Almost sure limit theorems for the maximum of stationary Gaussian ...

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Almost sure limit theorems for the maximum
of stationary Gaussian sequences
Endre Cs´aki
a,1
,Khurelbaatar Gonchigdanzan
b,2
a
A.R´enyi Institute of Mathematics,Hungarian Academy of Sciences,
P.O.Box 127,H-1364,Budapest,Hungary
b
Department of Mathematical Sciences,University of Cincinnati,
Cincinnati,OH 45221-0025,USA
Abstract.We prove an almost sure limit theorem for the maxima of stationary Gaussian
sequences with covariance r
n
under the condition r
n
log n(log log n)
1+ε
= O(1).
Key words:almost sure central limit theorem,logarithmic average,stationary Gaussian se-
quence.
Introduction.The early results on the almost sure central limit theorem (ASCLT) dealt
mostly with partial sums of random variables.A general pattern of these investigations
is that if X
1
,X
2
,...is a sequence of random variables with partial sums S
n
=
￿
n
k=1
X
k
satisfying a
n
(S
n
− b
n
)
D
−→ G for some numerical sequences (a
n
),(b
n
) and distribution
function G,then under some additional mild conditions we have
lim
n→∞
1
log n
n
￿
k=1
1
k
I (a
k
(S
k
−b
k
) < x) = G(x) a.s.
for any continuity point x of G,where I is indicator function.
For more discussions about ASCLT we refer to the survey papers by Berkes (1998),
and Atlagh and Weber (2000).Recently Fahrner and Stadtm¨uller (1998) and Cheng et
al.(1998) have extended this principle by proving ASCLT for the maxima of independent
random variables.
1
Supported by the Hungarian National Foundation for Scientific Research Grant No.T 029621
2
Supported by a TAFT Fellowship at the University of Cincinnati
Typeset by A
M
S-T
E
X
1
THEOREM A.Let X
1
,X
2
,...be i.i.d.random variables and M
k
= max
i≤k
X
i
.If
a
k
(M
k
−b
k
)
D
−→G for a nondegenerate distribution G and some numerical sequences (a
k
)
and (b
k
),then we have
lim
n→∞
1
log n
n
￿
k=1
1
k
I (a
k
(M
k
−b
k
) < x) = G(x) a.s.
for any continuity point x of G.
Berkes and Cs´aki (2001) extended the ASCLT for general nonlinear functionals of in-
dependent random variables.For strong invariance principles improving Theorem A see
Berkes and Horv´ath (2001) and Fahrner (2001).
Throughout this paper Z
1
,Z
2
,...is a stationary Gaussian sequence and we denote
its covariance function by r
n
= Cov(Z
1
,Z
n+1
),and M
n
= max
1≤i≤n
Z
i
and M
k,n
=
max
k+1≤i≤n
Z
i
.Here a b and a ∼ b stand for a = O(b) and a/b →1 respectively.Φ(x)
is the standard normal distribution function and φ(x) is its density function.
For notational convenience let R(n) = r
n
log n(loglog n)
1+ε
.
1.Main Result.The main result is an almost sure central limit theoremfor the maximum
of stationary Gaussian sequences.
THEOREM 1.1.Let Z
1
,Z
2
,...be a standardized stationary Gaussian sequence with
R(n) = O(1) as n →∞.Then
(i) If n(1 −Φ(u
n
)) →τ for 0 ≤ τ < ∞,then
lim
n→∞
1
log n
n
￿
k=1
1
k
I(M
k
≤ u
k
) = e
−τ
a.s.,
(ii) If a
n
= (2 log n)
1/2
and b
n
= (2 log n)
1/2

1
2
(2 logn)
−1/2
(log log n +log 4π),then
lim
n→∞
1
log n
n
￿
k=1
1
k
I(a
k
(M
k
−b
k
) ≤ x) = exp(−e
−x
) a.s..
2.Auxiliary Results.The main weak convergence result for the maximumof stationary
Gaussian sequence is summarized in the following theorem.
2
THEOREM 2.1.(Theorem 4.3.3 in Leadbetter et al.(1983)).Let Z
1
,Z
2
,...be a
standardized stationary Gaussian sequence with r
n
log n →0.Then
(i) For 0 ≤ τ < ∞,P(M
n
≤ u
n
) →e
−τ
if and only if n(1 −Φ(u
n
)) →τ
(ii) P(a
n
(M
n
−b
n
) ≤ x) →exp(−e
−x
),
where a
n
= (2 logn)
1/2
and b
n
= (2 logn)
1/2

1
2
(2 log n)
−1/2
(log log n +log 4π).
We need the following lemmas for the proof of our main result.
LEMMA 2.1.Let Z
1
,Z
2
,...be a standardized stationary Gaussian sequence.Assume
that R(n) = O(1) and n(1 −Φ(u
n
)) is bounded.Then
sup
1≤k≤n
k
n
￿
j=1
|r
j
| exp
￿

u
2
k
+u
2
n
2(1 +|r
j
|)
￿
(log log n)
−(1+ε)
.
PROOF OF LEMMA 2.1:Under the condition r
n
→0 we have sup
n≥1
|r
n
| = σ < 1 (cf.,
Leadbetter et al.,1983).By assumption,n(1 − Φ(u
n
)) ≤ K.Let the sequence (v
n
) be
defined by v
n
= u
n
if n ≤ K and n(1 −Φ(v
n
)) = K,if n > K.Then clearly u
n
≥ v
n
and
hence
k
n
￿
j=1
|r
j
| exp
￿

u
2
k
+u
2
n
2(1 +|r
j
|)
￿
≤ k
n
￿
j=1
|r
j
| exp
￿

v
2
k
+v
2
n
2(1 +|r
j
|)
￿
.
Thus it would be enough to prove the lemma for the sequence (v
n
).By the well known
fact
1 −Φ(x) ∼
φ(x)
x
,x →∞
we can see that
(2.1) exp
￿

v
2
n
2
￿

K

2πv
n
n
,v
n
∼ (2 log n)
1/2
.
Define α to be 0 < α < (1 −σ)/(1 +σ).Note that
k
n
￿
j=1
|r
j
| exp
￿

v
2
k
+v
2
n
2(1 +|r
j
|)
￿
=
= k
￿
1≤j≤n
α
|r
j
| exp
￿

v
2
k
+v
2
n
2(1 +|r
j
|)
￿
+k
￿
n
α
<j≤n
|r
j
| exp
￿

v
2
k
+v
2
n
2(1 +|r
j
|)
￿
=
=:T
1
+T
2
.
3
Using (2.1)
T
1
≤ kn
α
exp
￿

v
2
k
+v
2
n
2(1 +σ)
￿
= kn
α
￿
exp
￿

v
2
k
+v
2
n
2
￿￿
1/(1+σ)

kn
α
￿
v
k
v
n
kn
￿
1/(1+σ)
k
1−1/(1+σ)
n
α−1/(1+σ)
(log k log n)
1/2(1+σ)

≤ n
1+α−2/(1+σ)
(log n)
1/(1+σ)
.
Since 1+α−2/(1+σ) < 0,we get T
1
≤ n
−δ
for some δ > 0,uniformly for 1 ≤ k ≤ n.Now
we estimate the second term T
2
.Setting σ
n
= sup
j≥n
|r
j
| and counting on R(n) = O(1)
as n →∞
(2.2) σ
n
log n(log log n)
1+ε
≤ sup
j≥n
|r
j
| log j(log log j)
1+ε
= O(1),n →∞.
Set p = [n
α
].By (2.1) and (2.2) we have
σ
p
v
k
v
n

[n
α
]
(log k log n)
1/2

[n
α
]
log n
α

(log log n
α
)
−(1+ε)
∼ (log log n)
−(1+ε)
(2.3)
and similarly,for 1 ≤ k ≤ n
(2.4) σ
p
v
2
k
(log log n)
−(1+ε)
.
Hence using (2.1),(2.3) and (2.4)
T
2
≤ kσ
p
exp
￿

v
2
k
+v
2
n
2
￿
￿
p≤j≤n
exp
￿
(v
2
k
+v
2
n
)|r
j
|
2(1 +|r
j
|)
￿

≤ knσ
p
exp
￿

v
2
k
+v
2
n
2
￿
exp
￿
(v
2
k
+v
2
n

p
2
￿
(log log n)
−(1+ε)
.
The proof is completed.
LEMMA 2.2.Let Z
1
,Z
2
,...be a standard stationary Gaussian sequence.Suppose that
sup
n≥1
|r
n
| < 1.Then for k < n
|P(M
k
≤ u
k
,M
k,n
≤ u
n
) −P(M
k
≤ u
k
)P(M
k,n
≤ u
n
)| 
k
n
￿
j=1
|r
j
| exp
￿

u
2
k
+u
2
n
2(1 +|r
j
|)
￿
.
PROOF OF LEMMA 2.2.We use the following
4
THEOREM 2.2.(Theorem 4.2.1,Normal Comparison Lemma in Leadbetter et al.
(1983)).Suppose ξ
1
,...,ξ
n
are standard normal variables with covariance matrix Λ
1
=

1
ij
),and η
1
,...,η
n
with covariance matrix Λ
0
= (Λ
0
ij
),and let ρ
ij
= max(|Λ
1
ij
|,|Λ
0
ij
|).
Further,let u
1
,...,u
n
be real numbers.Then
|P(ξ
j
≤ u
j
,j = 1,...,n) −P(η
j
≤ u
j
,j = 1,...,n)| ≤
≤ K
￿
1≤i<j≤n

1
ij
−Λ
0
ij
| exp
￿

u
2
i
+u
2
j
2(1 +ρ
ij
)
￿
.
Apply this Theorem with (ξ
i
= Z
i
,i = 1,...,n),(η
j
= Z
j
,j = 1,...,k;η
j
=
˜
Z
j
,j =
k +1,...,n),where (
˜
Z
k+1
,...,
˜
Z
n
) has the same distribution as (Z
k+1
,...,Z
n
),but it is
independent of (Z
1
,...,Z
k
).Further,u
i
= u
k
,i = 1,...,k and u
i
= u
n
,i = k +1,...,n.
Then Λ
1
ij
= Λ
0
ij
= r
j−i
if either 1 ≤ i < j ≤ k,or k + 1 ≤ i < j ≤ n.Otherwise
Λ
1
ij
= r
j−i

0
ij
= 0.Hence we have
|P(M
k
≤ u
k
,M
k,n
≤ u
n
) −P(M
k
≤ u
k
)P(M
k,n
≤ u
n
)| 

k
￿
i=1
n
￿
j=k+1
|r
j−i
| exp
￿

u
2
k
+u
2
n
2(1 +|r
j−i
|)
￿
≤ k
n
￿
m=1
|r
m
| exp
￿

u
2
k
+u
2
n
2(1 +|r
m
|)
￿
.
This completes the proof of LEMMA 2.2.
LEMMA 2.3.Let Z
1
,Z
2
,...be a standardized stationary Gaussian sequence.Assume
that R(n) = O(1) and n(1 −Φ(u
n
)) is bounded.Then for 1 ≤ k < n
Cov(I(M
k
≤ u
k
),I(M
k,n
≤ u
n
)) (log log n)
−(1+ε)
.
PROOF OF LEMMA 2.3:It follows simply from LEMMA 2.1 and LEMMA 2.2.
LEMMA 2.4.Let Z
1
,Z
2
,...be a standardized stationary Gaussian sequence.Assume
that R(n) = O(1) and n(1 −Φ(u
n
)) is bounded,then
E|I(M
n
≤ u
n
) −I(M
k,n
≤ u
n
)| 
k
n
+(log log n)
−(1+ε)
.
5
PROOF OF LEMMA 2.4:Note that
E|I(M
n
≤ u
n
) −I(M
k,n
≤ u
n
)| = P(M
k,n
≤ u
n
) −P(M
n
≤ u
n
) ≤
≤ |P(M
k,n
≤ u
n
) −Φ
n−k
(u
n
)| +|P(M
n
≤ u
n
) −Φ
n
(u
n
)|+
+|Φ
n−k
(u
n
) −Φ
n
(u
n
)| =:D
1
+D
2
+D
3
.
From the elementary fact that
x
n−k
−x
n

k
n
,0 ≤ x ≤ 1
we have D
3
≤ (k/n).By Corollary 4.2.4 in Leadbetter et al.(1983),p.84
D
i
n
n
￿
j=1
|r
j
| exp
￿

u
2
n
1 +|r
j
|
￿
i = 1,2.
Thus by LEMMA 2.1 we have D
i
(log log n)
−(1+ε)
,i = 1,2.
3.Proof of Main Result.We now give the proof of THEOREM 1.1.We need the
following lemma for the proof.
LEMMA 3.1.Let η
1

2
,...be a sequence of bounded random variables.If
Var
￿
n
￿
k=1
1
k
η
k
￿
log
2
n(log log n)
−(1+ε)
for some ε > 0,
then
lim
n→∞
1
log n
n
￿
k=1
1
k

k
−Eη
k
) = 0 a.s..
PROOF OF LEMMA 3.1:Setting
µ
n
=
1
log n
n
￿
k=1
1
k

k
−Eη
k
)
and n
k
= exp(exp(k
ν
)) for some
1
1+
< ν < 1,we have

￿
k=3

2
n
k


￿
k=3
(log log n
k
)
−(1+)


￿
k=3
k
−ν(1+)
< ∞
6
implying
￿

k=3
µ
2
n
k
< ∞a.s.Thus
µ
n
k
→0 a.s..
Since
(k +1)
ν
−k
ν
→0 as k →∞ if ν < 1,
we have
log n
k+1
log n
k
= e
(k+1)
ν
−k
ν
→1 as k →∞.
Obviously for any given n there is an integer k such that n
k
< n ≤ n
k+1
.Therefore

n
| ≤
1
log n
￿
￿
￿
￿
￿
￿
n
￿
j=1
1
j

j
−Eη
j
)
￿
￿
￿
￿
￿
￿


1
log n
k
￿
￿
￿
￿
￿
￿
n
k
￿
j=1
1
j

j
−Eη
j
)
￿
￿
￿
￿
￿
￿
+
1
log n
k
n
k+1
￿
j=n
k
+1
1
j

j
−Eη
j
| 
|µ
n
k
| +
1
log n
k
(log n
k+1
−log n
k
) |µ
n
k
| +
￿
logn
k+1
logn
k
−1
￿
and thus
lim
n→∞
µ
n
= 0 a.s..
PROOF OF THEOREM1.1:First,we claimthat under the assumptions that R(n) = O(1)
and n(1 −Φ(u
n
)) is bounded,we have
(3.1) lim
n→∞
1
log n
n
￿
k=1
1
k
(I(M
k
≤ u
k
) −P(M
k
≤ u
k
)) = 0 a.s..
In order to show this,by LEMMA 3.1 it is sufficient to show
(3.2) Var
￿
n
￿
k=1
1
k
I(M
k
≤ u
k
)
￿
(log log n)
−(1+ε)
log
2
n for some ε > 0.
Let η
k
= I(M
k
≤ u
k
) −P(M
k
≤ u
k
).Then
Var
￿
n
￿
k=1
1
k
I(M
k
≤ u
k
)
￿
= E
￿
n
￿
k=1
1
k
η
k
￿
2
=
=
n
￿
k=1
1
k
2
E|η
k
|
2
+2
￿
1≤k<l≤n
|E(η
k
η
l
)|
kl
=:L
1
+L
2
.(3.3)
7
Since |η
k
| ≤ 2,it follows that
(3.4) L
1


￿
k=1
1
k
2
< ∞.
To estimate L
2
,note that for l > k
￿
￿
E(η
k
η
l
)
￿
￿
=
￿
￿
Cov
￿
I(M
k
≤ u
k
),I(M
l
≤ u
l
)
￿
￿
￿

￿
￿
Cov
￿
I(M
k
≤ u
k
),I(M
l
≤ u
l
)−
−I(M
k,l
≤ u
l
)
￿
￿
￿
+
￿
￿
Cov
￿
I(M
k
≤ u
k
),I(M
k,l
≤ u
l
)
￿
￿
￿

E|I(M
l
≤ u
l
) −I(M
k,l
≤ u
l
)| +
￿
￿
Cov
￿
I(M
k
≤ u
k
),I(M
k,l
≤ u
l
)
￿
￿
￿
.(3.5)
By LEMMA 2.3 and LEMMA 2.4 we get
￿
￿
Cov
￿
I(M
k
≤ u
k
),I(M
k,l
≤ u
l
)
￿
￿
￿
(log log l)
−(1+ε)
and
E|I(M
l
≤ u
l
) −I(M
k,l
≤ u
l
)| 
k
l
+(log log l)
−(1+ε)
.
Hence for l > k
(3.6) |E(η
k
η
l
)| 
k
l
+(log log l)
−(1+ε)
and consequently
L
2

￿
1≤k<l≤n
1
kl
￿
k
l
￿
+
￿
1≤k<l≤n
1
kl(log log l)
1+ε
=
=:L
21
+L
22
.(3.7)
For L
21
and L
22
we have the following estimates:
L
22

n
￿
l=3
1
l(log log l)
1+ε
l−1
￿
k=1
1
k

n
￿
l=3
log l
l(log log l)
1+ε

log n
n
￿
l=3
1
l(log log l)
1+ε
log
2
n(loglog n)
−(1+ε)
(3.8)
8
and
(3.9) L
21

￿
1≤k<l≤n
1
kl
￿
k
l
￿
log n.
Thus (3.3)–(3.9) together establish (3.1).
PROOF OF (i):Note that R(n) = O(1) implies r
n
log n →0.By THEOREM 4.3.3(i) in
Leadbetter et al.(1983),we have P(M
n
≤ u
n
) →e
−τ
.Clearly this implies
lim
n→∞
1
log n
n
￿
k=1
1
k
P(M
k
≤ u
k
) = e
−τ
which is,by (3.1),equivalent to
lim
n→∞
1
log n
n
￿
k=1
1
k
I(M
k
≤ u
k
) = e
−τ
a.s..
PROOF OF (ii):By THEOREM 2.1 we have n(1 −Φ(u
n
)) → e
−x
for u
n
= x/a
n
+b
n
.
Thus the statement of (ii) is a special case of (i).
Acknowledgement
We thank a referee for some useful comments.
REFERENCES
1.Atlagh,M.and Weber,M.(2000),Le th´eor`eme central limite presque sˆur.Expositiones
Mathematicae 18,097–126.
2.Berkes,I.(1998),Results and problems related to the pointwise central limit theorem.
Asymptotic results in Probability and Statistics,(Avolume in honour of Mikl´os Cs¨org˝o),
59–60,Elsevier,Amsterdam.
3.Berkes,I.and Cs´aki,E.(2001),A universal result in almost sure central limit theory.
Stoch.Process.Appl.94,105–134.
9
4.Berkes,I.and Horv´ath,L.(2001),The logarithmic average of sample extremes is asymp-
totically normal.Stoch.Process.Appl.91,77–98.
5.Cheng,S.,Peng,L.and Qi,Y.(1998),Almost sure convergence in extreme value theory.
Math.Nachr.190,43–50.
6.Fahrner,I.and Stadtm¨uller,U.(1998),On almost sure max-limit theorems.Stat.Prob.
Letters 37,229–236.
7.Fahrner,I.(2001),A strong invariance principle for the logarithmic average of sample
maxima.Stoch.Process.Appl.93,317–337.
8.Hurelbaatar,G.(1997),Almost sure limit theorems for dependent random variables.
Studia Sci.Math.Hung.33,167–175.
9.Leadbetter,M.R.,Lindgren,G.,and Rootz´en,H.(1983),Extremes and Related Prop-
erties of Random Sequences and Processes.Springer–Verlag,New York.
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