A note for extension of almost sure central limit theory

0.14 MB·English

by  Yu Miao

A NOTE FOR EXTENSION OF ALMOST SURE CENTRAL LIMIT
THEORY
7
0
YU MIAO AND GUANGYU YANG
0
2

n Abstract. Ho¨rmann(2006)gaveanextensionofalmostsurecentrallimittheorem
a for bounded Lipschitz 1 function. In this paper, we show that his result of almost
J
sure central limit theorem is also hold for any Lipschitz function under stronger
6
conditions.

]
R
P
.
h
t 1. Introduction
a
m
The classical results on the almost sure central limit theorem (ASCLT) dealt with
[
partial sums of random variables. A general pattern is that, if X1,X2,... be a
1 sequence of independent random variables with partial sums S = X + + X
n 1 n
v ···
3 satisfying (S b )/a L H for some sequences a > 0, b R and some distribution
n n n n n
8 function H, th−en under−→some mild conditions we have ∈
1
1
n
0 1 1
lim I (S b )/a x = H(x) a.s.
7 k k k
0 n→∞ logn X k n − ≤ o
k=1
/
h
for any continuity point x of H.
t
a
Several papers have dealt with logarithmic limit theorems of this kind and the
m
above relation has been extended in various directions. Fahrner and Stadtmu¨ller
:
v [5] gave an almost sure version of a maximum limit theorem. Berkes and Horv´ath
i
X [2] obtained a strong approximation for the logarithmic average of sample extremes.
r Berkes and Cs´aki [1] showed that not only the central limit theorem, but every weak
a
limittheoremforindependent randomvariableshasananalogousalmost sureversion.
For stationary Gaussian sequences with covariance r , Cs´aki and Gonchigdanzan [3]
n
proved an almost sure limit theorem for the maxima of the sequences under the con-
dition r logn(loglogn)1+ε = O(1). For some dependent random variables, Peligrad
n
and Shao [7] and Dudzin´ski [4] obtained corresponding results about the almost sure
central limit theorem.
Date: May 26, 2006.
2000 Mathematics Subject Classification. 60F05.
Key words and phrases. Almost sure central limit theorem; Lipschitz function; logarithmic limit
theorems.
1
2 YU MIAO ANDGUANGYUYANG
Recently, H¨ormann [6] gave an extension of almost sure central limit theory under
some regularity condition as the following form:
N S ∞
lim D−1 d f k b = f(x)dH(x) a.s. (1.1)
N→∞ N X k (cid:16)ak − k(cid:17) Z−∞
k=1
where f is a bounded Lipschitz 1 function and D = N d , d is a sequence
N k=1 k { k}k≥1
of positive constants. Using his method, we will showPthat for any Lipschitz function
f, (1.1) holds under some additional conditions.
At first, we give our main result.
Theorem 1.1. Let X ,X ,... be independent random variables with partial sums S
1 2 n
and assume that
S
: (C ) For some numerical sequences a > 0 and b , we have n b L H,
1 n n n
a − −→
n
where H is some (possibly degenerate) distribution function.
: (C ) kd = O(1) and for some 0 < α < 1, d kα is eventually non-increasing.
2 k k
D
: (C ) For some ρ > 0, d = O k .
3 k (cid:16)k(logDk)ρ(cid:17)
: (C ) There exist positive constants C, β, such that a /a C(k/l)β (1 k
4 k l
≤ ≤ ≤
l). Furthermore, for some 0 < r < ρ,
S µ
E n b = O(1), for some integer µ (2 4/(ρ r)). (1.2)
n
(cid:12)a − (cid:12) ≥ ∨ −
n
(cid:12) (cid:12)
(cid:12) (cid:12)
Then for any Lipschitz function f on the real line, we have (1.1).
Remarks 1.2. Obviously, for any bounded Lipschitz 1 function f, under the above
assumptions, the equation (1.1) holds, i.e. we can obtain Theorem 1 in [6].
2. Proof of Theorem 1.1
In this section, we will give the proof of Theorem 1.1, according to the process of
H¨ormann in [6].
Lemma 2.1. (See Lemma 1 in [6] ) Let (D ) be a summation procedure, then the
N
condition (C ) of Theorem 1.1 implies that D = o(Nε) for any ε > 0.
3 N
Lemma 2.2. Assume that condition (C ) of Theorem 1.1 is satisfied and b = 0.
4 n
Then for every Lipschitz function f : R R there exists constant c > 0 such that
→
S S
Cov f k ,f l c(k/l)β (1 k l), (2.1)
(cid:12) (cid:16) (cid:16)ak(cid:17) (cid:16)al(cid:17)(cid:17)(cid:12) ≤ ≤ ≤
(cid:12) (cid:12)
(cid:12) (cid:12)
where β is the same as in (C ).
4
A NOTE FOR EXTENSION OF ALMOST SURE CENTRAL LIMIT THEORY 3
Proof. Firstly, we assume f(0) = 0. Denoting f the Lipschitz constant of f, we
k k
get, by using the independence of S and S S ,
k l k
−
S S S S S S
k l k l l k
Cov f ,f = Cov f ,f f −
(cid:12) (cid:16) (cid:16)ak(cid:17) (cid:16)al(cid:17)(cid:17)(cid:12) (cid:12) (cid:16) (cid:16)ak(cid:17) (cid:16)al(cid:17)− (cid:16) al (cid:17)(cid:17)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) S S (cid:12)S (cid:12) S S S S(cid:12) S
E f k f l f l − k +E f k E f l f l − k
≤ (cid:12) (cid:16)ak(cid:17)h (cid:16)al(cid:17)− (cid:16) al (cid:17)i(cid:12) (cid:12) (cid:16)ak(cid:17)(cid:12) (cid:12) (cid:16)al(cid:17)− (cid:16) al (cid:17)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) a S2 a S (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
f 2 kE k + f 2 k(E k )2
≤k k al ha2ki k k al haki
a S2 S2
2 f 2 kE k 2C f 2E k (k/l)β,
≤ k k al ha2ki ≤ k k ha2ki
where the last inequality is due to condition (C ). Since the equation (1.2), we can
4
S2
take c > 0 such that for any k 1, 2C f 2E k c. So (2.1) is obtained.
≥ k k ha2i ≤
k
If f(0) = 0, we can define a function g, such that g(x) = f(x) f(0), then g is a
6 −
Lipschitz function and g(0) = 0. And noting that,
S S S S
k l k l
Cov f ,f = Cov g ,g
(cid:16) (cid:16)ak(cid:17) (cid:16)al(cid:17)(cid:17) (cid:16) (cid:16)ak(cid:17) (cid:16)al(cid:17)(cid:17)
we complete the proof of the lemma.
(cid:3)
Remarks 2.3. It is obvious to see that if we replace β by any 0 < β′ < β, the
Lemma 2.2 also holds. Hence, without loss of generality, we can assume that β is the
same as α in condition (C ) of Theorem 1.1.
2
Next we will use the following notations,
S S S S S S
ξ := f l Ef l , ξ := f l − k Ef l − k . (2.2)
l k,l
(cid:16)al(cid:17)− (cid:16)al(cid:17) (cid:16) al (cid:17)− (cid:16) al (cid:17)
Lemma 2.4. Assume that condition (C ) of Theorem 1.1 is satisfied and b = 0,
4 n
and define ξ and ξ as in (2.2). If d ,k 1 are arbitrary positive weights, then
l k,l k
we have for any k m n and p N{, p ≥µ, }
≤ ≤ ∈ ≤
n n
p p/2
E d (ξ ξ ) E d2l ,
(cid:12)X l l − k,l (cid:12) ≤ p(cid:16)X l (cid:17)
(cid:12)l=m (cid:12) l=m
(cid:12) (cid:12)
2κ 1 p/2
where E = c Cp f p 1+ and κ = 2β.
p p
k k h(cid:16) κ (cid:17)∨(cid:16) κ(cid:17)i
4 YU MIAO ANDGUANGYUYANG
Proof. Let Q(l) = Q(k,l) = ξ ξ , then
l k,l
−
S S S S S S p
E Q(l) p =E f l f l − k E f l f l − k
| | (cid:12) (cid:16)al(cid:17)− (cid:16) al (cid:17)− h (cid:16)al(cid:17)− (cid:16) al (cid:17)i(cid:12)
(cid:12) (cid:12)
(cid:12) S S p (cid:12)
f p(a /a )pE | k| +E | k|
k l
≤k k (cid:16) ak (cid:16) ak (cid:17)(cid:17)
S S p
Cp f pE | k| +E | k| (k/l)pβ
≤ k k (cid:16) ak (cid:16) ak (cid:17)(cid:17)
c Cp f p(k/l)pβ,
p
≤ k k
where C is the same as in condition (C ) and c is a positive constant such that for
4 p
S S p
all k, E | k| +E | k| c . Thus by the H¨older inequality, we get
p
(cid:16) ak (cid:16) ak (cid:17)(cid:17) ≤
n n n
p
E d (ξ ξ ) d d (E Q(l ) p E Q(l ) p)1/p
(cid:12)X l l − k,l (cid:12) ≤ X ··· X l1··· lp | 1 | ··· | p |
(cid:12)l=m (cid:12) l1=m lp=m
(cid:12) (cid:12)
n n
c Cp f pkpβ d d l−β l−β
≤ p k k ··· l1··· lp 1 ··· p
X X
l1=m lp=m
n
p
=c Cp f pkpβ d l−β
p l
k k (cid:16)X (cid:17)
l=m
n n
p/2 p/2
c Cp f pmpβ d2l l−2β−1 .
≤ p k k (cid:16)X l (cid:17) (cid:16)X (cid:17)
l=m l=m
For m 2, it is easy to see that
≥
n ∞
p/2 p/2
mpβ l−2β−1 mpβ l−2β−1dl
(cid:16)X (cid:17) ≤ (cid:16)Zm−1 (cid:17)
l=m
m pβ 1 p/2 2κ p/2
,
≤(cid:16)m 1(cid:17) (cid:16)2β(cid:17) ≤ (cid:16) κ (cid:17)
−
where κ := 2β. Similarly, we get for m = 1
n
p/2 1 p/2
l−2β−1 1+ .
(cid:16)X (cid:17) ≤ (cid:16) κ(cid:17)
l=1
Hence, we have
n n
p p/2
E d (ξ ξ ) c Cp f pτ(κ)p/2 d2l ,
(cid:12)X l l − k,l (cid:12) ≤ p k k (cid:16)X l (cid:17)
(cid:12)l=m (cid:12) l=m
(cid:12) (cid:12)
where τ(κ) := 2κ 1+ 1 . This completes the proof of our result. (cid:3)
h(cid:16) κ (cid:17)∨(cid:16) κ(cid:17)i
A NOTE FOR EXTENSION OF ALMOST SURE CENTRAL LIMIT THEORY 5
Lemma 2.5. Assume that conditions (C ) (C ) of Theorem 1.1 are satisfied. Fur-
2 4
−
ther let b = 0 in condition (C ) and f be a Lipschitz function. Then for every p µ
n 4
and p N we have ≤
∈
N
S S p k β p/2
E d f k Ef k C d d , (2.3)
k p k l
(cid:12)X (cid:16) (cid:16)ak(cid:17)− (cid:16)ak(cid:17)(cid:17)(cid:12) ≤ (cid:16) X (cid:16)l(cid:17) (cid:17)
(cid:12)k=1 (cid:12) 1≤k≤l≤N
(cid:12) (cid:12)
where C > 0 is a constant and β is the same as in (C ).
p 4
Proof. At first, we set C = (4γ)p2 and
p
n l
V := d l−β d kβ , (1 m n).
m,n l k
X (cid:16)X (cid:17) ≤ ≤
l=m k=1
For obtaining our result, it is enough to show that the following claim,
”if the number γ is chosen large enough, then
n
p
E d ξ C (V )p/2, for all 1 m n, (2.4)
k k p m,n
(cid:12)X (cid:12) ≤ ≤ ≤
(cid:12)k=m (cid:12)
(cid:12) (cid:12)
where ξ is defined as in (2.2).”
k
We will use induction on p to show (2.4). By Lemma 2.2, we have
n
2
E d ξ 2 d d Eξ ξ 2c d d (k/l)β 2cV .
k k k l k l k l m,n
(cid:12)X (cid:12) ≤ X | | ≤ X ≤
(cid:12)k=m (cid:12) m≤k≤l≤n m≤k≤l≤n
(cid:12) (cid:12)
Hence if we choose γ so large that (4γ)4 2c, then (2.4) holds for p = 2.
≥
Assume now that (2.4) is true for p 1 2. From kd = O(1) it follows that there
k
− ≥
is a positive constant A such that l d kβ Alβ. Then we get for V γ as
k=1 k ≥ m,n ≤
the proof of Lemma 2.4, there existPs a constant A such that
p
n n n
p
E d ξ d d (E ξ p E ξ p)1/p
(cid:12)X k k(cid:12) ≤ X ··· X k1··· kp | k1| ··· | kp|
(cid:12)k=m (cid:12) k1=m kp=m
(cid:12) (cid:12)
n n
A f p d d
≤ pk k ··· k1··· kp
X X
k1=m kp=m
n
p
=A f p d
p k
k k (cid:16)X (cid:17)
k=m
n k
p
A f pA−p d k−β d lβ .
p k l
≤ k k (cid:16)X (cid:16)X (cid:17)(cid:17)
k=m l=1
Now choose γ so large that the C (A 1/p f /A)pγp/2. In the case of V γ, we
p p m,n
≤ k k ≤
have shown (2.4) is valid.
6 YU MIAO ANDGUANGYUYANG
We now want to show that if for any given X γ and the inequality (2.4) holds
≥
for V X, then it will also hold for V 3X/2 and this will show that (2.4)
m,n m,n
≤ ≤
holds for any value of V , i.e. complete the induction step.
m,n
Assume V 3X/2 and set
m,n
≤
w n n
S +S := d ξ + d ξ , T := d ξ , (m w n).
1 2 k k k k 2 k w,k
≤ ≤
X X X
k=m k=w+1 k=w+1
From the discussion of Lemma 4 in H¨ormann, S. [6] (2006), and Lemma 2.1, for a
fixed m and n we choose w in such a way that
V
w+1,n
V X, V X and = λ [1/2,1].
m,w w+1,n
≤ ≤ V ∈
m,w
From the mean value theorem we get
Sj Tj j S T ( S j−1 + T j−1) (j 1). (2.5)
| 2 − 2| ≤ | 2 − 2| | 2| | 2| ≥
Since condition (C ) and Remarks 2.3, there exists a constant B > 0 such that for
2
all l 1,
≥
l
B d kβ l1+βd .
k l
≥
X
k=1
This also shows that
n
ld2 BV , for all 1 m n.
l ≤ m,n ≤ ≤
X
l=m
By Lemma 2.4, we get for all j 1,
≥
E S T j F (V )j/2,
2 2 j w+1,n
| − | ≤
where F = Bj/2E and E is the constant in Lemma 2.4.
j j j
From the induction hypothesis in the case of 1 j p 1 and from the validity
≤ ≤ −
of (2.4) for V X in the case of j = p, we have
m,n
≤
E S j C (V )j/2, (1 j p) (2.6)
1 j m,w
| | ≤ ≤ ≤
and
E S j C (V )j/2 C λj/2(V )j/2, (1 j p). (2.7)
2 j w+1,n j m,w
| | ≤ ≤ ≤ ≤
The remains of the proof are the same as in Lemma 4 in H¨ormann, S. [6] (2006), but
for completeness, we still give the proof. By C inequality, we have
r
E T j 2jC λj/2(V )j/2, (1 j p). (2.8)
2 j m,w
| | ≤ ≤ ≤
Furthermore, from H¨older inequality the following result is easy,
E S j S T S p−j−1 (E S p)j/p(E S T p)1/p(E S p)(p−j−1)/p
1 2 2 2 1 2 2 2
| | | − || | ≤ | | | − | | |
C(p−1)/pF1/pλ(p−j)/2(V )p/2. (2.9)
≤ p p m,w
A NOTE FOR EXTENSION OF ALMOST SURE CENTRAL LIMIT THEORY 7
The last inequality remains valid, with an extra factor 2p−j−1 on the right hand
side, if S p−j−1 on the left hand side is replaced by T p−j−1. Since S and T are
2 2 1 2
| | | |
independent, we get
p−1
E S +S p E S p +E S p + Gj(E S j Sp−j Tp−j +E S jE T p−j),
| 1 2| ≤ | 1| | 2| p | 1| | 2 − 2 | | 1| | 2|
X
j=1
where Gj denote the combination, i.e., Gj = p![j!(p j)!]−1. Substituting (2.5) (2.9)
p p − −
(using also the analogue of (2.9) with T p−j−1) in the above inequality and get
2
| |
p−1
E S +S p C (V )p/2 1+λp/2 +C−1/pF1/p 2p−jGj(p j)λ(p−j)/2
| 1 2| ≤ p m,w (cid:16) p p X p −
j=1
p−1
+C−1 2p−jGjλ(p−j)/2C C .
p p j p−j
X (cid:17)
j=1
Note that
C−1/pF1/p const τ(κ)1/2c1/p(4γ)−p
p p ≤ · p
and
C C /C (4γ)−p, (1 j p 1),
j p−j p
≤ ≤ ≤ −
thus, by λ 1, we have
≤
p−1
C−1/pF1/p 2p−jGj(p j)λ(p−j)/2 const τ(κ)1/2pc1/pγ−p
p p p − ≤ · p
X
j=1
and
p−1
C−1 2p−jGjλ(p−j)/2C C const γ−p.
p p j p−j ≤ ·
X
j=1
Since λ 1/2 we have shown that for a large γ the relation E S + S p C (1 +
1 2 p
≥ | | ≤
λ)p/2(V )p/2 = C (V )p/2 is true, i.e., for V 3X/2, (2.4) is valid. (cid:3)
m,w p m,n m,n
≤
Lemma 2.6. (See Lemma 5 in [6]) Assume the condition (C ) of Theorem 1.1 is
3
satisfied. Then for any α > 0 and any η < ρ, we have
k α D2
d d = O N .
X k l(cid:16)l(cid:17) (cid:16)(logDN)η(cid:17)
1≤k≤l≤N
Proof of Theorem 1.1. Without loss of generality, from Lemma 2.5 and Lemma
2.6, we have, for any ε > 0, p µ and p N,
≤ ∈
N
P d ξ > εD c(p,ε)(logD )−pη/2, for N N .
k k N N 0
(cid:16)(cid:12)X (cid:12) (cid:17) ≤ ≥
(cid:12)k=1 (cid:12)
(cid:12) (cid:12)
8 YU MIAO ANDGUANGYUYANG
Since µ (2 4/(ρ r)) for some 0 < r < ρ, we can choose suitable η < ρ and p
≥ ∨ −
such that pη > 4. By (C ), we have D /D 1, thus we can choose (N ) such
3 N+1 N j
→
that D exp √j . Applying Borel-Cantelli lemma, we get
Nj ∼ { }
Nj
lim D−1 d ξ = 0 a.s..
j→∞ Nj X k k
k=1
For N N N , we have
j j+1
≤ ≤
N Nj
D−1 d ξ D−1 d ξ +2(D /D 1) a.s..
N | k k| ≤ Nj| k k| Nj+1 Nj −
X X
k=1 k=1
The convergence of the subsequence implies that whole sequence converges a.s., since
D /D 1. This complete the proof of the theorem. (cid:3)
Nj+1 Nj →
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DepartmentofMathematicsandStatistics,WuhanUniversity,430072Hubei,China
and College of Mathematics and Information Science, Henan Normal University,
453007 Henan, China.
E-mail address: [email protected]
DepartmentofMathematicsandStatistics,WuhanUniversity,430072Hubei,China.
E-mail address: study [email protected]