Volume 2010, Article ID 168081,11pages doi:10.1155/2010/168081
Research Article
Convergence Theorems for Partial Sums of
Arbitrary Stochastic Sequences
Xiaosheng Wang and Haiying Guo
College of Science, Hebei University of Engineering, Handan 056038, China
Correspondence should be addressed to Xiaosheng Wang,[email protected] Received 27 May 2010; Revised 24 September 2010; Accepted 20 October 2010 Academic Editor: Jewgeni Dshalalow
Copyrightq2010 X. Wang and H. Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By using Doob’s martingale convergence theorem, this paper presents a class of strong limit theorems for arbitrary stochastic sequence. Chow’s two strong limit theorems for martingale-difference sequence and Lo`eve’s and Petrov’s strong limit theorems for independent random variables are the particular cases of the main results.
1. Introduction
Let {Xn,Fn, n ≥ 1}be a stochastic sequence on the probability space Ω,F, P that is, the
sequence ofσ-fields{Fn, n≥1}inFis increasing innthat isFn↑, and{Fn}are adapted to
random variables{Xn}.
Almost sure behavior of partial sums of random variables has enjoyed both a rich classical period and a resurgence of research activity. Some famous researchers, such as Borel, Kolmogorov, Khintchine, Lo`eve, Chung, and so on, were interested in convergence theorem of partial sums of random variables and obtained lots of classical results for sequences of independent random variables and martingale differences. For a detailed survey of strong limit theorems of sequences for random variables, interested readers can refer to the books
1,2.
sequences in 2003. In 2008, W. Yang and X. Yang8proved two strong limit theorems for stochastic sequences, which generalized results by Freedman9, Isaac,10and Petrov2. Qiu and Yang11established another type strong limit theorem for stochastic sequence in 1999. Then, Wang and Guo12extended the main result of Qiu and Yang in 2009. In addition, Wang and Yang13established a strong limit theorem for arbitrary stochastic sequences in 2005, which generalized Chow’s5series convergence theorem for sequence of martingale differences. Then, Qiu14extended the result of Wang and Yang in 2008.
The purpose of this paper is to discuss further the strong limit theorems for arbitrary stochastic sequences. By using Doob’s 1 convergence theorem for martingale-difference sequence, we establish a class of new strong limit theorems for stochastic sequences. Chow’s two strong limit theorems for martingale-difference sequence, Lo`eve’s series convergence theorem, and Petrov’s strong law of large numbers for sequences of independent random variables are the particular cases of this paper. In addition, the main theorems of this paper extend the main results by Wang and Guo in 2009, Qiu and Yang in 1999, and the result by Wang and Yang in 2005, respectively. The remainder of this paper is organized as follows. InSection 2, we present the main theorems of this paper. InSection 3, the proofs of the main theorems in this paper are presented.
2. Main Theorems
In this section, we will introduce the main results of this paper.
Let{cn, n ≥ 1}be a positive real numbers sequence andaxandbxtwo positive
real-valued functions on0,∞satisfyingax≥ a > 0 whenx∈ 0, cnandbx≥ b >0
whenx∈cn,∞.
Theorem 2.1. Let{Xn,Fn, n≥1}be a stochastic sequence defined as inSection 1and{φnx, n≥1}
a sequence of nondecreasing and nonnegative Borel functions on0,∞. For some1≤p≤2, suppose that
hnx axxpI0,cnx bxIcn,∞x, n≥1, 2.1
whereax, bxandcndefined as above. Assume that
φnx≥hnx, x∈0,∞. 2.2
Set
A
ω:
∞
n 1
Eφn|Xn|| Fn−1<∞
. 2.3
iIf there exists somec >0such that
holds whenx∈0, cn, then
∞
n 1
Xn converges a.e.onA. 2.5
iiIf there exists somec >0such that2.4holds whenx∈cn,∞, then
∞
n 1
Xn−EXn| Fn−1 converges a.e.onA. 2.6
Corollary 2.2Chow. Let{Xn,Fn, n≥1}be aLPmartingale-difference sequence and{an, n≥1}
be an increasing sequence of positive numbers. For1≤p≤2, let
A
ω:
∞
n 1
a−npE
|Xn|p| Fn−1<∞
. 2.7
Ifan↑ ∞, then
lim
n→ ∞
1
an n
i 1
Xi 0 a.e.on A. 2.8
Proof. By using Kroncker’s lemma, it is a special case of Theorem 2.1 when the random
variablesXnare replaced byXn/anandφnx |x|p.
Theorem 2.3. Let{Xn, n≥1}be a sequence of arbitrary random variables. LetFn σX0, . . . , Xn
andF0 {Ω,Φ}, n≥1. Letφnandhnbe defined asTheorem 2.1. If
∞
n 1
Eφn|Xn|
<∞, 2.9
then∞n 1Xnand∞n 1Xn−EXn| Fn−1converge a.e. under the same conditions (i) and (ii) as in
Theorem 2.1, respectively.
Corollary 2.4Lo`eve. Let{Xn, n ≥ 1}be a sequence of independent random variables, and0 <
rn≤2. Suppose that
∞
n 1
E|Xn|rn <∞. 2.10
If0< rn≤1, then
∞
n 1Xn converges a.e.If1< rn≤2, then
∞
Corollary 2.5Petrov. Let{Xn, n≥1}be a sequence of independent random variables. If0< an↑
∞and
∞
n 1
E|Xn|rn
arn
n
<∞, 2.11
thenlimn→ ∞1/an
n
i 1Xi 0a.e.when 0 < rn < 1, andlimn→ ∞1/an
n
i 1Xi−EXi
0a.e.when1≤rn≤2.
Theorem 2.6. Let{Xn,Fn, n≥1}be a stochastic sequence defined as inSection 1and{φnx, n ≥
1}a sequence of nondecreasing and nonnegative Borel functions withφnx/y≤ φnx/φnyon
0,∞. Lethnxbe defined asTheorem 2.1and
φnx
φndn ≥hnx, x∈
0,∞, 2.12
where{dn, n≥1}is a sequence of positive real numbers. Set
B
ω:
∞
n 1
Eφn|Xn|| Fn−1
φndn <∞
. 2.13
Under the same conditions (i) and (ii) as inTheorem 2.1,∞n 1d−1n Xn and
∞
n 1dn−1Xn −EXn |
Fn−1converge a.e. onB, respectively.
Remark 2.7. By using Kronecker’s lemma, ifdn↑ ∞, we have
lim
n→ ∞
1
dn n
k 1
Xk 0 a.e.onB, 2.14
lim
n→ ∞
1
dn n
k 1
Xk−EXk| Fk−1 0 a.e.onB, 2.15
respectively.
Theorem 2.8. Let{Xn,Fn, n ≥ 1}be a stochastic sequence defined as inSection 1,{ξn, n ≥ 1}a
sequence of nonzero random variables such thatξnisFn−1-measurable, andcn≥1,n≥1a sequence
of real numbers. Letφnx, ϕnxbe two sequences of nonnegative Borel functions onR. Suppose that
forp≥2,φnx/xpdoes not decrease asx >0, and for0< x1< x2,
ϕnx1
xp1 ≤ φnx2
holds. Let
A
ω:
∞
n 1
ξ2
n
ϕn|ξn| E
φn|Xn|| Fn−1<∞
,
B
ω:
∞
n 1
ξ2
n
c2
n
<∞
.
2.17
Then,
∞
n 1
c−1n Xn−EXn| Fn−1 converges a.e.on AB. 2.18
Furthermore, ifcn↑ ∞, one has
lim
n→ ∞
1
cn n
k 1
Xk−EXk| Fk−1 0 a.e.on AB. 2.19
Corollary 2.9Chow. Let{Xn,Fn, n≥1}be a sequence of martingale differences, and let{an, n≥
1}be a sequence of positive real numbers with∞n 1an<∞. Forp≥2, let
∞
n 1
a1−np/2E
|Xn|p| Fn−1<∞. 2.20
Then,
∞
n 1
Xn converges a.e. 2.21
Corollary 2.10. Let{Xn,Fn, n≥1}be an arbitrary stochastic sequence. Forp≥2, let
A
ω:
∞
n 1
nlognp/2−1E|Xn|p| Fn−1<∞
. 2.22
Then,
∞
n 1
1
nXn−EXn| Fn−1converges a.e.on A,
lim
n→ ∞
1
n
n
k 1
Xk−EXk| Fk−1 0 a.e.on A,
2.23
Proof. It is a special case ofTheorem 2.8whenξn
logn,cn n,φnx |x|p, andϕnx
|x|p/np/2−1lognp−2here, we setϕ
1x |x|p.
3. Proofs of Theorems
We first give a lemma.
Lemma 3.1see1. Let{Sn
n
i 1Xi,Fn, n≥1}be a martingale. Then, for some1≤p≤2,Sn
converges a.e. on the set{∞i 1EXip| Fi−1<∞}.
Proof ofTheorem 2.1. LetX∗n XnI|Xn|≤cnandka positive integer number. LetZn φn|Xn|,
Ak
ω:
∞
n 1
EZn| Fn−1≤k
,
τk min
n:n≥1,
n1
i 1
EZi| Fi−1> k
,
3.1
whereτk ∞, if the right-hand side of18is empty. Then,
τk
n 1Zn
∞
n 1Iτk≥nZn. Since Iτk≥nis measurableFn−1, andZnis nonnegative, we have
E
τ
k
n 1
Zn E
∞
n 1
Iτk≥nZn
E
∞
n 1
EIτk≥nZn| Fn−1
≤E
∞
n 1
EZn| Fn−1
≤k.
3.2
SinceAk {τk ∞}, we have by3.2
∞
n 1
Ak ZndP
∞
n 1
EIAkZn
≤E
τ
k
n 1
Zn
≤k.
By2.1,2.2, and3.3, we obtain
∞
n 1
PAkXn∗/Xn
∞
n 1
AkXn∗/Xn dP
≤∞
n 1
1
b
Ak|Xn|>cn
b|Xn|dP
≤ 1 b
∞
n 1
Ak|Xn|>cn ZndP
≤ 1 b
∞
n 1
Ak ZndP
≤ k b.
3.4
It follows from Borel-Cantelli lemma and3.4thatPAkXn∗/Xni.o. 0 holds. Hence, we
have
∞
n 1
Xn−X∗n converges a.e.onAk. 3.5
SinceA kAk, it follows from3.5that
∞
n 1
Xn−Xn∗ converges a.e.onA. 3.6
Let
Yn Xn∗−EXn∗| Fn−1. 3.7
It is clear that{Yn,Fn, n≥1}is a sequence for martingale difference. By using Cr inequality,
we have
EYnp| Fn−1
≤ 2pEX∗
np| Fn−1≤2pE|Xn∗|p| Fn−1 a.e. 3.8
By using2.1and2.2, we have
|X∗
n|p≤
1
a|X∗|φn|Xn∗|≤
1
aφn|X
∗
n|. 3.9
Thus, the following inequality holds from2.3,3.8, and3.9
∞
n 1
EYnp| Fn−1
By usingLemma 3.1, we obtain
∞
n 1
Yn converges a.e.onA. 3.11
Hence, it follows from3.6,3.7, and3.11that
∞
n 1
Xn−EXn∗| Fn−1 converges a.e.onA. 3.12
The following argument breaks down into two cases.
Case 1. If there exists somec >0 such that2.4holds when 0≤x≤cn, then
∞
n 1
EX∗
n| Fn−1≤
1
c
∞
n 1
Eφn|X∗n|| Fn−1
≤ 1 c
∞
n 1
Eφn|Xn|| Fn−1 a.e.
3.13
By using2.3and3.13, we obtain
∞
n 1
EX∗
n| Fn−1 converges a.e.on A. 3.14
It follows from3.12and3.14that2.5holds.
Case 2. If there exists somec >0 such that the inequality2.4holds whenx > cn, then
|EXn| Fn−1−EX∗n| Fn−1|
≤E|Xn−X∗n| | Fn−1
≤E|Xn| | Fn−1
≤ 1 cE
φn|Xn|| Fn−1 a.e.
3.15
By using2.3and3.15, we obtain that
∞
n 1
EXn| Fn−1−EXn∗| Fn−1 converges a.e.on A. 3.16
Proof ofTheorem 2.3. Since∞n 1Eφn|Xn| n∞1E{Eφn|Xn|| Fn−1}, we have by2.9
∞
n 1
EEφn|Xn|| Fn−1<∞. 3.17
It follows from the nonnegative property ofφnxthat
∞
n 1
Eφn|Xn|| Fn−1 converges a.e. 3.18
That is PA 1. By Theorem 2.1, the conclusion of Theorem 2.3 holds. The theorem is proved.
Proof ofTheorem 2.6. It is a similar way withTheorem 2.1exceptZn φn|Xn|/φndn.
Proof ofTheorem 2.8. Forn≥1, letZn Xn/cn, Yn Zn−EZn| Fn−1. Then,{Yn,Fn, n≥1}is
a martingale-difference sequence. It follows fromp≥2 and Jensen’s inequality that
EYn2 | Fn−1
EZ2n| Fn−1
−E2Zn| Fn−1
≤EZ2n| Fn−1
E|Zn|p·2/p| Fn−1
≤E2/p|Zn|p| Fn−1 a.e.
3.19
Furthermore,
E2/p|Zn|p| Fn−1
E2/p|Zn|p| Fn−1
IE2/p|Z
n|p|Fn−1≤ξn2/c2nIE2/p|Zn|p|Fn−1>ξ2n/c2n
≤ ξ2n
cn2
E2/p|Zn|p| Fn−1I
E
2/p|Z
n|p| Fn−1
ξ2
n/c2n
>1
≤ ξ2n
cn2
ξn2
c2n
E2/p|Z
n|p| Fn−1
ξ2n/c2n
p/2
I
E
2/p|Z
n|p| Fn−1
ξ2
n/cn2
>1
≤ ξ2n
c2
n
ξn2
c2
n
E
|X
n|p
|ξn|p| Fn−1
a.e.
3.20
It follows from2.16that
|Xn|p
|ξn|p
≤ φn|Xn|
ϕn|ξn|
holds when|ξn|<|Xn|. By3.21, we have
E
|X
n|p
|ξn|p | Fn−1
E
|X
n|p
|ξn|p
I|Xn|≤|ξn|| Fn−1
E
|X
n|p
|ξn|p
I|Xn|>|ξn|| Fn−1
≤1E
φn|Xn|
ϕn|ξn| | Fn−1
a.e.
3.22
Note thatcn≥1, it follows from3.19,3.21, and3.22that
EYn2| Fn−1
≤2ξ
2
n
cn2
ξn2
c2n
E
φn|Xn|
ϕn|ξn| | Fn−1
≤2ξ
2
n
cn2
ξ2
n
ϕn|ξn| E
φn|Xn|| Fn−1 a.e.
3.23
And it follows from2.17and3.23that
∞
n 1
EYn2| Fn−1
≤2 ∞ n 1 ξ2 n c2 n ∞ n 1 ξ2 n
ϕn|ξn| E
φn|Xn|| Fn−1<∞ a.e.onAB. 3.24
It follows from Lemma 3.1that 2.18 holds. Furthermore, when cn ↑ ∞, it follows from
Kronecker’s lemma that2.19holds. The theorem is proved.
Acknowledgment
This work was supported by National Natural Science Foundation of China no. 11071104 and Hebei Natural Science Foundation no. F2010001044.
References
1 W. F. Stout,Almost Sure Convergence, Probability and Mathematical Statistics, Vol. 2, Academic Press, London, UK, 1974.
2 V. V. Petrov,Sums of Independent Random Variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 8, Springer, New York, NY, USA, 1975.
3 W. Liu and W. Yang, “A class of strong limit theorems for the sequences of arbitrary random variables,”Statistics & Probability Letters, vol. 64, no. 2, pp. 121–131, 2003.
4 K. L. Chung,A Course in Probability Theory, Probability and Mathematical Statistics, Vol. 2, Academic Press, London, UK, 2nd edition, 1974.
5 Y. S. Chow and H. Teicher,Probability Theory: Independence, Interchangeability, Martingales, Springer Texts in Statistics, Springer, New York, NY, USA, 2nd edition, 1988.
6 W. Yang, “Strong limit theorem for arbitrary stochastic sequences,”Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1445–1451, 2007.
7 C. Jardas, J. Peˇcari´c, and N. Sarapa, “A note on Chung’s strong law of large numbers,”Journal of Mathematical Analysis and Applications, vol. 217, no. 1, pp. 328–334, 1998.
9 D. Freedman, “A remark on the strong law,”Annals of Probability, vol. 2, pp. 324–327, 1974.
10 R. Isaac, “On equitable ratios of Dubins-Freedman type,”Statistics & Probability Letters, vol. 42, no. 1, pp. 1–6, 1999.
11 D. H. Qiu and X. Q. Yang, “A class of strong limit theorems for sequences of arbitrary random variables,”Acta Sciences Natural University Normal Hunan, vol. 22, no. 4, pp. 9–15, 1999Chinese.
12 X. Wang and H. Guo, “Some strong limit theorems for arbitrary stochastic sequence,” inProceeding of the 2nd International Joint Conference on Computational Sciences and Optimization, pp. 423–426, Sanya, China, April 2009.
13 X. S. Wang and W. G. Yang, “Strong convergence theorems for arbitrary stochastic sequence series,”
Mathematics in Practice and Theory, vol. 35, no. 8, pp. 229–232, 2005Chinese.
