Moderate deviation principle for autoregressive processes

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Journal of Multivariate Analysis

journal homepage:www.elsevier.com/locate/jmva

Moderate deviation principle for autoregressive processes

Yu Miao

a,∗

, Si Shen

b

a_{College of Mathematics and Information Science, Henan Normal University, 453007 Henan, China} b_{Department of Statistics, College of Science, Central University for Nationalities, 100081 Beijing, China}

a r t i c l e i n f o Article history:

Received 20 April 2007 Available online 21 June 2009 2000 MR Subject Classifications: 60F10 60G10 62J05 Keywords: Moderate deviation Autoregressive processes Least squares estimator Yule–Walker estimator

a b s t r a c t

A moderate deviation principle for autoregressive processes is established. As statistical applications we provide the moderate deviation estimates of the least square and the Yule–Walker estimators of the parameter of an autoregressive process. The main assumption on the autoregressive process is the Gaussian integrability condition for the noise, which is weaker than the assumption of Logarithmic Sobolev Inequality in [H. Djellout, A. Guillin, L. Wu, Moderate deviations of empirical periodogram and nonlinear functionals of moving average processes, Ann. I. H. Poincaré-PR 42 (2006) 393–416].

©2009 Elsevier Inc. All rights reserved.

1. Introduction

Consider the linear autoregressive model in_Rd_,

Xn

=

θ

Xn−1

+

ξ

n

,

(1.1)

where

θ

∈

Md (the space ofd

×

dmatrices) is unknown,

(ξ

n

)

n∈Z is a sequence of centered i.i.d. r.v. valued in Rd representing the noise and which is independent ofX0, and

(

Xn

)

n≥0is observed. Assume that the law ofX0is invariant (or equivalently

(

Xn

)

n≥0is stationary), there are two important issues: (i) the estimate of the covariance matrix Cov

(

X0

,

Xl

)

:=

E

(

X0

−

EX0

)(

Xl

−

EXl

)

t(hereXnis regarded as column vector andAtdenotes the transposition of matrixA); (ii) estimate of

θ

. It is quite easy (and well known) to check a stationary solution to(1.1), which is given by

Xn

=

∞

X

p=0

θ

p

_ξ

n−p

,

n

≥

0

once if

k

θ

k :=

sup|x|≤1

|

θ

x

|

<

1. So it is a special moving average process. A general moving average process is given by

Xn

:=

+∞

X

j=−∞ aj−n

ξ

j

=

+∞

X

j=−∞ aj

ξ

n+j

,

∀

n

∈

Z

,

where

(ξ

n

)

n∈Zis i.i.d.,

(

an

)

n∈Zbe a sequence of real numbers such that

X

n∈Z

|

an

|

2

<

∞

.

(1.2)

∗_{Corresponding author.}

E-mail addresses:[email protected](M. Yu),[email protected](S. Si). 0047-259X/$ – see front matter©2009 Elsevier Inc. All rights reserved.

The most natural estimator ofΓ

(

Xl

,

X0

)

:=

(

Cov

(

X0i

,

X j

l

))

1≤i,j≤d

(

l

≥

0

)

is given by the empirical covariance (with the given sample

(

Xk

)

0≤k≤n+l) C_n∗_,l

=

1 n n

X

k=1 Xk+lXkt

,

(1.3)

and for estimating

θ

, the following two estimators are widely used: (i) Least Square Estimator:

b

θ

n

=

n

X

k=1 XkXkt−1

!

_n

X

k=1 Xk−1Xkt−1

!

−1

.

(1.4)

(ii) Yule–Walker Estimator:

e

θ

n

=

n

X

k=1 XkXkt−1

!

_n

X

k=0 XkXkt

!

−1

.

(1.5)

In this paper we are interested in the moderate deviation behavior of the empirical covariance and of

b

θ

n

,

e

θ

n. There is a rich literature on the central limit theorem and iterated logarithmic law about those three estimators.

The study on large deviations and moderate deviations are relatively recent.

Gaussian case(i.e., the noise

ξ

is assumed Gaussian). This subject is opened by Donsker and Varadhan [1] who proved

the level-3 large deviation principle for general stationary Gaussian processes under the continuity of the spectral function. Bryc and Dembo [2] (1993) proved for the first the large and moderate deviation principles for the empirical varianceC_n∗_,0

even for general stationary Gaussian processes. Bercu, Gamboa and Rouault [3] proved the large deviation principle forC_n∗_,l,

l

≥

0 (which is much more delicate thanC∗

n,0) and for

b

θ

n

,

e

θ

n.

Non-Gaussian case. Wu [4] first extended Donsker–Varadhan’s theorem on large deviations of level-3 from stationary

Gaussian processes to general moving average processes under the Gaussian integrability condition on the driven variable

ξ

. Djellout–Guillin–Wu [5] established, in the one-dimensional case (i

.

e

.,

d

=

1), moderate deviation principle for nonlinear functionals of a general moving average processes covering the case ofC∗

n,land for the periodogram, but under the assumption that the law of the driven random variable

ξ

satisfies the log-Sobolev inequality, stronger than the Gaussian integrability in [4]. The main contribution of this paper is to remove the assumption of log-Sobolev inequality on the driven variable, for the particular but important autoregression model.

For the Hilbertian autoregressive model with driven r.v.

ξ

satisfying the Gaussian integrability condition, in which

{

ξ

_k

,

Xk

}

k∈Ztake values in some separable Hilbert spaceH, Mas and Menneteau [6] established large and moderate deviation for the empirical meanXn

=

1_n

P

n

k=1Xk, and moderate deviation for the empirical variance matrix1_n

P

n

k=1Xk

⊗

Xk, where

x

⊗

y(x

,

y

∈

H) denotes the linear operator fromHtoH,

x

⊗

y

:

h

∈

H

→ h

x

,

h

i

y

,

extending the result of Bryc–Dembo [2] from_Rd_{toH, and especially from Gaussian case to general sub-Gaussian case.} Furthermore, Menneteau [7] obtained some laws of the iterated logarithm in Hilbertian autoregressive models for the empirical covariance 1_n

P

n

k=1Xk

⊗

Xk. Our main purpose is to extend this result to the MDP of the empirical covariance matrices 1_n

P

n

k=1Xk

⊗

Xk+l

0≤l≤M. The difficulty in this generalization is similar for the passage from empirical variance to empirical covariance (i.e. from Bryc–Dembo [2] to Bercu, Gamboa and Rouault [3]) in the Gaussian case.

2. Main results 2.1. Assumptions

Let

{

ξ

_n

}

_n∈Zbe a sequence ofR

d_{-valued centered i.i.d. random variables, and suppose the following conditions hold:} (C₁) the moderate deviation scale

(

b_n

)

is a sequence of positive numbers satisfying 1

b_n

√

n, i.e., asn

→ ∞

,

bn

→ ∞;

bn

√

n

→

0

.

(C2)E

ξ

0

=

0 and

ξ

0satisfies the Gaussian integrability condition, i.e., there exists

α >

0 such that

Eeα|ξ0| 2

<

∞

.

(C3)Kθ

:=

P

∞ k=0

k

θ

k

_k

_<

_+∞

_where

_k

_θ

_{k :=}

_sup

2.2. Main results

The space ofd

×

dreal matricesMdis provided with the inner product

h

A

,

B

i :=

tr

(

ABt

)

where tr

(

·

)

is the trace. Let

Uk,l

=

θ

Xk+l−1

ξ

kt

+

ξ

k+lXkt−1

θ

t

₊

_ξ

k+l

ξ

kt

−

θ

l_Γ

_(ξ

0

, ξ

0

),

−

→

Uk

:=

(

Uk,l

)

0≤l≤M

.

Theorem 2.1. Assume that the conditions

(

C1

)

,

(

C2

)

and

(

C3

)

are satisfied, then

1 bn √ n

P

n k=1

(

Xk+lXkt

−

EXk+lXkt

)

0≤l≤Msatisfies

the large deviation principle onM_dM+1with speed b2

nand with the rate function given by

I

(

Γ

)

=

sup Λ∈MM_d+1

(

_M

X

l=0

h

Λ_l

,

Γ_l

−

θ

Γ_l

θ

t

i −

1 2Σ 2

₍

_Λ

₎

)

,

∀

Γ

=

(

Γ₀

, . . . ,

Γ_M

)

∈

MM_d+1

.

(2.1) Here

h

Λ

,

Γ

i :=

P

M k=0

h

Λk

,

Γk

i

and Σ2

₍

_Λ

₎

₌

E

h

Λ

,

−

→

U1

i

2

+

2E M+1

X

k=2

h

Λ

,

−

→

U1

ih

Λ

,

−

→

Uk

i

.

(2.2)

In particular for every l

≥

0fixed,_b1

n

√

n

P

n

k=1

(

Xk+lXkt

−

EXk+lXkt

)

satisfies the large deviation principle onMdwith speed b2nand

with the rate function given by I

(

Γl

)

=

sup Λl∈Md

h

Λ_l

,

Γ_l

−

θ

Γ_l

θ

t

i −

1 2E

h

Λl

,

U1,l

i

2

,

∀

Γ_l

∈

Md

.

(2.3) 2.3. Applications

In the subsection, we provide a statistical application. More precisely we shall applyTheorem 2.1to the least square estimator

θ

ˆ

nand the Yule–Walker estimator

θ

˜

n:

Proposition 2.2.√ Suppose the conditions

(

C1

)

,

(

C2

)

and

(

C3

)

and assume that the covariance matrixEX0X0tis non-singular. Then n bn

(

ˆ

θ

n

−

θ)

as well as √ n bn

(

˜

θ

n

−

θ)

satisfies the large deviation principle onMdwith speed b2nand the rate function given by

I

(

Γ

)

=

sup Λ∈Md

h

Λ

,

Γ

i −

1 2E

h

Λ

, ξ

1X t 0

[

EX0X0t

]

−1

_i

2

.

In particular, when d

=

1, the rate function above can be identified as

I

(

x

)

=

x 2

2

(

1

−

θ

2

)

.

Remark 2.3. Under our conditions it is quite easy to see that the Yule–Walker estimator

θ

˜

n shares the same MDP as

the least square estimator

θ

ˆ

. A curious phenomena was found by Bercu, Gamboa and Rouault [3]: in the Gaussian noise and one-dimensional case, they proved the large deviations of

θ

ˆ

and

θ

˜

n, which possess two different rate functions. The MDP above was proved by Worms [8], where the rate function is identified basing on the Kronecker product of matrices. Djellout–Guillin–Wu [5] derived it as a consequence of their general results on the MDP of moving average processes, but with an extra and strong condition that the law of

ξ

0satisfies a log-Sobolev inequality (though their method go far beyond the regression model).

3. Proofs

3.1. Proof ofTheorem 2.1

Letmbe a given positive integer, a sequence

(

Zn

)

n≥1of strictly stationary random variables is calledm-dependent if for everyk

≥

1 the two collections

{

Z1

, . . . ,

Zk

}

and

{

Zk+m

,

Zk+m+1

, . . .

}

are independent. We have the following:

Lemma 3.1 (Chen [9]).Let

(

Zn

)

n≥1be a stationary sequence of m-dependent random variables taking values inRd, such that

E

(

eα|Z1|

) <

∞

,

for some

α >

0

.

lim n→∞ 1 b2 n log_E





e b2nhλ,√_nbn1 n P k=1 (Zk−EZk)i





=

1 2nlim→∞ 1 nE

*

λ,

n

X

k=1

(

Zk

−

EZk

)

+

2

=

1 2 E

h

λ,

Z1

i

2

₊

₂ m+1

X

k=2 E

h

λ,

Z1

ih

λ,

Zk

i

!

.

Step1:(Autoregressive Representation for the Covariance Process)

By the stationarity of

{

Xn

}

, for everyk

∈

Z, the distribution law ofXk+lXkt is the same withXlX0t. LetCl

:=

EXk+lXkt

=

Γ

(

Xl

,

X0

)

and it is easy to see that

Cl

=

θ

lΓ

(

X0

,

X0

)

=

θ

l ∞

X

k=0

θ

k_Γ

_(ξ

0

, ξ

0

)(θ

k

)

t

=

ECn∗,l

,

(3.1) whereC∗

n,lis defined in(1.3). In addition, let

Zk,l

=

Xk+lXkt

−

Cl

,

Uk,l

=

θ

Xk+l−1

ξ

kt

+

ξ

k+lXkt−1

θ

t

₊

_ξ

k+l

ξ

kt

−

θ

l_Γ

_(ξ

0

, ξ

0

),

(3.2) and

−

→

Uk

=

(

Uk,l

)

0≤l≤M

.

We have the following autoregressive representation for the covariance process.

Lemma 3.2. Under the above notions, for any k

∈

_Z, l

∈

_Z+

_{∪ {}

₀

_}

_{, we have}

Zk,l

=

θ

Zk−1,l

θ

t

+

Uk,l

,

(3.3) and C_n∗_,l

−

Cl

=

(

1

−

R

)

−1

(

U

¯

n,l

)

+

(

1

−

R

)

−1R

Z0,l

−

Zn,l n

,

(3.4)

whereU

¯

n,l

=

n−1

P

nk=1Uk,land the linear operator R

:

s

∈

M

(

d

×

d

)

7→

θ

s

θ

t.

Proof. Easy calculus, so omitted.

The following result shows that the main part ofC_n∗_,l

−

Clis

(

1

−

R

)

−1U

¯

n,lin the sense of moderate deviation.

Lemma 3.3. Under the conditions

(

C₁

)

and

(

C₂

)

, there exists

α >

0such that, for all r

>

0,

P

k

Z0,l

−

Zn,l

k

HS

>

rbn

√

n

≤

4 exp

−

α

rbn

√

n 2K_θ2

E

eα|ξ0|2

,

(3.5) where K_θ is given by K_θ

:=

∞

X

k=0

k

θ

k

k

.

(3.6)

Consequently for any r

>

0,

lim sup n→∞ 1 b2 n log_P

√

n bn

k

(

1

−

R

)

−1_R

₍

_Z 0,l

−

Zn,l

)

k

HS n

>

r

= −∞

.

(3.7)

Here

k

A

k

_HS

=

√

h

A

,

A

i =

√

tr

(

AAt

)

_{is the Hilbert–Schmidt norm.}

Proof. From the stationarity of

{

Xn

}

n≥0, we get

P

(

k

Z0,l

−

Zn,l

k

HS

>

rbn

√

n

)

=

_P

(

k

XlX0t

−

Xn+lXnt

k

HS

>

rbn

√

n

)

≤

2P

(

k

XlX0t

k

HS

>

rbn

√

n

/

2

)

≤

4_P

(

|

X0

|

2

>

rbn

√

n

/

2

)

where the last inequality follows from Cauchy–Schwarz:

k

XlX0t

k

HS

≤

12

(

|

Xl

|

2

+ |

X0

|

2

)

. Since the remainders of the proof is similar to Lemma 13 of Mas and Menneteau in [6], here we omit them.

Step2:(Asymptotic Term and Moderate Deviation)

For allk

≥

1 andm

≥

2,m

>

l, set

Xk−1,m

=

ξ

k−1

+

θξ

k−2

+ · · · +

θ

m −2

_ξ

k−m+1

=

m−2

X

j=0

θ

j

_ξ

k−1−j

,

Uk,l,m

=

θ

Xk+l−1,m

ξ

kt

+

ξ

k+lXkt−1,m

θ

t

₊

_ξ

k+l

ξ

kt

−

θ

l_Γ

_(ξ

0

, ξ

0

)

=

m−1

X

j=1

θ

j

_ξ

k+l−j

ξ

kt

+

m−1

X

j=1

ξ

k+l

ξ

kt−j

(θ

j

₎

t

₊

_ξ

k+l

ξ

kt

−

θ

l_Γ

_(ξ

0

, ξ

0

),

and

E

Uk,m

=

(

Uk,l,m

)

0≤l≤M

.

It is easy to see that

{ E

Uk,m

}

k≥1is a strictly stationary sequence with (M

+

m)-dependent structure. Furthermore for each

l

≥

0 fixed,

{

Uk,l,m

,

k

≥

0

}

is a martingale difference sequence w.r.t.

(

Fk

)

k≥0, whereFn

:=

σ(ξ

k

; −∞

<

k

≤

n

)

. Set

¯

Un,l,m

=

1 n n

X

k=1 Uk,l,m and Qn,m

=

(

U

¯

n,l,m

)

0≤l≤M

.

Applying Chen’sLemma 3.1, we have

Lemma 3.4. Under the conditions

(

C1

)

,

(

C2

)

and

(

C3

)

, for all m

≥

1,Λ

=

(

Λ0

, . . . ,

ΛM

)

∈

MM

+1 d , writing

h

Λ

,

Qn,m

i :=

P

M l=0

h

Λl

,

U

¯

n,l,m

i

we have lim n→∞ 1 b2 n log_Eexp

bn

√

n

h

Λ

,

Qn,m

i

=

1 2Σ 2 m

(

Λ

),

(3.8) whereΣ2 m

(

Λ

)

:=

E

h

Λ

,

−

→

U_1,m

i

2

₊

₂

P

M+1 k=2 E

h

Λ

,

−

→

U_1,m

i · h

Λ

,

−

→

U_k,m

i

. Step3: (The Asymptotic Negligibility of 1

bn

√

n

{ E

Un

− E

Un,m

}

n≥1as m

→ ∞

)

This is, of course, the crucial and difficult step. Without loss of generality, we only consider

{ ¯

Un,l

− ¯

Un,l,m

}

n≥1,

∀

0

≤

l

≤

M. In the next result, we establish an exponential inequality for

{ ¯

Un,l

− ¯

Un,l,m

}

n≥1and we obtain that

n

√ n bn

¯

Un,l,m

o

n≥1,m≥2is a b2

n-exponentially good approximation of the sequence

n

√ n bn

¯

Un,l

o

n≥1 . For allp

≥

0 andk

≥

1, set

Wk,p

=

ξ

k

θ

p

k

θ

p

_k

ξ

k−p

t

.

Lemma 3.5. Assume the conditions

(

C1

)

,

(

C2

)

and

(

C3

)

.

(i) There exist

α

0and

β

0such that, for all p

≥

1, n

≥

1and t

≥

0,

P max j≤n

j

X

k=1 Wk,p

HS

≥

t

!

≤

36 exp

−

t 2

α

0n

+

β

0t

.

(3.9)

(ii) For all r

>

0, there exist N

≥

1and A

,

B

>

0such that, for all n

≥

N and m

≥

1,

P max j≤n

j

X

k=1

(

Uk,l

−

Uk,l,m

)

HS

>

rbn

√

n

!

≤

72

1

−

exp

−

b 2 nr2

(

Ar

+

B

)

k

θ

m

_k

−1 exp

−

r 2_b2 n

(

Ar

+

B

)

k

θ

m

_k

.

(3.10)

(iii) For all r

>

0,

lim sup m→∞ lim sup n→∞ 1 b2 n logP

√

n bn

k ¯

Un,l

− ¯

Un,l,m

k

HS

>

r

= −∞

.

(iv) For every

λ >

0,

lim sup m→∞ lim sup n→∞ 1 b2 n log_Eexp

λ

√

nbn

k ¯

Un,l

− ¯

Un,l,m

k

HS

=

0

.

Proof. (i) This is Lemma 17 in [6].

(ii) The approach to prove the inequality stems from Lemma 17 of Mas and Menneteau in [6]. For the sake of completeness, we state the proof. We have

n

X

k=1

(

Uk,l

−

Uk,l,m

)

HS

=

n

X

k=1

[

θ(

Xk+l−1

−

Xk+l−1,m

)ξ

kt

+

ξ

k+l

(

Xk−1

−

Xk−1,m

)

t

θ

t

]

HS

≤

n

X

k=1

θ(

Xk+l−1

−

Xk+l−1,m

)ξ

kt

HS

+

n

X

k=1

ξ

k+l

(

Xk−1

−

Xk−1,m

)

t

θ

t

HS

.

(3.11) Moreover, Xk+l−1

−

Xk+l−1,m

=

∞

X

p=m−1

θ

p

_ξ

k+l−1−p

=

θ

m−1 ∞

X

p=0

θ

p

_ξ

k+l−m−p

!

.

Hence using

k

AB

k

_HS

≤ k

A

kk

B

k

_HS, we have

n

X

k=1

θ(

Xk+l−1

−

Xk+l−1,m

)ξ

kt

HS

=

∞

X

p=0 n

X

k=1

θ

m

_(θ

p

_ξ

k+l−m−p

)ξ

kt

HS

≤ k

θ

m

k

∞

X

p=0

k

θ

p

k

n

X

k=1 Wk,m+p−l

HS

.

Now we need the following fact: under

(

C3

)

,

K1

:=

∞

X

p=0

(

p

+

1

)

k

θ

p

k

<

∞

.

Therefore, if we set tm,p

(

r

)

=

r

(

p

+

1

)

2K1

k

θ

m

k

,

we have, by(3.9), P max j≤n

j

X

k=1

θ(

Xk+l−1

−

Xk+l−1,m

)ξ

kt

HS

>

rbn

√

n

/

2

!

≤

_P ∞

X

p=0

(

p

+

1

)

k

θ

p

_k

p

+

1maxj≤n

j

X

k=1 Wk,m+p−l

HS

>

∞

X

p=0

(

p

+

1

)

k

θ

p

k

rbn

√

n 2K1

k

θ

m

k

!

≤

∞

X

p=0 P max j≤n

j

X

k=1 Wk,m+p−l

HS

>

(

p

+

1

)

rbn

√

n 2K1

k

θ

m

k

!

≤

36 ∞

X

p=0 exp

−

b 2 ntm2,p

(

r

)

α

0

+

β

0tm,p

(

r

)

bn

/

√

n

!

.

(3.12)

By the assumption ofbn, there exists constantsN

∈

N∗

,

A

,

B

>

0, such that for alln

≥

N,m

≥

1 andl

≥

0,

√ n bn

≥

1 and we obtain t2 m,p

(

r

)

α

0

+

β

0tm,p

(

r

)

bn

/

√

n

≥

c

(

r

)

p

+

1

k

θ

m

_k

,

c

(

r

)

:=

r2 Ar

+

B

.

(3.13)

Hence, by(3.12)and(3.13), we get

P max j≤n

j

X

k=1

θ(

Xk+l−1

−

Xk+l−1,m

)ξ

kt

HS

>

rbn

√

n

/

2

!

≤

36 ∞

X

p=0 exp

−

b2_n c

(

r

)

k

θ

m

_k

(

p

+

1

)

.

(3.14)

For the same reason, we have

n

X

k=1

ξ

k+l

(

Xk−1

−

Xk−1,m

)

t

θ

t

HS

=

∞

X

p=0 n

X

k=1

ξ

k+l

ξ

kt−m−p

(θ

p+m

₎

t

HS

≤ k

θ

m

k

∞

X

p=0

k

θ

p

k

n

X

k=1 Wk+l,m+p+l

HS

,

and for alln

≥

N,

P max j≤n

j

X

k=1

ξ

k+l

(

Xk−1

−

Xk−1,m

)

t

θ

t

HS

>

rbn

√

n

/

2

!

≤

36 ∞

X

p=0 exp

−

b2_n c

(

r

)

k

θ

m

_k

(

p

+

1

)

.

(3.15)

So, from(3.14)and(3.15), we obtain

P max j≤n

j

X

k=1

(

Uk,l

−

Uk,l,m

)

HS

>

rbn

√

n

!

≤

_P max j≤n

j

X

k=1

θ(

Xk+l−1

−

Xk+l−1,m

)ξ

kt

HS

>

rbn

√

n

/

2

!

+

_P max j≤n

j

X

k=1

ξ

k+l

(

Xk−1

−

Xk−1,m

)

t

θ

t

HS

>

rbn

√

n

/

2

!

≤

72 ∞

X

p=0 exp

−

b2_n c

(

r

)

k

θ

m

_k

(

p

+

1

)

=

K

(

r

)

exp

−

b2_n c

(

r

)

k

θ

m

_k

where K

(

r

)

=

72

1

−

exp

−

b 2 nc

(

r

)

k

θ

m

_k

−1

the desired inequality.

(iii) It follows obviously by(3.10).

(iv) The inequality

≥

in (iv) is obvious. Below we prove the converse inequality. Let

Z

=

Zn,m

:=

1 bn

√

n

n

X

k=1

(

Uk,l

)

−

Uk,l,m

HS and

δ >

0 be an arbitrary positive constant. We have

Eeλb 2 nZn,m

₌

E

Z

Z 0

λ

b2_neλbn2r_dr

₊

₁

₌

₁

₊

Z

∞ 0

λ

b2_neλb2nr P

(

Z

>

r

)

dr

≤

1

+

λ

b2_neλb2nδ

₊

Z

∞ δ

λ

b2_neλb2nr P

(

Z

>

r

)

dr

.

Choosingm0

≥

1 so that _(Ar+Br)k_θm_k

≥

λ

+

1 for allr

≥

δ

, we have by (ii),

P

(

Zn,m

>

r

)

≤

72

1

−

exp

−

b 2 nr2

(

Ar

+

B

)

k

θ

m

_k

−1 exp

−

b2_n r 2

(

Ar

+

B

)

k

θ

m

_k

≤

C

(δ)

exp

(

−

(λ

+

1

)

b2_nr

)

whereC

(δ)

is some constant (independent ofm

,

n). Consequently

Z

∞ δ

λ

b2_neλb2nr P

(

Z

>

r

)

dr

≤

λ

b2n

R

∞ δ C

(δ)

e−b 2 nr_dr

=

λ

C

(δ)

e−b2nδ

≤

λ

C

(δ).

Thus lim sup m→∞ lim sup n→∞ 1 b2 n logEexp

λ

√

nbn

k ¯

Un,l

− ¯

Un,l,m

k

HS

≤

lim sup n→∞ 1 b2 n log

1

+

λ

b2_neλb2nδ

₊

λ

_C

(δ)

=

λδ.

As

δ >

0 is arbitrary, we obtain the desired claim.

Step4:(The Identification of Rate Function)

At firstΣ_m2

(

Λ

)

→

Σ2

(

Λ

)

asmgoes to infinity, whereΣ_m2

(

Λ

)

is given inLemma 3.4, andΣ2

(

Λ

)

is given inTheorem 2.1. For everyk

≥

M

+

2, by considering the case:l

=

0 and 0

<

l

≤

Mseparately, we obtain

E

(

Uk,l

|

F1+M

)

=

E

[

θ

Xk+l−1

ξ

kt

+

ξ

k+lXkt−1

θ

t

₊

_ξ

k+l

ξ

kt

−

θ

l_Γ

_(ξ

0

, ξ

0

)

|

F1+M

] =

0

.

From the properties of conditional expectation,

EU1,iUk,l

=

E

[

U1,iE

(

Uk,l

|

F1+M

)

] =

0

,

for any 0

≤

i

,

l

≤

Mandk

≥

M

+

2.

LetQn

= {

(

U

¯

n,l

)

}

0≤l≤M, we will establish the moderate deviation ofQn, namely, for anyΛ

∈

MMd+1, lim n→∞ 1 b2 n logEexp

bn

√

n

h

Λ

,

Qn

i

=

1 2Σ 2

₍

_Λ

_).

_(3.16)

For any fixedp

,

q

>

1 with1_p

+

1

q

=

1, by the Hölder inequality we have that log_Eexp

bn

√

n

h

Λ

,

Qn

i

≤

1_plogEexp

pbn

√

n

h

Λ

,

Qn,m

i

+

1_qlogEexp

qbn

√

n

h

Λ

,

Qn

−

Qn,m

i

for allΛ

∈

MM_d+1. FromLemma 3.4and (iv) inLemma 3.5, we have

lim sup n→∞ 1 b2 n logEexp

bn

√

n

h

Λ

,

Qn

i

≤

p 2Σ 2

₍

_Λ

_).

_(3.17)

Similarly, by the Hölder inequality, we have for everyΛ, 1 b2 n logEexp

p−1bn

√

n

h

Λ

,

Qn,m

i

≤

1 b2 n

1 plogEexp

bn

√

n

h

Λ

,

Qn

i

+

1 qlogEexp

(

q

/

p

)

bn

√

n

h

Λ

,

Qn,m

−

Qn

i

.

Taking first lim infn→∞and next limm→∞we get fromLemma 3.4and (iv) inLemma 3.5

1 2p2Σ 2

₍

_Λ

₎

_≤

lim inf n→∞ 1 b2 n Eexp

bn

√

n

h

Λ

,

Qn

i

.

(3.18)

Lettingp

→

1 in(3.17)and(3.18)yields(3.16).

By the Ellis–Gärtner theorem ([10], Section2.3),(3.16)implies that_P

√

n bnQn

∈ ·

satisfies the large deviation principle onMM_d+1with speedb2

nand with the rate function given by

J

(

Γ

)

=

sup Λ∈MM_d+1

h

Λ

,

Γ

i −

1 2Σ 2

₍

_Λ

₎

.

ByLemmas 3.2and3.3and the contraction principle,_P

√ n bn

(

C ∗ n,l

−

Cl

)

∈ ·

satisfies the large deviation principle onMM_d+1

with speedb2

nand with the rate function

I

(

Γ

)

=

J

((

1

−

R

)

Γ

)

which is exactly the expression(2.1)ofIby the definition ofJabove.

3.2. Proof ofProposition 2.2 Let us introduce rn

:=

√

n bn

(

ˆ

θ

n

−

θ)

and Rn

=

1

√

nbn n

X

i=1

(

XiXit−1

−

θ

Xi−1Xit−1

)(

EX0X0t

)

−1

_.

ByTheorem 2.1,Rnsatisfies the moderate deviation principle. Before identifying its rate function let us first show thatrn

−

Rn is negligible with respect to the moderate deviation principle, i.e., for anyr

>

0,

lim n→∞ 1 b2 n log_P

(

k

rn

−

Rn

k

HS

>

r

)

= −∞

.

(3.19)

To that end, note rn

=

√

n bn n

X

i=1

(

XiXit−1

−

θ

Xi−1Xit−1

)

!

_n

X

i=1 Xi−1Xit−1

!

−1

=

1 bn

√

n n

X

i=1

(

XiXit−1

−

θ

Xi−1Xit−1

)

!

×

n

×

n

X

i=1 Xi−1Xit−1

!

−1

.

Thus rn

−

Rn

=

1 bn

√

n n

X

i=1

(

XiXit−1

−

θ

Xi−1Xit−1

)

!

(

EX0X0t

)

−1

_×

EX0X0t

−

1 n n

X

i=1 Xi−1Xit−1

!

×

n

×

n

X

i=1 Xi−1Xit−1

!

−1

.

For anyr

>

0,L

>

0 and

δ >

0, using

k

AB

k

_HS

≤ k

A

k

_HS

k

B

k

we have

P

(

k

rn

−

Rn

k

HS

>

r

)

≤

P

1 bn

√

n n

X

i=1

(

XiXit−1

−

θ

Xi−1Xit−1

)(

EX0X0t

)

−1

HS

≥

L

√

δ

r

!

+

_P

EX0X0t

−

1 n n

X

i=1 Xi−1Xit−1

HS

≥

√

δ

r L

!

+

_P





1 n n

X

i=1 Xi−1Xit−1

!

−1

>

1

_δ





.

For

δ

,rsufficiently small but fixed, the first term at the r.h.s. above is negligible by the moderate deviation principle ofRn by lettingL

→ ∞

. The second one is bounded from above by (fornlarge enough)

P 1

√

nbn

n

X

i=1

Xi−1Xit−1

−

EX0X0t

HS

≥

√

n bn

!

which is clearly negligible by the moderate deviation principle of 1

√

nbn n

X

i=1

X_i−1Xit−1

−

EX0X0t

.

For the third term, asE

(

X0X0t

)

is non-degenerate, then there is some

δ

0

>

0 such that for anyd

×

dmatrixAsuch that

k

A

−

_E

(

X0X0t

)

k ≤

δ

0,A−1exists and

k

A−1

k ≤

2

k

(

E

(

X0X0t

))

−1

_k

_{. Thus for}

_{δ >}

_{0 so that 2}

_k

₍

E

(

X0X0t

))

−1

_k

_<

1 δ, P





1 n n

X

i=1 Xi−1Xit−1

!

−1

>

1

_δ





≤

P

1 n n

X

i=1 Xi−1Xit−1

−

EX0X0t

> δ

0

!

where the r.h.s. is negligible by the same argument as for the second term.

In conclusion we have proven(3.19), and sornsatisfies the same moderate deviation principle asRn. It remains to identify the rate function governing the moderate deviation principle ofRn. Noting that

P

n

i=1

(

XiXit−1

−

θ

Xi−1Xit−1

)(

EX0X0t

)

−1_{is a} martingale with stationary differences, by the similar proof ofTheorem 2.1,Rnsatisfies the MDP with the speedb2nand with the rate function

I

(

Γ

)

=

sup Λ∈Md

h

Λ

,

Γ

i −

1 2E

h

Λ

, (

X1X t 0

−

θ

X0X0t

)(

EX0X0t

)

−1

_i

2

.

(3.20)

ButX1

−

θ

X0

=

ξ

1, so the expression above coincides with the claimed rate function. Whend

=

1, then it is easy to see that

E

(

X02

)

=

1 1

−

θ

2E

ξ

2 0 and E

(

X1X0

−

θ

X02

)

2

₌

E

(ξ

1X0

)

2

=

E

(ξ

02

)

2 1 1

−

θ

2

,

then from(3.20), we have

I

(

x

)

=

sup λ∈R

λ

x

−

1 2

λ

2

₍

₁

₋

_θ

2

₎

=

x 2 2

(

1

−

θ

2

)

which is the desired result.

Acknowledgments

Here the authors want to thank their supervisor – Professor Liming Wu of Université Blaise Pascal and Wuhan University – for many suggestions and helpful discussions during the preparation of the paper. Furthermore, the authors are very grateful to the referee for his/her valuable reports which improved the presentation of this work.

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