Internat. J. Math. & Math. Sci.
Vol. 24, No. 11 (2000) 737–747
S0161171200000867
© Hindawi Publishing Corp.
ON COMPLETE CONVERGENCE FOR Lp -MIXINGALES
YIJUN HU
(Received 31 May 1996 and in revised form 16 March 1998)
Abstract. We provide in this paper sufficient conditions for the complete convergence
for the partial sums and the random selected partial sums of B-valued Lp -mixingales.
Keywords and phrases. Complete convergence, p-smooth Banach space, Lp -mixingale.
2000 Mathematics Subject Classification. Primary 60F15.
1. Introduction and results. McLeish [7] introduced first the concept of mixingales,
a generalization of the concepts of mixing sequences and martingale differences,
where the mixingale convergence theorems and the strong laws of large numbers
have been proved. Furthermore, McLeish [6, 8] studied the invariance principles for
mixingales. Yin [9] generalized McLeish’s concept of mixingales to operator-valued
mixingales, and proved the operator-valued mixingale convergence theorems. Hall and
Heyde [2] also pointed out that mixingales include martingale differences, lacunary
functions, linear processes, and uniformly mixing processes (also called Φ-mixing).
On the other hand, up till now, there have been an extensive literatures in complete
convergence for independent and dependent random sequences (especially, martingale differences and various mixing sequences), see partially the references here. However, there are few papers reported on the complete convergence for mixingales; see,
for example, Liang and Ren [5].
Preceding observations stir us to investigate the complete convergence for mixingales. In the present paper, we first generalize slightly McLeish’s definition of mixingales to B-valued Lp -mixingales, and then give some general results about complete
convergence for B-valued Lp -mixingales.
Next, we introduce some notations. Let (B, · ) be a Banach space. B is said to be
q-smooth (1 ≤ q ≤ 2) if there exists a constant Cq > 0 such that for every B-valued
Lq -integrable martingale difference sequence {Di ; i ≥ 1},
q
n
n
D
EDi q ,
E
i ≤ Cq
i=1 i=1
n ≥ 1.
(1.1)
Let {Xn ; n ≥ 1} be B-valued Lp -integrable (1 ≤ p ≤ 2) random variables on a probability space (Ω, Ᏺ, P ), and let {Ᏺn ; −∞ < n < ∞} be an increasing sequence of sub
σ -fields of Ᏺ. Then {Xn , Ᏺn } is called a Lp -mixingale if there exist sequences of nonnegative constants Cn and ψ(n), where ψ(m) ↓ 0 as m → ∞, which satisfy the following properties:
738
YIJUN HU
(i) E(Xn | Ᏺn−m )p ≤ ψ(m)Cn ;
(ii) Xn − E(Xn | Ᏺn+m )p ≤ ψ(m + 1)Cn , for all n ≥ 1 and m ≥ 0, where Xp =
(EXp )1/p .
Let {Xn ; n ≥ 1} be B-valued random variables, and X0 be a real nonnegative random
variable. We call that {Xn } is bounded in probability by X0 (abbreviated {Xn } < X0 ) if
∀t > 0.
P Xn > t ≤ P X0 > t
(1.2)
Given a positive function l(x) defined on (0, +∞), we say that l(x) is a slowly variable
function as x → ∞, if for all c > 0,
lim
x→+∞
l(cx)
= 1,
l(x)
(1.3)
see also Laha and Rohatgi [4].
From now on, we use C to denote finite positive constants whose value may change
from statement to statement. For real numbers x, y, [x] denotes the largest integer
k ≤ x, and x ∧ y means min(x, y).
The following are the main results of this paper.
Theorem 1.1. Let 1 ≤ t < q ≤ 2, 0 < δ < 1 ∧ 3(q/t − 1), 1 ≤ p ≤ 2. B is a q-smooth
Banach space. Suppose {Xn , Ᏺn } is a B-valued Lp -mixingale, and X is a real nonnegative
random variable satisfying {Xn } < X. Suppose l(x) is an increasing slowly variable
function as x → ∞. If E(X t+δ l(X t+δ )) < ∞ and there exist λ (1 ≤ λ ≤ p) and β > 0 such
that t + (1 − t)λ > 0, β < min(δ/2t, δ/(t + qt), (q − t)/(t + qt)), and
∞
n=1
ψλ nβ max Ciλ < ∞,
1≤i≤n
(1.4)
then
∞
l(n) P Sn ≥ n1/t ε < ∞
n
n=1
for ε > 0, where Sn =
(1.5)
n
i=1 Xi .
Theorem 1.2. Let 1 ≤ t < q ≤ 2, 0 < δ < 1 ∧ 3(q/t − 1), and 1 ≤ p ≤ 2. B is a
q-smooth Banach space. Suppose {Xn , Ᏺn } is a B-valued Lp -mixingale, and X is a real
nonnegative random variable satisfying {Xn } < X. Suppose l(x) is an increasing slowly
variable function as x → ∞. If E X t+δ l(X t+δ ) ln+ X < ∞ and there are λ (1 ≤ λ ≤ p)
and β > 0 satisfying β < min(δ/2t, δ/(t + qt), (q − t)/(t + qt)) and
∞
j2(1+2λ)j ψλ 2βj
max Ciλ < ∞,
(1.6)
∞
Sk l(n)
≥ε <∞
P sup 1/t n
k≥n k
n=1
(1.7)
j=1
then
for ε > 0, where Sn =
n
i=1 Xi .
1≤i≤2j+1
ON COMPLETE CONVERGENCE FOR Lp -MIXINGALES
739
Based on Theorem 1.2, we can now obtain the analogue to random selected partial
sums of Lp -mixingales.
Theorem 1.3. Let 1 < t < q ≤ 2, 0 < δ < 1 ∧ 3(q/t − 1), 1 ≤ p ≤ 2. B is a qsmooth Banach space. Suppose {Xn , Ᏺn } is a B-valued Lp -mixingale and X is a real
nonnegative random variable satisfying {Xn } < X. Suppose {νn ; n ≥ 1} are random
variables which only take positive integer values and are defined on the same probability space as {Xn }. Suppose l(x) is an increasing slowly variable function as x → ∞.
If E X t+δ l(X t+δ ) ln+ X < ∞, and there exist positive constants λ, β, and η such that
β < min(δ/2t, δ/(t + qt), (q − t)/(t + qt)),
∞
νn
l(n)
P
< η < ∞,
n
n
n=1
∞
j=1
j2(1+2λ)j ψλ 2βj
max Ciλ < ∞,
1≤i≤2j+1
(1.8)
(1.9)
then for ε > 0,


νn
∞
l(n) 
1/t

P Xi ≥ νn · ε < ∞.
n
i=1 n=1
(1.10)
Remark 1.4. To our best knowledge, even if B = R (the real numbers), the
results here are new. Furthermore, conditions (1.4), (1.6), and (1.9) are reasonable.
For this purpose, we now particularize the general situation as follows. Let B = R.
In return, q = 2. Let t = 1, p = 2 and {Xn , Ᏺn } be a L2 -mixingale (coinciding
with mixingale of McLeish [7] or Hall and Heyde [2]). Consequently, 1 ∧ 3(q/t − 1)
= 1. Given 0 < δ < 1, then min(δ/2t, δ/(t + qt), (q − t)/(t + qt)) = min(δ/2, δ/3,
1/3) = δ/3.
∞
Suppose that {Ci ; i ≥ 1} is bounded or i=1 Ci < ∞ and that ψ(m) = o(m−θ )
for some constant θ satisfying θ · δ > 3, then condition (1.4) is satisfied with
λ = 1 and each β ∈ (1/θ, δ/3), as can be easily verified. Moreover, in addition to
the above assumptions, suppose that ψ(m) = o(m−θ ) for some constant θ satisfying θ · δ > 9, then conditions (1.6) and (1.9) are satisfied with λ = 1 and each
β ∈ (3/θ, δ/3).
On the other hand, condition ψ(m) = o(m−θ ) is implied with summability condi∞
tions such as m=1 ψ1/θ (m) < ∞ (see also [6, 7]).
∞
λ
Remark 1.5. In general, if {Ci ; i ≥ 1} is bounded or
i=1 Ci < ∞ for some
1 ≤ λ ≤ p satisfying t + (1 − t)λ > 0, and ψ(m) = o(m−θ ) for some sufficiently
large θ satisfying (1 + 2λ)/θλ < min(δ/2t, δ/(t + qt), (q − t)/(t + qt)) then conditions (1.4), (1.6), and (1.9) are satisfied with the above λ and each β ∈ ((1 + 2λ)/θλ,
min(δ/2t, δ/(t + qt), (q − t)/(t + qt))). Meanwhile, condition ψ(m) = o(m−θ ) can
∞
1/θ
also be implied with summability conditions such as
(m) < ∞ (see
m=1 ψ
also [6, 7]).
740
YIJUN HU
∞
In any case, roughly speaking, conditions such as {Ci ; i ≥ 1} is bounded or i=1 Ciλ <
∞ for some 1 ≤ λ ≤ p satisfying t + (1 − t)λ > 0, plus a specific rate of convergence of
ψ(m) to 0, ensure conditions (1.4), (1.6), and (1.9).
Remark 1.6. Condition (1.8) is just one which is usually employed in literatures.
2. Proofs of the main results. For the sake of convenience, we begin with two
lemmas, which will be needed below.
Lemma 2.1. Suppose that l(x) is a slowly variable function as x → ∞, then we have
(1) limx→∞ l(x + u)/l(x) = 1, ∀u > 0;
(2) limk→+∞ sup2k ≤x≤2k+1 l(x)/l(2k ) = 1;
(3) limx→+∞ x δ l(x) = +∞, limx→+∞ x −δ l(x) = 0, ∀δ > 0;
k
(4) C · 2kr l(η · 2k ) ≤ j=1 2jr l(η · 2j ) ≤ C · 2kr l(η · 2k ) for every positive r , η and
integer k;
∞
∞
(5) C ·2kr l(η·2k ) ≤ j=k 2jr l(η·2j ) ≤ C ·2kr l(η·2k ), C ·k2kr l(2k ) ≤ j=k j2jr l(2j )
≤ C · k2kr l(2k ) for every r < 0, η > 0 and integer k.
We refer to Bai and Su [1] and Hu [3] for a proof of Lemma 2.1.
By applying integration by parts, it is easy to prove the following lemma.
Lemma 2.2. Let X be a real random variable, then
E |X|r I(|X| ≤ a) ≤ r
a
t (r −1) P (|X| > t) dt,
∞
P (|X| > t) dt
E |X|I(|X| > a) = aP (|X| > a) +
0
(2.1)
a
for r ≥ 1, a > 0, where and elsewhere I(|X| ≤ a) means the indicator of {|X| ≤ a}.
Proof of Theorem 1.1. We write α = 1/t. Notice first that
Sn =
n i=1
n Xi − E Xi | Ᏺi+[nβ ] +
E Xi | Ᏺi+[nβ ] − E Xi | Ᏺi+[nβ ]−1
i=1
n n
+···+
E Xi | Ᏺi−[nβ ]+1 − E Xi | Ᏺi−[nβ ] +
E Xi | Ᏺi−[nβ ] .
i=1
(2.2)
i=1
By denoting
Part 1 =
i=1
Part 2 =
Xi − E Xi | Ᏺi+[nβ ] ,
n β
[n
]
n
E Xi | Ᏺi+l − E Xi | Ᏺi+l−1 ,
l=−[nβ ]+1 i=1
Part 3 =
n
E Xi | Ᏺi−[nβ ] ,
i=1
(2.3)
ON COMPLETE CONVERGENCE FOR Lp -MIXINGALES
741
it is sufficient for us to prove
∞
l(n) P Part 1 ≥ nα ε < ∞,
n
n=1
(2.4)
∞
l(n) P Part 2 ≥ nα ε < ∞,
n
n=1
(2.5)
∞
l(n) P Part 3 ≥ nα ε < ∞.
n
n=1
(2.6)
By Chebyshev inequality, Cr -inequality, Lemma 2.1, and Lp -mixingale property
we have

λ
∞
∞
n 

l(n) α
−(1+αλ)
Xi − E Xi | Ᏺi+[nβ ] P Part 1 ≥ n ε ≤ C ·
l(n)n

λ
n
n=1
n=1
≤C·
∞
n=1
≤C·
∞
n=1
i=1
l(n)nλ−1−αλ ψλ nβ max Ciλ
1≤i≤n
(2.7)
ψλ nβ max Ciλ < ∞,
1≤i≤n
which proves (2.4).
Similarly, we obtain

λ
∞
∞
n 

l(n) E Xi | Ᏺi−[nβ ] P Part 3 ≥ nα ε ≤ C ·
l(n)n−1−αλ

λ
n
n=1
n=1
i=1
≤C·
∞
n=1
≤C·
∞
n=1
l(n)nλ−1−αλ ψλ nβ max Ciλ
1≤i≤n
(2.8)
ψλ nβ max Ciλ < ∞,
1≤i≤n
which is exactly (2.6).
To prove (2.5), let Yi,n = Xi I(Xi ≤ nα ), Zi,n = Xi − Yi,n , Wl,i = E(Xi | Ᏺi+l ) −
E(Xi | Ᏺi+l−1 ), Ul,i = E(Yi,n | Ᏺi+l ) − E(Yi,n | Ᏺi+l−1 ), Vl,i = E(Zi,n | Ᏺi+l ) − E(Zi,n |
Ᏺi+l−1 ), n ≥ 1, 1 ≤ i ≤ n, −[nβ ]+1 ≤ l ≤ [nβ ]. Clearly, Xi = Yi,n +Zi,n , Wl,i = Ul,i +Vl,i .
For fixed l, {Ul,i , Ᏺi+l , 1 ≤ i ≤ n} and {Vl,i , Ᏺi+l , 1 ≤ i ≤ n} are martingale difference
sequences. Hence,


β
[n
∞
]
n
l(n)
ε
α−β
the left hand side (LHS) of (2.5) ≤
P 
· 
Wl,i ≥ n
n
2
i=1
n=1
l=−[nβ ]+1


β
[n
∞
]
n
l(n)
ε
α−β
≤
P 
· 
Ul,i ≥ n
n
4
i=1
(2.9)
n=1
l=−[nβ ]+1


β
[n
∞
]
n
l(n)
α−β ε 
+
P 
V
·
l,i ≥ n
n
4
β
n=1
i=1
l=−[n ]+1
= Part 4 + Part 5.
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YIJUN HU
By Kolmogorov inequality and Lemma 2.2,
∞
l(n)
Part 4 ≤ C ·
n
n=1
β
[n
]
n
n(β−α)q
EYi,n q
i=1
l=−[nβ ]+1
n
n
∞
l(n) β (β−α)q n ·n
αq ·
s αq−1 P Xi t > s ds
n
0
n=1
i=1
n
∞
l(n)n(β−α)q+β
s αq−1 P X t > s ds
≤C·
=C·
0
n=1
≤C·
∞
(2.10)
2(1+β+βq−αq)j l 2j
2j
0
j=1
s αq−1 P X t > s ds.
Observing
2j
1
j 2k
s αq−1 P X t > s ds =
s αq−1 P X t > s ds + s αq−1 P X t > s ds
0
2k−1
k=1
≤C+
j
0
2αqk P X t > 2k−1 ,
(2.11)
(2.12)
k=1
we get
Part 4 ≤ C ·
∞
j
∞
αqk t
2(1+β+βq−αq)j l 2j + C ·
2(1+β+βq−αq)j l 2j
2
P X > 2k−1 .
j=1
j=1
k=1
(2.13)
Since 1 + β + βq − αq < 0, from Lemma 2.1(5) we know that
∞
2(1+β+βq−αq)j l 2j < ∞.
(2.14)
j=1
Now from Lemma 2.1(5) again, it follows that
j
∞
∞
αqk t
2(1+β+βq−αq)j l 2j
2
P X > 2k−1 ≤ C ·
2(1+β+βq)k l 2k P X t > 2k−1
j=1
k=1
k=1
≤ C · E X t+(1+q)βt l X t+(1+q)βt
≤ C · E X t+δ l X t+δ < ∞.
(2.15)
By Kolmogorov inequality and Lemma 2.2 again, we get
Part 5 ≤ C ·
∞
l(n)
n
n=1
β
[n
]
l=−[nβ ]+1
n(β−α)
n
EZl,i i=1
∞
∞
l(n) β
n · nβ−α · n nα P X t > n +
P (X > s) ds
n
nα
n=1
∞
∞
∞
n2β l(n)P X t > n + C ·
n2β−α l(n)
P (X > s) ds.
≤C·
≤C·
n=1
n=1
nα
(2.16)
ON COMPLETE CONVERGENCE FOR Lp -MIXINGALES
743
Keeping Lemma 2.1 in mind, we obtain
∞
∞
n2β l(n)P X t > n ≤ C ·
2(2β+1)j l 2j P X t > 2j−1
n=1
∞
j=1
≤ C · E X t+2βt l X t+2βt
≤ C · E X t+δ l X t+δ < ∞,
n2β−α l(n)
∞
nα
n=1
P (X > s) ds ≤ C ·
∞
2(1+2β−α)j l 2j
∞
j=1
2αj
(2.17)
P (X > s)ds
∞ (1+2β)t (1+2β)t X
X
E
l
ds
s
s
1
t+δ t+δ ≤ C ·E X
l X
< ∞.
≤C·
(2.18)
Hence, equation (2.5) follows from (2.9), (2.13), (2.14), (2.15), (2.16), (2.17), and (2.18).
Theorem 1.1 is proved.
Proof of Theorem 1.2. By Lemma 2.1, we know first that
∞
∞
Sk Sk j
l(n)
P sup 1/t ≥ ε ≤ C ·
l 2 P sup 1/t ≥ ε
n
k≥n k
k≥2j k
n=1
j=1
∞
m
m/t
ml 2 P
ε .
max
Sk ≥ 2
≤C·
m=1
(2.19)
2m ≤k<2m+1
Hence, it is enough to show that
∞
j
jl 2 P
j=1
max
2j ≤n<2j+1
Sn ≥ 2j/t ε < ∞
∀ε > 0.
(2.20)
Observe that for 2j ≤ n < 2j+1 ,
Sn =
Xi − E Xi | Ᏺi+[2βj ]
n i=1
+
βj
[2
]
n
n
E Xi | Ᏺi−[2βj ]
E Xi | Ᏺi+l − E Xi | Ᏺi+l−1 +
(2.21)
i=1
l=−[2βj ]+1 i=1
= I1 + II2 + III3 ,
we only have to prove
∞
j=1
jl 2j P
max
2j ≤n<2j+1
I1 ≥ 2j/t ε < ∞,
(2.22)
744
YIJUN HU
∞
jl 2j P
j=1
∞
jl 2j P
j=1
max
2j ≤n<2j+1
max
2j ≤n<2j+1
II2 ≥ 2j/t ε < ∞,
(2.23)
III3 ≥ 2
j/t
ε < ∞.
(2.24)
To this end, we write α = 1/t. By Lemma 2.1 and (1.6), we have


n (α−1)j
jl 2
P 
ε
Xi − E Xi | Ᏺi+[2βj ] LHS of (2.22) ≤
≥2
i=1
j=1
2j ≤n<2j+1

λ
∞
n j

(1−α)λj 
Xi − E Xi | Ᏺi+[2βj ] jl 2
2
≤C·
∞
j
j=1
≤C·
∞
j
jl 2
j
·2 ·2
(1−α)λj
·2
λj
≤C·
j=1
· ψ 2βj
max Ciλ
(2.25)
λ
1≤i≤2j+1
j=1
∞
λ
i=1
2j ≤n<2j+1
j2(1+2λ)j ψλ 2βj
max Ciλ < ∞.
1≤i≤2j+1
Similarly, we can get
LHS of (2.24) ≤ C ·
∞
j=1
≤C·
∞
jl 2j · 2j · 2(λ−αλ)j · 2λj · ψλ 2βj
max Ciλ
1≤i≤2j+1
j2
(1+2λ)j
ψ
λ
βj
2
j=1
max
1≤i≤2j+1
Ciλ
(2.26)
< ∞.
Now, all that remains is to prove (2.23). For this purpose, we denote Yi,j = Xi I(Xi t
≤ 2j ), Zi,j = Xi − Yi,j , Wl,i = E(Xi | Ᏺi+l ) − E(Xi | Ᏺi+l−1 ), Ui,l = E(Yi,j | Ᏺi+l ) − E(Yi,j |
Ᏺi+l−1 ), Vl,i = E(Zi,j | Ᏺi+l ) − E(Zi,j | Ᏺi+l−1 ) for 2j ≤ n < 2j+1 , 1 ≤ i ≤ n, −[2βj ] + 1 ≤
l ≤ [2βj ]. Therefore,

n
(α−β)j
P  max ε
LHS of (2.23) ≤
Wl,i ≥ 2
2j ≤n<2j+1 i=1
j=1
l=−[2βj ]+1


βj
∞
n
j [2 ]
ε
(α−β)j

jl 2
P  max ≤
Ul,i ≥ 2
2
2j ≤n<2j+1 i=1
j=1
l=−[2βj ]+1


βj
∞
n
j [2 ]
ε
(α−β)j

jl 2
P  max +
Vl,i ≥ 2
2
2j ≤n<2j+1 i=1
βj
j=1
∞
jl 2j
l=−[2
= IV4 + V5 .

βj
[2
]
]+1
(2.27)
ON COMPLETE CONVERGENCE FOR Lp -MIXINGALES
745
By Kolmogorov inequality and Lemma 2.2,
V I4 ≤ C ·
≤C·
∞
2(β−α)qj
j=1
l=−[2βj ]+1
∞
[2βj ]
j=1
≤C·
βj
[2
]
jl 2j
∞
jl 2j
j+1
2
EUl,i q
i=1
2(β−α)qj
j+1 j
2
2
i=1
l=−[2βj ]+1
0
s (αq−1) P Xi t > s ds

j 2k
j βj
t
(β−α)qj j
(β−α)qj j
(αq−1)

jl 2 2
2 +2
2
s
P X > s ds 
2

j=1
≤C·
∞
k=1
(2.28)
2k−1
jl 2j 2(1+β+βq−αq)j
j=1
+C ·
∞
j
jl 2j 2(1+β+βq−αq)j
P X t > 2k−1 2αqk .
j=1
k=1
Since 1 + β + βq − αq < 0, by Lemma 2.2(5) we know
∞
jl 2j 2(1+β+βq−αq)j < ∞.
(2.29)
j=1
Furthermore, from Lemma 2.1, we get
∞
j
jl 2j 2(1+β+βq−αq)j
2αqk P X t > 2k−1
j=1
k=1
≤C·
∞
2αqk P X t > 2k−1
k=1
≤C·
∞
∞
jl 2j 2(1+β+βq−αq)j
j=k
2(1+β+βq)k l 2k P X t > 2k−1
(2.30)
k=1
+C ·
∞
(k − 1)2(1+β+βq)(k−1) l 2(k−1) P X t > 2k−1
j=1
≤ C · E X t+(β+βq)t l X t+(β+βq)t + C · E X t+(β+βq)t l X t+(β+βq)t ln+ X
≤ C · E X t+δ l X t+δ + C · E X t+δ l X t+δ ln+ X < ∞,
since (1 + q)βt < δ. Consequently, V I4 < ∞. Now all that remains is to prove V5 < ∞.
In fact, by Chebyshev inequality and Lemma 2.2, we have
V5 ≤ C ·
∞
∞
jl 2j 2βj 2(β−α)j 2j 2αj P X > 2αj +
P (X > s)ds
2αj
j=1
≤C·
∞
j=1
∞
j2(1+2β)j l 2j P X t > 2j + C ·
j2(1+2β−α)j l 2j
j=1
∞
2αj
(2.31)
P (X > s) ds.
746
YIJUN HU
Since 2βt < δ,
∞
j2j(1+2β) l 2j P X t > 2j ≤ C · E X t+2βt l X t+2βt ln+ X
j=1
≤ C ·E X
t+δ
l X t+δ ln+ X < ∞.
(2.32)
At the same time, by Fubini’s theorem,
∞
j2(1+2β−α)j l 2j
j=1
∞
2αj
P (X > s) ds =
∞
j2(1+2β)j l 2j
j=1
∞
1
P X t > 2j s ds
∞ t+2βt t+2βt X
X
X
ds
E
l
ln+
≤C·
s
s
s
1
∞
≤ C · E X t+2βt l X t+2βt ln+ X
s −(t+2βt) ds
1
≤ C · E X t+δ l X t+δ ln+ X < ∞,
(2.33)
which, together with (2.32), implies V5 < ∞. This completes the proof.
Proof of Theorem 1.3. Because




k
νn X
ν
j
n
1/t 
≥ ε ,

P 
Xi ≥ νn ε ≤ P n < η + P sup 1/t i=1 k≥ηn j=1 k
(2.34)
keeping in mind (1.8), it is enough to show
∞
l(n)
P
n
n=1
where Sk =
k
i=1 Xi .
Sk sup 1/t ≥ ε < ∞,
k
k≥ηn
(2.35)
Indeed Theorem 1.2 implies (2.35), hence Theorem 1.3 is proved.
Acknowledgement. I am grateful to Professor George L. O’Brien for his assistance. This work was supported in part by the National Natural Science Foundation of
China and Natural Sciences and Engineering Research Council of Canada.
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Yijun Hu: Department of Mathematics, Wuhan University, Hubei 430072, China
E-mail address: yijunhu@public.wh.hb.cn