IJMMS 29:3 (2002) 125–131
PII. S0161171202011626
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ALMOST SURE CENTRAL LIMIT THEOREMS FOR STRONGLY
MIXING AND ASSOCIATED RANDOM VARIABLES
KHURELBAATAR GONCHIGDANZAN
Received 24 January 2001 and in revised form 9 June 2001
We prove an almost sure central limit theorem (ASCLT) for strongly mixing sequence of
random variables with a slightly slow mixing rate α(n) = O((log log n)−1−δ ). We also show
that ASCLT holds for an associated sequence of random variables without a stationarity
assumption.
2000 Mathematics Subject Classification: 60F05, 60F15.
1. Introduction and main results. The almost sure central limit theorem (ASCLT)
has been first introduced independently by Schatte [11] and Brosamler [4]. Since then,
many interesting results have been discovered in this field. For further results on
ASCLT we refer to Berkes [1].
An interesting direction is to prove ASCLT for weakly dependent cases, namely, α,
ρ, φ-mixing, and associated random variables. Among the results in this direction we
refer to Peligrad and Shao [9], Hurelbaatar [5], and Matuła [7].
Let X1 , X2 , . . . be a sequence of random variables on some probability space (Ω, Ᏺ, P ),
and let σab be the σ -algebra generated by the random variables Xa , Xa+1 , . . . , Xb . For
any two σ -algebras Ꮽ, Ꮾ ⊂ Ᏺ, define
α(Ꮽ, Ꮾ) = sup P(AB) − P(A)P(B); A ∈ Ꮽ, B ∈ Ꮾ
(1.1)
∞
.
α(n) = sup α σ1k , σk+n
(1.2)
and put
k≥1
The sequence X1 , X2 , . . . is called strongly mixing if α(n) → 0 as n → ∞.
A sequence of random variables X1 , X2 , . . . is called associated if for every n ≥ 1 and
any coordinatewise increasing functions f , g : Rn → R1 ,
Cov f X1 , X2 , . . . , Xn , g X1 , X2 , . . . , Xn ≥ 0
(1.3)
whenever the covariance is defined.
We set Sn = X1 + X2 + · · · + Xn and the notation an bn means an = O(bn ). The
function IA (·) denotes an indicator function on the set A.
Peligrad and Shao [9] proved ASCLT for stationary Gaussian sequences as well as
stationary associated random variables. Their main results are as follows.
126
KHURELBAATAR GONCHIGDANZAN
Theorem 1.1 (see [9, Theorem 3]). Let X1 , X2 , . . . be a stationary α-mixing sequence
with EX1 = 0, EX12 < ∞, a2n = ESn2 → ∞ as n → ∞ and α(n) log−γ n, for some γ > 0.
Assume that
Sn Ᏸ
→
N(0, 1) as n → ∞.
an
(1.4)
Then ASCLT holds, that is,
n
1 1
Sk
1
2
IA
e−t /2 dt
=√
n→∞ log n
k
a
2π
A
k
k=1
lim
a.s.
(1.5)
for all Borel sets A ⊂ R with λ(∂A) = 0.
Theorem 1.2 (see [9, Theorem 2]). Let X1 , X2 , . . . be a stationary associated sequence
∞
with EX1 = 0, and k=1 EX1 Xk < ∞. Then (1.5) holds.
In this paper, we prove almost sure limit theorems which generalize Theorems 1.1
and 1.2. We also show that under the stationarity assumption, Theorems 1.4 and 1.6
imply Theorems 1.1 and 1.2, respectively. For strong mixing we impose much slower
mixing rate. Our main results are as follows.
Theorem 1.3. Let X1 , X2 , . . . be a sequence of random variables with zero mean.
Assume that
n
1 Sk (1.6)
Var
f
(log log n)−1−
log2 n
k
ak
k=1
for all bounded Lipschitz functions f (x). Then (1.5) holds if and only if
n
1 1
Sk
1
2
P
∈A = √
e−t /2 dt
n→∞ log n
k
a
2π
A
k
k=1
lim
(1.7)
for all Borel sets A ⊂ R with λ(∂A) = 0.
Theorem 1.4. Let X1 , X2 , . . . be a strongly mixing sequence of random variables with
mean zero, and let an > 0 be a numerical sequence such that ESn2 ≤ a2n and for n ≥ k
an
≥
ak
n
k
γ
,
γ > 0.
(1.8)
Assume that
α(n) (log log n)−1−δ .
(1.9)
Then (1.5) and (1.7) are equivalent.
Berkes, Dehling, and Móri [3] gave an example of independent random variables
such that (1.5) was satisfied, but (1.4) failed. This shows that the class of sequences
satisfying ASCLT is larger than the class of sequences satisfying CLT. Berkes and
Dehling [2] proved the equivalence of (1.5) and (1.7) for independent random variables
not necessarily identically distributed, under mild technical condition on generalized
ALMOST SURE CENTRAL LIMIT THEOREMS . . .
127
moments of the partial sums. This result has been generalized by Hurelbaatar [5]
for strongly mixing and associated random variables. In Theorem 1.4, concerning the
limit distributional behavior of Sn /an , we require more restrictive moment condition
than the one in Hurelbaatar [5] and Berkes and Dehling [2].
Remark 1.5. For stationary sequence of α-mixing random variables, it is known
that an = nL(n), where L(n) is a slowly varying function at infinity, provided the
central limit theorem holds (see [6, page 316]). By the representation theorem of slowly
varying function, L(s)/L(t) (s/t)−
for any > 0 and s ≥ t ≥ n0 (
) and we see
that condition (1.8) is satisfied for the an and γ = 1/2 − . Therefore, for stationary
sequence of random variables Theorem 1.4 implies Theorem 1.1.
Theorem 1.6. Let X1 , X2 , . . . be a sequence of associated random variables with
mean zero satisfying
u(n) < ∞
(1.10)
for all n ≥ 1 and where u(n) = supk≥1 j:|k−j|≥n Cov(Xk , Xj ).
Assume that (1.8) is satisfied for some γ > 0 and
lim inf VarXk > 0.
(1.11)
k
Then (1.5) and (1.7) are equivalent.
Remark 1.7. In stationary case, the assumption of Theorem 1.2 implies the one of
Theorem 1.6. We can easily verify that
∞
Cov X1 , Xk < ∞,
k=1
u(n) = sup
(1.12)
Cov Xk , Xj < ∞
(1.13)
k≥1 j:|k−j|≥n
are equivalent for a stationary sequence of associated random variables. It is well
known that if (1.12) holds, then
∞
Var Sn
= EX12 + 2
EX1 Xk .
n→∞
n
k=1
lim
(1.14)
Choosing a2k as Var(Sk ), we see that (1.8) is satisfied with γ = 1/2. By Newman and
Wright [8, Theorem 3], (1.4) is true under the assumption of Theorem 1.2 and implies
(1.7). Thus Theorem 1.2 is a stationary case of Theorem 1.6.
2. Proofs
Proof of Theorem 1.3. It suffices to show that (see [2])
µn =
n
1 1
ξk → 0
log n k=1 k
a.s.
for any bounded Lipschitz function f , where ξk = f (Sk /ak ) − Ef (Sk /ak ).
(2.1)
128
KHURELBAATAR GONCHIGDANZAN
By (1.6) we have
2
Eµn
2
2
n
1
E
=
ξk
k
log2 n
k=1
n
1 Sk 1
Var
=
f
(log log n)−1−
,
k
a
log2 n
k
k=1
n
1 1
=E
ξk
log n k=1 k
1
(2.2)
and setting nk = exp(exp(kγ )),
2
Eµn
k
log log nk
−1−
k−γ(1+
) .
(2.3)
Hence
∞
2
Eµn
<∞
k
(2.4)
k=1
for any γ > 1/(1 + ).
By the well-known result that
∞
EXk < ∞
implies
k=1
∞
Xk < ∞ a.s.,
(2.5)
k=1
we have
µnk → 0
a.s.
(2.6)
It is easy to see that
(1 + k)
− k
→ 0
as k → ∞ for any < 1,
(2.7)
and thus
log nk+1
γ
γ
= e(1+k) −k → 1
log nk
as k → ∞.
(2.8)
Obviously, for any given n there always exists k such that nk ≤ n ≤ nk+1 and we have
µ n ≤
nk
nk+1
n
1 1
1 1
1
ξk ≤
ξk + 1
ξk log n k=1 k
log nk k=1 k
log nk k=n k
k
µnk +
1 lognk+1
log nk + 1 − log nk µnk +
−1 .
log nk
lognk
(2.9)
It follows that
lim µn = 0
n →∞
a.s.
(2.10)
ALMOST SURE CENTRAL LIMIT THEOREMS . . .
129
Proof of Theorem 1.4. According to Theorem 1.3, it suffices to show that for all
bounded Lipschitz,
n
Sk
1 1
f
(log log n)−1−
.
Var
log n k=1 k
ak
(2.11)
Here ξk is the same variable as we defined in the proof of Theorem 1.3. For l > 2k,
we have
E ξk ξl = Cov f Sk , f Sl
ak
al Sl
Sl − S2k Sk
≤
Cov f a , f a − f
al
k
l
Sl − S2k Sk
.
+
Cov f a , f
a
k
(2.12)
l
Since f is bounded, by [10, Theorem 1.1], we get
Cov f Sk , f Sl − S2k α(k),
a
a
k
(2.13)
l
and by Cauchy-Schwarz inequality, Lipschitz property of f , the facts that ESn2 ≤ a2n ,
and by (1.8) we have
Cov f Sk , f Sl − f Sl − S2k ak
al
al
γ
1/2
S2k a2k
k
S2k 2
.
E
E
al
al
al
l
(2.14)
Noting that
n
1
ξk
E
k
k=1
2
n
E ξk ξl E ξk ξl 2
1 ≤
E ξk + 2
+2
k2
kl
kl
1≤k<l≤n
1≤k<l≤n
k=1
2k≥l
2k<l
(2.15)
= T 1 + T2 + T3
and since ξk is bounded, we have the following estimations for the first two terms
T1 ∞
1
< ∞,
2
k
k=1
T2 2k
n
1
log n.
kl
k=1 l=k+1
(2.16)
By (2.12), (2.13), and (2.14) the third term is estimated as
T3 α(k)
1 k γ
= T31 + T32 ,
+
kl l
kl
1≤k<l≤n
1≤k<l≤n
2k<l
2k<l
(2.17)
130
KHURELBAATAR GONCHIGDANZAN
and furthermore
n
l−1
n
n
1 α(k)
α(k) 1
l k=1 k
k l=k l
l=2
k=1
T32 (2.18)
n
log k
(log log n)−1−δ log2 n,
1+δ
k(log
log
k)
k=1
T31 ≤
n
n
l
1 k γ
1 1
1
log n.
1+γ
1−γ
kl
l
l
k
l
1≤k<l≤n
l=1
k=1
l=1
(2.19)
2k<l
Now it can be easily shown that (2.15), (2.16), (2.17), (2.18), and (2.19) together give
(1.6) which is equivalent to (2.11).
Proof of Theorem 1.6. The general idea of the proof is similar to the method in
the proof of Theorem 1.4. We will verify (2.11). From (2.15) we see that
n
2
n
1 Sk 1
f
ξk
=E
Var
k
ak
k
k=1
k=1
≤
n
E ξk ξl E ξk ξl 1 ξk 2 + 2
+
2
E
k2
kl
kl
1≤k<l≤n
1≤k<l≤n
k=1
2k≥l
(2.20)
2k<l
= T 1 + T2 + T3 ,
and moreover,
T3 =
E ξk ξl kl
1≤k<l≤n
2k<l
=
1
Cov f Sk , f Sl
kl
a
a
k
l
1≤k<l≤n
2k<l
1
Cov f Sk , f Sl − f Sl − S2k ≤
kl
ak
al
al
1≤k<l≤n
2k<l
+
1≤k<l≤n
(2.21)
1
Cov f Sk , f Sl − S2k kl
a
a
k
l
2k<l
= T31 + T32 .
For a bounded Lipschitz function f , it is shown, by Peligrad and Shao [9], that
Sl − S2k
Sk Sl − S2k
Sk
,f
Cov
.
(2.22)
,
Cov f
ak
al
ak
al
Hence
T32 1 ku(k)
1
Cov Sk , Sl − S2k .
kl
ak
al
kl ak al
1≤k<l≤n
1≤k<l≤n
2k<l
(2.23)
2k<l
For associated random variables, clearly
VarSn ≥
n
k=1
VarXk2 .
(2.24)
ALMOST SURE CENTRAL LIMIT THEOREMS . . .
131
By the assumption that ESn2 ≤ a2n and (1.11), we get
a2n n
VarXk2 n,
(2.25)
k=1
therefore,
T32 n
n
ku(k)
u(k) 1
3/2
3/2
1/2
k l
k
l3/2
1≤k<l≤n
k=1
l=k
2k<l
(2.26)
n
log k
(log log n)−1−δ log2 n.
1+δ
k(log
log
k)
k=1
By (2.14) and (2.19)
T31 log n,
(2.27)
and (2.16), (2.26), and (2.27) follow (2.11).
Acknowledgements. I am very grateful to Prof. Magda Peligrad for her helpful discussion. Also I would like to thank the referees for their helpful comments
and remarks. This work was supported by a Research Fellowship at the University of
Cincinnati.
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Khurelbaatar Gonchigdanzan: Department of Mathematical Sciences, University
of Cincinnati, Cincinnati, OH 45221-0025, USA
E-mail address: hurlee@math.uc.edu