bounded on some interval [c,∞). Now by a Karamata result (see [3, Th. 1.5.3] or [11, pp.4546]), for everyδ >0,x^−δL(x)is asymptotic to a non increasing function. Fixingδ∈(0,1), there exists then somen₀such that for every n≥n₀,b_n≥cand

L(x)

x^δ ≤ 2L(bn)

b^δ_n , x≥bn. (16)

Using (15) and (16), we obtain n

b_n Z ∞

bn

P(|1|> x) dx≤ 2nL(bn) b^1+δn

Z ∞ bn

g(x)

x^2−δ dx=2nL(bn) b²_n

Z ∞ 1

g(bnt) t^2−δ dt.

This last estimate tends to0applying (12) and the bounded convergence theo- rem (recall thatg is bounded on[c,∞)).

Lemma 5 (see Lemma 9 in [19]). If ₁∈DANthen sup

t∈[0,1]

V_[nt]² () V_n²() −t

−−−−→Pr n→∞ 0.

Theorem 6 (see Th. 2.1 in [18]). The convergence V_n⁻¹()ξ_n⁽⁾−−−−→^D

n→∞ W in the spaceC[0,1]

holds if and only if 1∈DAN.

Lemma 7 (see Lemma 4 in [10]). If 1∈DANthen U_n²()

V_n²()

−−−−→Pr n→∞ 1.

The following key lemma is essentially an adaptation of Lemma 1 of Peligrad and Utev [16]. It plays a key role to transfer some limit theorems from the innovations to the linear process.

Lemma 8 (see e.g. Lemma 1 in [17]). Let(ai)_i∈Zbe a collection of real numbers, satisfying

X

i∈Z

|ai|<∞. (17)

Assume that(Zn,i, n∈N, i∈Z)is a collection of random elements with values in a separable Banach space(E,|| · ||_E)satisfying the following conditions:

(i) sup_n∈N,i∈ZE||Zn,i||_E<∞,

(ii) For every xedi, j∈Z it holds ||Zn,i−Zn,j|| −−−−→^Pr 0.

Then for each n∈N the series P

i∈Za_iZ_n,i converge a.s. and for every index

`∈Z, the following convergence

X

i∈Z

aiZn,i−AZn,`

E

−−−−→Pr

n→∞ 0 (18)

holds, whereA=P

i∈Zai.

We state here for reference convenience the following special case of Lemma 8 whereE=`²(N)andZ_n,i= Z_n,i^(j)

j∈N.

Lemma 9. Let (ai)i∈Z satises (17). Assume that (Z_n,i^(j), n, j ∈ N, i ∈Z)is a collection of random variables satisfying the following conditions:

(i) sup_{n∈N, i∈Z}E P

j∈N[Z_n,i^(j)]²1/2

<∞;

(ii) for anyi, k∈Z,P

j∈N(Z_n,i^(j)−Z_n,k^(j))²−−−−→^Pr

n→∞ 0. Then with anyk∈Zit holds

∆_n:=

X

j∈N

X

i∈Z

a_iZ_n,i^(j)21/2

− A²X

j∈N

[Z_n,k^(j)]^21/2

−−−−→Pr n→∞ 0.

3 Proof of the limit theorems

Theorem 1 is an immediate consequence of Theorem 2 sinceS_n=ξ_n(1). Theo- rem 3 follows easily from Theorem 2 in view of the elementary estimate

kζ_n−ξ_nk_∞≤ max

1≤k≤n|X_k|.

Indeed combining Lemma 3 in [13] with Lemma 4 in [10] it is clear thatU_n⁻¹max_1≤k≤n|Xk| goes to zero in probability. So we only have to prove Theorem 2.

We shall prove that b⁻¹_n h

ξ_n, U_n

− Aξ⁽⁾_n ,|A|Un()i _Pr

−−−−→

n→∞ 0 (19)

as n → ∞, in the product space C[0,1]×R. If (19) is established, then Theorem 2 follows easily. Indeed, (19) yields that the asymptotic behavior of {U_n⁻¹ξ_n, n∈N} is the same as that of{(A/|A|)U_n⁻¹()ξn⁽⁾, n∈N}. By Lemma 6 and Lemma 7:

Aξn⁽⁾

|A|Un()

−−−−→D n→∞

A

|A|W, in the spaceC[0,1].

But(A/|A|)W has the same distribution asW. So the proof reduces to (19).

Evidently, (19) follows from b⁻¹_n max

t∈[0,1]|ξ_n(t)−Aξ_n⁽⁾(t)|−−−−→^Pr

n→∞ 0 (20)

and

b⁻¹_n |U_n− |A|U_n()|−−−−→^Pr

n→∞ 0. (21)

First we prove (20). To this aim, introducing the elementary polygonal lines e_n,k(t) = (nt−(k−1))1[(k−1)/n,k/n](t) +1_(k/n,+∞)(t), 1≤k≤n, we write for everyt∈[0,1], the expansion

ξn(t) =

n

X

k=1

Xken,k(t) =

n

X

k=1

X

i∈Z

ai_k−ien,k(t)

=X

i∈Z

ai n

X

k=1

_k−ien,k(t)

! ,

which leads naturally to introduce the partial sums processes ξ⁽⁾_n,i(t) :=

n

X

k=1

_k−ien,k(t), t∈[0,1], i∈Z, n≥1.

Thus

ξ_n =X

i∈Z

a_iξ_n,i⁽⁾. (22)

A priori, the series of functions (22) converges pointwise on [0,1]. In fact, this convergence holds also (almost surely) in the topology ofC[0,1], since by stationarity

X

i∈Z

|ai|E ξ⁽⁾_n,i

_∞=E ξ⁽⁾_n

_∞

X

i∈Z

|ai|<∞.

DenoteZ_n,i =b⁻¹_n ξ_n,i⁽⁾, i∈Z, n∈N. To this collection of random elements in the Banach spaceC[0,1]we shall apply Lemma 8. First we check Condition (i). We have by stationarity

sup

n≥1,i∈Z

EkZn,ik_∞= sup

n≥1

EkZn,0k_∞= sup

n≥1

b⁻¹_n ξ_n⁽⁾

∞.

To check the niteness of this last supremum, we note rst the stochastic bound- edness of(b⁻¹_n ξ⁽⁾n )_n≥1inC[0,1](combine (13) with Theorem 6). Then we apply Proposition 2 in [17] withbn=n^1/2L(n),Lslowly varying.

Next we check (ii) of Lemma 8. We have kZn,i−Zn,jk_∞=b⁻¹_n

n

X

k=1

(_k−i−_k−j)en,k

∞

=b⁻¹_n max

1≤k≤n|Ti,j,k|, where

Ti,j,k=

k

X

`=1

(_{`−i</sub>−<sub>`−j}).

Looking at the dierent possible congurations, we observe that

|T_i,j,k| ≤2|j−i| max |_`|.

As i and j are xed, we may assume thatn ≥max(|i|,|j|), in which case we obtain

1≤k≤nmax |Ti,j,k| ≤2|j−i| max

−n<`≤2n|`|.

By (11) b⁻¹_n max<sub>1≤`≤n</sub>|`| goes to zero in probability and by stationarity, the same holds true forb⁻¹_n max<sub>−n<`≤2n</sub>|`|, so Condition (ii) of Lemma 8 is satis- ed. Hence, by this lemma we have

X

i∈Z

aiZn,i−AZn,0

∞

−−−−→Pr n→∞ 0, which coincides with (20).

Next we prove (21). We have

U_n²=

N

X

j=1

B_m,j² =

N

X

j=1

X

i∈Z

aiVn,j,i

!2

, (23)

whereV_n,j,i=Pjm

`=(j−1)m+1<sub>`−i</sub>(recalling thatm=m(n)). We shall check the conditions of Lemma 9 with Z_n,i^(j) =b⁻¹_n Vn,j,i, n, j∈N, i∈Z. For (i) we have to prove

b⁻¹_n sup

n,i

E





N

X

j=1

V_n,j,i²





1/2

<∞.

By stationarity this will follow from

b⁻¹_n sup

n≥1

E





N

X

j=1

V_n,j,0²





1/2

<∞. (24)

Set for eachi∈Z,

⁰_i=i1_{|ei|≤bn}−Ei1_{|i|≤bn}, ⁰⁰_i =i1_{|i|>bn}−Ei1_{|i|>bn} and dene V_n,j,i⁰ ,V_n,j,i⁰⁰ by substituting respectivelyby⁰, ⁰⁰ in the denition ofVn,j,i. AsEi= 0, i=⁰_i+⁰⁰_i and we have

E





N

X

j=1

V_n,j,0²





1/2

≤T_n⁰ +T_n⁰⁰,

where

T_n⁰ =E





N

X

j=1

V_n,j,0⁰²





1/2

, T_n⁰⁰=E





N

X

j=1

V_n,j,0⁰⁰²





1/2

. Using Jensen inequality and recalling thatN m≤nwe get

b⁻¹_n T_n⁰ ≤b⁻¹_n





N

X

j=1

EV_n,j,0⁰²





1/2

≤b⁻¹_n (nE⁰²₁)^1/2.

Hence sup_nb⁻¹_n T_n⁰ <∞ by (12). Next, by comparison of the norms`¹(N)and

`²(N), we obtain

T_n⁰⁰≤

N

X

j=1

E V_n,j,0⁰⁰

≤nE|⁰⁰₁|.

As E|⁰⁰₁| ≤ 2E |1|1{|1| > b_n}, b⁻¹_n T_n⁰⁰ converges to zero by Lemma 4 which completes the verication of (24) and of Condition (i).

Next we check Condition (ii) of Lemma 4, that is we have to establish the convergence

b⁻²_n

N

X

j=1

(Vn,j,i−Vn,j,`)²−−−−→^Pr

n→∞ 0.

By stationarity and a simple chaining argument it is enough to prove that b⁻²_n

N

X

j=1

(Vn,j,1−Vn,j,0)²−−−−→^Pr

n→∞ 0.

This follows from

b⁻²_n

N

X

j=1

²_jm−−−−→^Pr

n→∞ 0.

As the random vectors (i)_1≤i≤N and (ejm)_1≤j≤N have the same distribution, it is enough to check that

b⁻²_n

N

X

i=1

²_i −−−−→^Pr

n→∞ 0. (25)

Accounting (13), this reduces to V_N²() V_n²()

−−−−→Pr n→∞ 0.

Noting that the non negative random variables Yn,i =V_n⁻²()²_i, i = 1, . . . , n have identical distribution and thatPn

i=1Yn,i= 1, it follows thatEYn,i= 1/n and consequently

EV_N²() V_n²() = N

n.

This implies the convergence (25) since N/n tends to zero. By Lemma 9 we conclude (21). This completes the proof of Theorem 2.

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