A Note on the Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Partial Sums of Certain Stationary Gaussian Sequences

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http://dx.doi.org/10.4236/am.2014.510153

How to cite this paper: Wang, Y.F. and Wu, Q.Y. (2014) A Note on the Almost Sure Central Limit Theorem in the Joint Ver-sion for the Maxima and Partial Sums of Certain Stationary Gaussian Sequences. Applied Mathematics, 5, 1598-1608. http://dx.doi.org/10.4236/am.2014.510153

A Note on the Almost Sure Central Limit

Theorem in the Joint Version for the Maxima

and Partial Sums of Certain Stationary

Gaussian Sequences

*

Yuanfang Wang, Qunying Wu

College of Science, Guilin University of Technology, Guilin, China Email: wangyuanfang89@126.com, wqy666@glut.edu.cn

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

Considering a sequence of standardized stationary Gaussian random variables, a universal result in the almost sure central limit theorem for maxima and partial sum is established. Our result ge-neralizes and improves that on the almost sure central limit theory previously obtained by Marcin Dudzinski [1]. Our result reaches the optimal form.

Keywords

Almost Sure Central Limit Theorem, Stationary Gaussian Sequence, Slowly Varying Functions at Infinity

1. Introduction

In this paper, we let

(

X X, _{n n}

)

_∈_ be a standardized stationary Gaussian sequence, also let

1

: max ,

n i

i n

M X

≤ ≤

= _,

1

: max

m n i m i n

M X

+ ≤ ≤

= ,

1

:

n n i

i

S X

=

∑

,

σ

n:= Var

( )

Sn ,

1

n n k

k

D d

=

∑

,

{ }

d_k

be some sequence of weights, I . and

( )

Φ

( )

. denote the indicator function and the standard normal distribution function, respectively.

*_{Supported by the National Natural Science Foundation of China (11361019) and Innovation Project of Guangxi Graduate Education}

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The ASCLT has been first introduced independently by Brosamler [2] and Schatte [3] for partial sum, since then the concept has already started to receive applications in many fields. For example, Fahrner and Sadtmuller [4] and Cheng et al. [5] extended this almost sure central limit theorem for partial sums to the case of maxima of i.i.d. random variables. Under some conditions, they proved as follows:

(

)

(

)

( )

1

1 1

lim ln

n

k k k n

k

a S b x G x

n k

→∞

∑

₌ Ι − ≤ = a.s.,

for all x∈R, where ak >0 and bk∈R satisfy

(

)

d

k k k

a S −b →G, where G(:) is some non-degenerate distribution function. Afterwards, Marcin Dudzinski [5] showed the ASCLT in its two-dimensional version, i.e.

(

)

( )

1

lim , e Φ

n

k k k k k n

k

n k

S

d I a M b x y y

D

τ

σ −

→∞ ₌

 

− ≤ ≤ =

 

 

∑

a.s. for ∀x y, ∈. (1.1)

In this paper, we extend the weight d_k 1 k

= to the weight

(

( )

)

exp ln

k

k d

k β

= , 0 1

2

β

≤ < .

Our purpose is to prove that if

(

X X, _{n n N}

)

_∈ is a standardized stationary Gaussian sequence, the covariance function r t

( )

fulfills

( )

L t

( )

r t tα

= , α>0 and t=1, 2,, (1.2)

where L t

( )

is a positive slowly varying function at infinity. Moreover if the numerical sequence

{ }

u_n satisfies the following relation

( )

(

)

lim 1 _n

n→∞n − Φ u =τ for some 0≤ < ∞τ

, (1.3)

then we have

(

)

( )

1

lim , e

n

k k k k k

n k

S

d I a M b x y y

D

τ σ

−

→∞ ₌

 

− ≤ ≤ = Φ

 

 

∑

a.s. for any x y, ∈R, (1.4)

where

1

n n k

k

D d

=

∑

and

(

( )

)

exp ln

k

k d

k β

= , 0 1

2

β

≤ < .

In the following, denote by a_n ∼b_n if n 1

n a

b → as n→ ∞, by anbn if there exists a constant c>0 such that a_n≤cb_n for sufficiently large n. The c stands for a constant, which may vary from one line to another.

2. Main Results

Theorem 2.1. Assume that

(

X X, _{n n}

)

_∈_ be a standardized stationary Gaussian sequence, the covariance function r t

( )

satisfies (1.2) for some α >0. If the numerical sequence

{ }

u_n satisfies (1.3), then (1.4) holds with

( )

(

)

exp ln

k

k d

k β

= , 0 1

2

β

≤ < .

Let

{ }

_n n

D= D _∈_ be a positive non-decreasing sequence with lim n

n→∞D = ∞. We say that

{ }

xk k∈ is D-

summable to a finite limit x if

1 1 lim

n

k k n

k n

d x x

D

→∞

∑

₌ = , (2.1)

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Remark 2.1. By a classical theorem of Harly (see Chandrasekharan and Minakshisundaram [6]), if two sequences D and D* satisfy D_n∗=O D

( )

_n , then D-summability implies the D*-summability, i.e., if a sequence

{ }

_n

n N

x _∈ is D-summable to x, then it is also *

D -summable to x. These results show that if (2.1)

holds with the weight

{ }

_k k

d _∈_, then for 0 dk dk

∗

≤ ≤ , D_n∗ → ∞, (2.1) also holds with the weight

{ }

_k k d∗

∈. So,

if ASCLT holds with

{ }

_k k

d _∈_, then for 0 dk dk

∗

≤ ≤ , D_n∗→ ∞, ASCLT also holds with

{ }

_k k d∗

∈. So, by this,

if we use larger weights, we should expect to get stronger results.Theorem 2.1 remains valid if we extend the

weights from d_k 1 k

= to

(

( )

)

exp ln

k

k d

k β

= , 0 1

2

β

< < . When

β

=0, the weights d_k 1 k = .

Lemma 2.2. Under the same assumptions as in Theorem 2.1, if the numerical sequence

{ }

u_n fulfills (3), then there exists some

γ

>0 and for any x y, ∈ and m<n,

,

, n , n

n n m n n

n n

S S m

E I M u y I M u y

n γ

σ

 _≤ _≤ ₋  _≤ _≤   

     _{ }

     , (2.2)

,

, m , , n ,

m m m n n

m n

S S m

Cov I M u y I M u y

n γ

σ

  _≤ _≤   _≤ _≤   

      

 _ _{ } _ _{ }

   (2.3)

where m m, n n.

m n

x x

u b u b

a a

= + = +

3. Proofs

3.1. Proof of Theorem 2.1

Under the assumptions of Theorem 1 on X X₁, ₂,, r t

( )

and

{ }

u_n , by Theorem 4.3.3 in Leadbetter et al. [7], we have lim

(

_n _n

)

e

n P M u

τ −

→∞ ≤ = for τ which is defined in (1.3). Let y be a real number, for each n≥1,

n

Y denotes a standard normal variable, which is independent of

(

X₁,,X_n

)

and has the same distribution as n

n S

σ . From the proof of Lemma 2.2, we get that

(

) (

)

1

, n 0

n n n n n

n S

P M u y P M u P Y y

nγ

σ

 

≤ ≤ − ≤ ≤ →

 

   , as

n→ ∞. Thus we have

(

) (

)

( )

lim , n lim e

n n n n n

n n

n

S

P M u y P M u P Y y τ y

σ −

→∞ →∞

 

≤ ≤ = ≤ ≤ = Φ

 

  ,

by Toeplitz Lemma, we obtain

( )

1

lim , e

n

k k k k n

k

n k

S

d P M u y y

D

τ σ

− →∞ ₌

 

≤ ≤ = Φ

 

 

∑

(3.1)

where

1

n n k

k

D d

=

∑

and

(

( )

)

exp ln

k

k d

k β

= , 0 1

2

β

< < .

To prove Theorem 2.1 for 0 1 2

β

< < . We should prove the following:

1 1

, , 0

n

k k

k k k k k

k

n k k

S S

d I M u y P M u y

D =

σ

    

≤ ≤ − ≤ ≤ →

    

    

 

∑

a.s. (3.2)

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[8]) for some ε>0

(

)

(1 )

2

1

Var , ln

n

k

k k k n n

k k

S

d I M u y D D ε

σ

− +

=

  

≤ ≤

  

 _ _



∑

 (3.3)

Set , k , k

k k k k k

k k

S S

I M u y P M u y

ξ

σ σ

   

= _ ≤ ≤ _− _ ≤ ≤ _

   , by Lemma 2.2, we have

(

)

,

, , ,

, , , ,

, , ,

,

k l

k l k k l l

k l

k l l

k k l l k l l

k l l

k l

k k k l l

k l

l

l l

l

S S

E Cov I M u y I M u y

S S S

Cov I M u y I M u y I M u y

S S

S

E I M u y

ξ ξ

σ σ

σ σ σ

σ σ

σ

    

= _  ≤ ≤   ≤ ≤ _

   

 

      

≤ _  ≤ ≤   ≤ ≤ _−  ≤ ≤ 

     

 

    

+ _  ≤ ≤   ≤ ≤ _

   

 



≤ ≤

 

 ,

, ,

, , ,

for 1 .

l k l l

l

k l

k k k l l

k l

S

I M u y

S S

k

k l l

γ

σ

σ σ

  

− ≤ ≤

  

  

    

+ _  ≤ ≤   ≤ ≤ _

   

 

  _{≤ <}

   



We know that

(

)

( )2 ( )2

2

1 1 1

1 2

1 , ln 1 , ln

Var , 2

:

n n

k

k k k k k k l k l

k k k k l n

k l k l n n

k k

k l n D k l n D

l l

S

d I M u y E d d d E

k

d d d d T T

l

γ γ

γ

ξ ξ ξ

σ

− −

= = ≤ ≤ ≤

≤ ≤ ≤ ≤ ≤ ≤ ≤ >

  _≤ _≤ _≤   _≤

    

 _ _ _ _

 

  + = +

   

∑



For T_n₁, we have the following estimate:

( ) 2 ( )2

(

)

(

)

(

)

2 2

1 2 2 1

1 , ln 1 , ln

1

.

ln ln ln

γ γ

γ

ε

− −

+

≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤

 

≤ _{ }

 

∑

n n

n k l k l

k k _n _n _n

k l n D k l n D

l l

D D

k

T d d d d

l  _D  _D  _D

By the elementary calculation, it is easy to see that D_n 1

( )

lnn 1 βexp ln

(

( )

n

)

β β

−

∼ , lnD_n∼

( )

lnn β,

ln lnDn ∼ln lnn.

For 0 1

2

β

< < , we have : 1 2 0 2

β ε

β −

= > and 1 1

2β = +ε. Then

( )

(

)

( )

(

)

(

)

(

) (

)

(

)

(

)

2 2

2

2 1 1

1 1

1 , ln ln

2 2 2

1 1 2 1 1

2 2 2

exp ln _{ln ln}

ln ln

ln

ln ln ln

γ λ

β

β β

β ε

β β β

− −

≤ ≤ ≤ =

≤ ≤ ≤ > ≤ <

− +

= =

∑

n n

n

n n n

n k l k k n

k l n k

k

k l n D k l D _n _n

l

n n n n

n

n n n

n _D _D _D

T d d d d D

l

D D

D D D D

D

D D D

   



.

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3.2. Proof of Lemma 2.2

We first consider that (1.2) holds with some 0< <α 1. Let 1≤ ≤i n, we have

(

)

(

)

( )

1

1 1 1

1

1 1

0 , , 1

2 2

for some 0 1

n i n

n

i i j

j j j i

n n n

n

t

n n

S

Cov X Cov X X r i j r j i

L t tα

σ σ σ

α

σ σ

−

= = = +

−

=

   

 

< _ _= _ _= _ + − + − _

     

+ < <

∑



. (4.1)

By the application of the Karamata’s theorem (see Mielniczuk [9]), we obtain

(

)(

)

( )

1

2 1 ₁

2 2 2

1 2

n L n n

α σ

α α

−

 

∼ _ _

− −

  . (4.2)

From (4.1) and (4.2), for some 0< <α 1, we have

( )

(

)

( )

(

)

1

1 ₂

1

1 ₁ ₁

1

2 2 2 2

2

1 ,

n n

i

t n

L n

L t L n n

S Cov X

t

L n n L n n n

α

α α α α

σ

− −

− ₌ −

 

=

 

 

∑

 .

Since L n

( )

is a slowly varying function at infinity, for any ε>0, L n

( )

nε. So

2 2

1 , n

i n

S n

Cov X

n n

ε

α _{α ε}

σ −

 

=

 

  for any ε>0 and some 0< <α 1. Hence,

1

0 sup , n

i i n _n

S Cov X

nβ σ

≤ ≤

 

<  

 

→

0

, as n→ ∞, for some

1

0 ,

2

β

< < (4.3)

so there exist λ, n₀ such that

1

0 sup , n 1

i i n _n

S

Cov X λ

σ

≤ ≤

 

< _ _< <

  , for any n>n0. (4.4)

By (1.2), lim

( )

lim

( )

0

t t

L t r t

tα

→∞ = →∞ = , thus there exist δ , n1 such that

( )

1

0 sup 1

t n

r t δ >

< = < . (4.5)

Let y be an arbitrary real number and m<n. Suppose that Y_n is a random variable, which has the same

distribution as n n S

σ , but Yn is independent of

(

X1,,Xn

)

, then we have

(

) (

)

(

)

(

)

(

)

(

)

,

, ,

,

1 2 3

, ,

,

:

n n

n n m n n

n n

m n n n n

n n

n

n n n n n

n n

m n n m n n n n

m n n n n

S S

E I M u y I M u y

S S

P M u y P M u y

S

P M u P M u

A A A

σ σ

σ

   

≤ ≤ − ≤ ≤

   

   

   

= _ ≤ ≤ _− _ ≤ ≤ _

   

 

≤ _ ≤ ≤ _− ≤ ≤

 

 

+ _ ≤ ≤ _− ≤ ≤

 

+ ≤ − ≤

= + +

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By Theorem 4.2.1 in Leadbetter et al. (1983) and (4.3), (4.4) and (4.5), we get that

(

)

(

)

2 2

1 2

1

, exp exp

2 1 2 1

n

n n n

i

i n

S u u

A A Cov X n

nβ

σ

λ

=

   

 

+   _− ₊ _ _− ₊ _

     

∑

  . (4.6)

As we know

{ }

u_n fulfills (1.3), which implies that

2 _2π

exp 2

n n

u c u

n

 

− ∼

 

  ,

(

)

1 2 2 ln

n

u ∼ n ,

(

)

( )

(1 )

2 _{2 1}

1 1 ln exp

2 1

n n

u

n λ

λ λ

+

 

−

 

 ₊ 

  ,

combine this and (4.6), we have

( )

2 1(1 )

( )

2 1(1 )

1 2 1 1

1

1 1

ln ln

1 n n

A A n

n

n n

λ λ

β _β

λ λ

+ +

+ −

+ +

+ _ = .

We set 1 1 0

1

β λ + − >

+ , thus

1 2

1 m

A A

n n

γ

 

+ _{ }

 

  for some 0 1 1

1

γ β λ < < + −

+ . (4.7) Next, we estimate A₃.

(

)

(

)

(

)

( )

(

)

( )

3 ,

,

1 1

1 2 3

:

m n n n n

n n

m n n n n n n

i m i

n n

i m i

A P M u P M u

P M u u P M u u

u u

B B B

= + =

= ≤ − ≤

≤ ≤ − Φ + ≤ − Φ

+ Φ − Φ

= + +

∏

. (4.8)

Let δ is defined as in (4.5), set

2

α δ

α <

− such that

2

2 0

1+δ + − >α . By Theorem 4.2.1 in Leadbetter et

al. and (1.3), we have

( ) ( )

( )

1 1

2 ₁ ₁

1 2 2 2

1 2

1 ₁ 1 ₁

ln ln

exp 1

n n

n

t t

n L t n L n

u

B B n r t

t

n n

δ δ

α _α

δ δ

δ

+ +

− + −

= ₊ = ₊

 

+ _− _

+

 

∑

   ,

since L n

( )

and lnn are slowly varying functions at infinity, we have

( )

1 1

lnn +δ L n nε for any ε>0, then

1 2

1

for some 0 m

B B

n n

γ

γ γ

 

+ _{ } >

 

  . (4.9)

Meanwhile, following from the elementary inequality xn m xn m n − ₋ _≤

, 0≤ ≤x 1. We get

3 for

m m

B m n

n n

γ  

≤ _{ } <

 

 . (4.10)

By (4.8), (4.9) and (4.10), if (1.2) holds with 0< <α 1, then (2.2) holds.

Provided (1.2) also holds with some α≥1, since r

( )

. is positive, we get

( )

1 2 Var

n Sn n

(7)

( )

1 1

1 2

2

0 ,

n n

i

t n

L t S

Cov X

t n

α σ

−

=

 

< _ _

 

∑

. As

( )

1

n

t L t

tα =

∑

is a slowly varying function infinity, we get

( )

1

n

t L t

n t

ε α =

∑

 for

any ε >0. We obtain that (4.3) and (4.4). By using Theorem 4.2.1 in Leadbetter et al. and by (4.6)-(4.10), let

1

A - A₃ be replaced by C₁ - C₃ in (4.3), we also get C₁ C₂ m n

γ  

+ _{ }

 

 , let D₁ - D₃ be replaced by B₁-

3

B in (4.8), we also obtain that

( ) ( )

1

2 ₁

1 2 2

1

1 ₁

ln exp

1

n n

t

n

u m

D D n r t n

n n

γ δ _ε

δ δ

+

−

= ₊

  _{ }

+ _− _ _{ }

+  

 

∑

   for some

γ

>0,

like (4.11), D₃ m m

n n

γ  

≤ _{ }

 

 , we get C₃ m

n γ      

 , at last , we obtain when (1.2) holds with α≥1, then (2.2)

also holds.

Next, we prove (2.3). First we consider the situation of some 0< <α 1. Let i≥ +m 1. Since L t

( )

is a positive slowly varying function, then we have

(

( )

)

1 2

L m mε for any ε >0, L t

( )

tα is monotonic decreasing,

so

( )

1

i m

t i m t

L t L t

tα tα

−

= −

∑



∑

=

, by (1.2) and (4.2), we obtain

( )

1 1

1 ₁ 2 2

1 1

2

1 ₁ 1 ₁

1

2 2 2

2 2

2

1 1

0 ,

1

1 1

for some 0 2

i i

m i

t i m t i m m m

m t

L t L t

S Cov X

t t

L m m

L t L m m L m

t

L m m L m m m

m

m m

α α α

α

α α α α

ε

α η

σ σ

η

− −

−

= − = −

−

− ₌ −

 

< _ _

 

=

< <

∑

 

,

then there exist numbers µ,m₀, such that

1

0 sup , m 1

i i m _m

S

Cov X µ

σ

≥ +

 

<  < <

  for all m>m0. By the Normal Comparison Lemma, we obtain

(

)

₍

₎

,

, ,

2 2 2

1 1 1

, , ,

, , , , ,

exp , exp

2 1

m n

m m m n n

m n

m n m n

m m m n n m m m n m

m n m n

m n m

m n n m

i

i j m i n

S S

S S S S

P M u y M u y P M u y P M u y

u u S u

r j i Cov X

r j i

σ σ

σ σ σ σ

σ

= = + =

    

≤ ≤ ≤ ≤

    

 _ _{ } _

 

     

= _ ≤ ≤ ≤ ≤ _− _ ≤ ≤ _{ } ≤ ≤ _

     

 ₊  _ _

− _− _+   −

+ − _ _

 

∑ ∑

∑



2

1

1 2 3 4

2 1 ,

, exp ,

2 1 ,

:

n i

n n

m n m n

i

i m m m m n

i m

S Cov X

S u S S

Cov X Cov

S Cov X

E E E E

σ

σ σ σ

σ

= +

 

 

 _ _ __

 _ ₊ _ __

 

 _ _ __

 

 

 

     

+ _ _ _− _+ _ _

  

  _ ₊ _  

  

 

 _ _ __

 

= + + +

∑

(8)

By (4.5), set 1, 2 α δ α < <

− then

2

2 0

1+δ + − >α , we have

(

)

₍

₎

₍

₎

( )

(

)

( )

(

)

( )

2 2 2 2

1

1 1 1 1

2 2 2 2

1 1 1 1 1 1 1 2 2 1 exp exp 2 1 2 1 exp exp

2 1 2 1

1

m n m n i

m n m n

i j m i t m i

n i n

m n m n

t m i t

u u u u

E r j i r t

r j i

L t L t

u u u u

m m t t m m L n n n n α α δ α δ δ δ δ − = = + = = + − − = + − = − − + − + +  +   +  = − _− _ _− _ + + − _ _    +   +  − −      ₊   ₊                 

∑ ∑

∑

  

  1 m for some 0 1

n γ δ γ +   < <      .

By (4.4), set 1

1 β λ

β < <

− , then

1

1 0

1

β

λ − + − <

+ , we have

(

)

(

)

( )

(

)

( ) 2 2 1 1 1

2 1 1

2 1

1

1 1

1

1 _{2 1}

, exp

2 1 ,

exp , ln

2 1

1

ln for 0

2

m

n m

i

i n _n

i n m m n i i n S u

E Cov X

S Cov X

u S m

Cov X m

n

m m m

m m n n n λ λ β β γ β _β λ _λ β σ σ λ σ γ β = − + + = − + − + ₊         = _ _ _− _      _ ₊ _       _ _ __       −      ₊  _ _  

    _{< < <}

       

∑

     . and

(

)

(

)

( )

1 1 1 1 1

1 1

,

1 1

n n m m n

m i

i m m mi m j m i j m

m n i m n n

i k m i i k k

m m m

S

Cov X r i j r j i

L k m

r k r k

kα

σ σ σ

= + = + = = = + − = = + − = = =   = − = −     =

∑

∑ ∑



∑∑



∑

(

)

(

)

( )

( ) ( )

(

)

( )

( ) 2 3 1 2 1 1 1 2

2 2 1 1 1 1 2 2 exp , 2 1 exp 2 1 1 ln for 0 2 n n m i i m m

n n

k m

u S

E Cov X

L k

u m

k

m

L n L m n

n n m m n n α α µ α µ α _γ µ σ µ σ α γ = + = + + − +     −      ₊  _ _     −    ₊         

    _{< <}

       

∑

     .

2 2 2 2

4

2 2

1 2 3

,

:

m n m m m n n m m

m n m n n

m n n m m

m n n

S S S S S S S

E Cov E E

S S S S

E E

F F F

σ σ σ σ σ

σ σ σ

+ + +  −     −  + −                −  − _ _{ } − _     = + + 

(9)

SinceL t

( )

tα is monotonic decreasing, so for 1≤ ≤i m, r j

(

− ≤i

) (

r j−m

)

, by (4.2), Karamata’s theorem and the property of the slowly varying function, we obtain

(

)

(

)

( )

( ) ( )

( )

2 2

1

1 2 1 1 2 1

2

1 2 1 1

1 2 2

1

1 1 ₁ ₁

2 2

2 4

1 1

,

1

( ) ( )

1 for 0

4 4

m m n m m n

i j

i j m i j m

m n m n

m m n n

i j m k

m n m n

F Cov X X r j i

m

r j m r k

m m L n

L n n

n L m

L m L n m n

m n m

n m n

α α

α α _γ

σ σ σ σ

α γ

+ +

== = + = = +

+

= = + =

− − −

 

= −

 

 

≤ −

 

 

   

      _{< <} _<

     

     

∑

∑ ∑

∑



 

.

By

σ

n = Var

( )

Sn , then we get

2

1

m

m S E

σ

 

=

 

  , by the stationary of the sequence

(

X X, n n

)

∈, we get

2 2

d n m n m

S₊ −S =S , thus E S

(

_n₊₂_m−S_n

)

2 =E S

( )

₂_m 2 and by the fact that EX X_i _j>0 for all i j, and (4.2), we deduce that

2 2

2 2 2 2

2

1 2

1 1

2 2

2 ( )

for some 1

( )

m n m n m m n m m

m n n m n

m

n

S S S S S S S

F E E E E

m L m m m

n L n n n

α α _γ

σ σ σ σ σ

σ _γ

σ

+ +

− −

   

− −

≤ − ≤ _ _ _ _

   

 

      _<

 

     

       

   

,

where the second inequality follows from Jensen inequality. For F₃, we have

2 2

3 0

m n n m m

m n n

S S S S

F E E

σ σ + σ

   − 

= − _ _{ } − _=

    , so we prove (2.3) for some 0< <α 1. Next, we prove (2.3) for some α>1.

, 1 2 3 4

, m , , n :

m m m n n

m n

S S

Cov I M u y I M u y G G G G

σ

    

≤ ≤ ≤ ≤ = + + +

    

 _ _{ } _

  ,

where G₁ - G₄ are defined as E₁ - E₄ in (4.11), but for (1.2) holds for α>1. Similarly as the proof of

1

E - E₄, it is easy to check that G₁ G₂ G₃ m n

γ  

+ + _{ }

 

 for some

γ

>0.

Define G₄=:H₁+H₂+H₃ as in (4.12), by Karamata’s theorem, we get

(

)

( )

2 2 1

1 1 1 1 1

1 2 1 1 1

2 2 2 2

1

2 ₂

1 1 1

1

2 2

1 1

for 0 2

m m n m m n

i j m i t m

m n

t m

H r j i r t

m n m n

m m m

r t L m

n m n

m n

γ

α γ

+ + −

= = + = = +

+

− = +

≤ −

    _{< <}

   

   

∑ ∑

∑



  

.

By the definition of σ_n and Karamata’s theorem, it is easy to obtain that

(

) ( )

( )

1 1 1 1

2

1 1 1 1 1

2 2 _α 2 2 _α 2 _α

σ − − − − ∞

= = = = =

≤ _n = +

∑

n − = +

∑

n −

∑

n × +

∑

n +

∑

+

t t t t t

L t L t L t

n n n t r t n n t r t n n n n n cn

(10)

then we have

( )

1 2

n O n

σ = . Similarly as the proof of F₂,

1 2 2

2

m

n

m m

H

n n

γ σ

σ

   

   

   

   for 0 1

2

γ

< < ,

by the stationary of the sequence

(

X X, _{n n}

)

_∈_, it is easy to see that 2 2

3 0

m n n m m

m n n

S S S S

H E E

σ σ + σ

   − 

= − _ _{ } − _=

    .

So we have proved (2.3) for the case of α>1. Finally we should prove (2.3) for α=1.

, 1 2 3 4

, m , , n :

m m m n n

m n

S S

Cov I M u y I M u y I I I I

σ

    

≤ ≤ ≤ ≤ = + + +

    

 _ _{ } _

  ,

where we replaced E₁ - E₄ by I₁ - I₄ as in (4.11), but for (1.2) holds for some α =1.

Similarly, it is easy to check that I₁ I₂ I₃ m n

γ  

+ + _{ }

 

 for some

γ

>0. Define I₄=:J₁+J₂+J₃. Since

( )

r t is monotonic decreasing, define

( )

1

ˆ _{1 2}n

t

L n r t

−

=

= +

∑

, by the application of the proposition (Potter’s TH)

of the slowly varying function, for any 0 ₁ 1 2

δ

< < , we have

( )

(

)

( )

(

)

(

( )

)

(

)

(

)

( )

(

)

(

)

(

)

( )

(

)

1 2

1 1

2 1 1

1 1 1 1 1

1 1 ₂ ₂

2 2

1 1

2 2

1 1

2 2

1 1

2 2

ˆ

ˆ ˆ

2 1

for some 0

m n m n

t m t

m n m n

mL n m

m m

J r t r t

L m L n m n

L n m L n m

m m n m n m

n n m n

L m L n

m n m m n m

n m n m n

δ δ

δ δ γ

σ σ σ σ

γ

+ + + +

= + =

+

+ + ₊ ₊

       

=  _{ }   _{  }  _{ } _

+

          _{< <}

         

         

∑

  



  

2 .

( )

1

1 1

2

2 2

2

2 1

2 ˆ ˆ

ˆ _ˆ

m

n

L m

m m m m

J

n n n n

L n

δ γ

σ σ

−

       

       

       

    for some 0 1

2

γ

< < .

Similarly as F₃ =0 when 0< <α 1, we have J₃ =0, So we have I₄ m n γ      

 . Finally, we get that

1 2 3 4

m

I I I I

n γ  

+ + + _{ }

 

 for some

γ

>0, when α=1.

References

[1] Dudzinski, M. (2008) The Almost Sure Central Limit Theorems in The Joint Version for The Maxima and Sums of Certain Stationary Gaussian Sequence. Statistics & Probability Letters, 78, 347-357.

http://dx.doi.org/10.1016/j.spl.2007.07.007

[2] Brosamler, G.A. (1988) An Almost Everywhere Central Limit Theorem. Mathematical Proceedings of the Cambridge Philosophical Society, 104, 561-574. http://dx.doi.org/10.1017/S0305004100065750

[3] Schatte, P. (1988) On Strong Versions of the Central Limit Theorem. Mathematische Nachrichten, 137, 249-256.

http://dx.doi.org/10.1002/mana.19881370117

[4] Fahrner, I. and Stadtmuller, U. (1998) On Almost Sure Central Max-Limit Theorems. Statistics & Probability Letters,

(11)

[5] Cheng, H., Peng, L. and Qi, Y.C. (1998) Almost Sure Convergence in Extreme Value Theory. Mathematische Nach-richten, 190, 43-50. http://dx.doi.org/10.1002/mana.19981900104

[6] Chandrasekharan, K. and Minakshisundaram, S. (1952) Typical Means. Oxford University Press, Oxford.

[7] Leadbetter, M.R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer, New York. http://dx.doi.org/10.1007/978-1-4612-5449-2

[8] Csaki, E. and Gonchigdanzan, K. (2002) Almost Sure Limit Theorems for The Maximum of Stationary Gaussian Se-quences. Statistics & Probability Letters, 58, 195-203. http://dx.doi.org/10.1016/S0167-7152(02)00128-1