Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2014, Article ID 412758, 5 pages
http://dx.doi.org/10.1155/2014/412758
Research Article
A Strong Law of Large Numbers for Weighted Sums of
i.i.d. Random Variables under Capacities
Defei Zhang and Ping He
School of Mathematics, Honghe University, Mengzi 661199, China
Correspondence should be addressed to Ping He; [email protected]
Received 16 February 2014; Accepted 3 June 2014; Published 16 June 2014
Academic Editor: Abdel-Maksoud A. Soliman
Copyright © 2014 D. Zhang and P. He. 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.
With the notion of independent identically distributed (i.i.d.) random variables under sublinear expectations initiated by Peng,
a strong law of large numbers for weighted sums of i.i.d. random variables under capacities induced by sublinear expectations is
obtained.
1. Introduction
The strong law of large numbers plays important role in
the development of probability theory and mathematical
statistics; many studies about the extension of it have been
completed by many authors. For example, Chow and Lai
[1], Stout [2], Choi and Sung [3], Cuzick [4], Rosalsky and
Sreehari [5], Wu [6], Bai and Cheng [7], Bai et al. [8], and
so forth investigated the almost sure limiting behavior of
weighted sums of i.i.d. random variables. In fact, the additivity of probability and expectations is not reasonable in many
areas of applications because many uncertain phenomena
cannot be well modeled using additive probabilities or linear
expectations (see, e.g., Chen and Epstein [9], Huber and
Strassen [10], and Wakker [11]). In the case of nonadditive
probabilities, Marinacci [12] proved several limit laws for
nonadditive probabilities and Maccheroni and Marinacci [13]
obtained a strong law of large numbers for totally monotone
capacities.
Recently, motivated by the risk measures, superhedge
pricing, and modelling uncertainty in finance, Peng [14]
introduced the notion of sublinear expectation space, which
is a generalization of probability space. Together with the
notion of sublinear expectation, Peng also introduced the
notions about i.i.d., 𝐺-normal distribution, and 𝐺-Brownian
motion. Under this framework, the weak law of large numbers and the central limit theorems under sublinear expectations were obtained in the studies by Peng in [15, 16]. Soon
thereafter, Denis et al. [17] introduced the function spaces
and capacity related to a sublinear expectation. Chen et al.
[18] proved a strong law of large numbers for nonadditive
probabilities.
A natural question is the following: can we investigate
strong laws of large numbers for weighted sums of random
variables under capacities? Indeed, the goal of this paper is
to discuss the strong laws of large numbers for weighted
sums of i.i.d. random variables under capacities. Under some
assumptions, we obtain a strong law of large numbers for
weighted sums of i.i.d. random variables under capacities.
The paper is organized as follows: in Section 2, we give
some definitions and lemmas that are useful in this paper. In
Section 3, we give our main results including the proofs.
2. Preliminaries
In this section, we present some preliminaries in the theory
of sublinear expectations and capacities. More details of this
section can be found in the studies by Chen et al. [18] and
Peng [19].
Let (Ω, F) be a measurable space, and let H be the set of
random variables on (Ω, F).
̂ is a functional E
̂ :
Definition 1. A sublinear expectation E
H → 𝑅 satisfying the following:
̂
̂
(i) monotonicity: E[𝑋]
≥ E[𝑌]
if 𝑋 ≥ 𝑌;
̂
(ii) constant preserving: E[𝐶]
= 𝐶 for 𝐶 ∈ 𝑅;
2
Journal of Applied Mathematics
̂ + 𝑌] ≤ E[𝑋]
̂
̂
(iii) subadditivity: E[𝑋
+ E[𝑌];
̂
̂
(iv) positive homogeneity: E[𝜆𝑋]
= 𝜆E[𝑋]
for 𝜆 ≥ 0.
Artzner et al. [20] showed that a sublinear expectation can
be expressed as a supremum of linear expectations. That is, if
̂ is a sublinear expectation on H, then there exists a set (say
E
P) of probability measures such that
̂ [𝑋] = sup 𝐸𝑃 [𝑋] ,
E
̂ [−𝑋] = inf 𝐸𝑃 [𝑋] ,
−E
𝑃∈P
𝑃∈P
∀𝑋 ∈ H.
(1)
For this P, following Huber and Strassen [10], we define a
pair (𝐶, 𝐶) of capacities denoted by
𝐶 (𝐴) := sup 𝑃(𝐴) ,
𝐶 (𝐴) := inf 𝑃(𝐴) ,
𝑃∈P
𝑃∈P
∀𝐴 ∈ F. (2)
𝐶 (𝑋 ∈ 𝐷, 𝑌 ∈ 𝐺) = 𝐶 (𝑋 ∈ 𝐷) 𝐶 (𝑌 ∈ 𝐺)
holds for both capacities 𝐶 and 𝐶.
Borel-Cantelli lemma is still true for capacities 𝐶 and 𝐶
under some assumptions.
Lemma 6 (see Chen et al. [18]). Let {𝐴 𝑛 , 𝑛 ≥ 1} be a sequence
of events in F.
∞
∞
(1) If ∑∞
𝑛=1 𝐶(𝐴 𝑛 ) < ∞, then 𝐶(⋂𝑛=1 ⋃𝑖=𝑛 𝐴 𝑖 ) = 0.
(3)
where 𝐴 is the complement set of 𝐴.
It is easy to check that 𝐶 and 𝐶 are two continuous
capacities in the sense of the following definition.
Definition 2. A set function 𝐶: F → [0, 1] is called a continuous capacity if it satisfies
(1) 𝐶(𝜙) = 0, 𝐶(Ω) = 1;
𝑖=1
𝑖=1
(7)
∞
∞
If ∑∞
𝑛=1 𝐶(𝐴 𝑛 ) = ∞, then 𝐶(⋂𝑛=1 ⋃𝑖=𝑛 𝐴 𝑖 ) = 1.
3. Main Results
Theorem 7. Let {𝑋𝑖 , 𝑖 ≥ 1} be a sequence of i.i.d. random
̂ 1 ], 𝜇 = −E[−𝑋
̂
variables in H satisfying 𝜇 = E[𝑋
1 ] and for
any ℎ, 𝑟 > 0
(3) 𝐶(𝐴 𝑛 ) ↑ 𝐶(𝐴), if 𝐴 𝑛 ↑ 𝐴;
(4) 𝐶(𝐴 𝑛 ) ↓ 𝐶(𝐴), if 𝐴 𝑛 ↓ 𝐴, where 𝐴 𝑛 , 𝐴 ∈ F.
𝑟
̂ [𝑒(ℎ|𝑋𝑖 | ) ] < ∞.
E
Definition 3 (see Peng [19]).
Identical Distribution. Let 𝑋1 and 𝑋2 be two 𝑛-dimensional
random vectors in H. They are called identically distributed,
𝑑
denoted by 𝑋1 = 𝑋2 , if, for each measurable function 𝜑 on
𝑅𝑛 such that 𝜑(𝑋1 ), 𝜑(𝑋2 ) ∈ H, one has
(5)
Remark 4. A sequence of random variables {𝑋𝑖 , 𝑖 ≥ 1} is
said to be i.i.d., if 𝑋𝑖 = 𝑋1 and 𝑋𝑖+1 is independent of 𝑌 :=
(𝑋1 , . . . , 𝑋𝑖 ) for each 𝑖 ≥ 1.
The following lemma shows the relation between Peng’s
independence and pairwise independence in the study by
Marinacci in [12].
(8)
Let {𝑎𝑛𝑖 , 1 ≤ 𝑖 ≤ 𝑛, 𝑛 ≥ 1} be an array of constants satisfying
𝐴 𝛼 = lim sup𝐴 𝛼,𝑛 < ∞,
𝑛→∞
𝐴𝛼𝛼,𝑛 =
(4)
Independence. A random vector 𝑌 := (𝑌1 , . . . , 𝑌𝑛 ), 𝑌𝑖 ∈ H,
is said to be independent of another random vector 𝑋 :=
̂ if, for each measurable
(𝑋1 , . . . , 𝑋𝑚 ), 𝑋𝑖 ∈ H, under E
function 𝜑 on 𝑅𝑚 × 𝑅𝑛 with 𝜑(𝑋, 𝑌) ∈ H and 𝜑(𝑥, 𝑌) ∈ H
for each 𝑥 ∈ 𝑅𝑚 , one has
𝑑
∞
In this section, we give our main results including the proofs.
(2) 𝐶(𝐴) ≤ 𝐶(𝐵), whenever 𝐴 ⊂ 𝐵 and 𝐴, 𝐵 ∈ F;
̂ [𝜑 (𝑋, 𝑌)] = E
̂ [E[𝜑
̂ (𝑥, 𝑌)] ] .
E
𝑥=𝑋
∞
𝐶 (⋂𝐴𝑐𝑖 ) = ∏𝐶 (𝐴𝑐𝑖 ) .
𝑐
̂ [𝜑 (𝑋1 )] = E
̂ [𝜑 (𝑋2 )] .
E
(6)
(2) Suppose that {𝐴 𝑛 , 𝑛 ≥ 1} are pairwise independent
with respect to 𝐶; that is,
Obviously,
𝐶 (𝐴) + 𝐶 (𝐴𝑐 ) = 1,
Lemma 5 (see Chen et al. [18]). Suppose that 𝑋, 𝑌 ∈ H are
̂ is a sublinear expectation and (𝐶, 𝐶)
two random variables. E
̂ If random variable 𝑋 is
is the pair of capacities induced by E.
̂ then 𝑋 is also pairwise independent
independent of 𝑌 under E,
of 𝑌 under capacities 𝐶 and 𝐶; that is, for all subsets 𝐷 and
𝐺 ⊂ 𝑅,
󵄨 󵄨𝛼
∑𝑛𝑖=1 󵄨󵄨󵄨𝑎𝑛𝑖 󵄨󵄨󵄨
,
𝑛
(9)
(1 < 𝛼 ≤ 2) .
Then, for 0 < 𝑟 ≤ 1, if
𝜇 (1 − lim sup
𝑛→∞
∑𝑛𝑖=1 𝑎𝑛𝑖
) ≥ 0,
𝑏𝑛
(10)
we have
𝐶 (𝜇 ≤ lim inf
𝑛→∞
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
𝑏𝑛
∑𝑛 𝑎 𝑋
≤ lim sup 𝑖=1 𝑛𝑖 𝑖 ≤ 𝜇) = 1,
𝑏𝑛
𝑛→∞
where 𝑏𝑛 = 𝑛1/𝛼 log1/𝑟 𝑛.
(11)
Journal of Applied Mathematics
3
Moreover, for 𝑟 > 1, if 𝜇(1−lim sup𝑛 → ∞ (∑𝑛𝑖=1 𝑎𝑛𝑖 /𝑏𝑛 )) ≥ 0,
we have
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
𝐶 (𝜇 ≤ lim inf
𝑛→∞
𝑏𝑛
𝑛
𝐶 (lim sup∑𝑎𝑛𝑖 𝑋𝑛𝑖 > 𝜀) = 0 ∀𝜀 > 0.
𝑛 → ∞ 𝑖=1
∑𝑛 𝑎 𝑋
≤ lim sup 𝑖=1 𝑛𝑖 𝑖 ≤ 𝜇) = 1,
𝑏𝑛
𝑛→∞
(12)
(1+(𝛼−1)(𝑟−1))/(1+𝛼(𝑟−1))
1/𝛼
Using Lemma 6, we have
where 𝑏𝑛 = 𝑛 (log 𝑛)
.
In order to prove Theorem 7, we need the following lemma.
Lemma 8. Let {𝑋𝑖 , 𝑖 ≥ 1} be a sequence of i.i.d. random
variables satisfying (8), and let {𝑎𝑛𝑖 , 1 ≤ 𝑖 ≤ 𝑛, 𝑛 ≥ 1} be an
array of constants. Truncate |𝑋𝑖 − 𝜇| at Δ 𝑛 and denote 𝑋𝑛𝑖 :=
(𝑋𝑖 − 𝜇)𝐼{|𝑋𝑖 −𝜇|≤Δ 𝑛 } . Suppose that the following conditions hold:
(1) |𝑎𝑛𝑖 𝑋𝑛𝑖 | ≤ 𝐶|𝑋𝑖 |𝛽 / log 𝑛 𝑎.𝑠. 𝐶, for some 0 < 𝛽 ≤ 𝑟
and some constant 𝐶 > 0;
2
2
∑𝑛𝑖=1 𝑎𝑛𝑖
≤ 𝑢𝑛 |𝑋𝑖 |𝛿 / log 𝑛 𝑎.𝑠. 𝐶, for some 𝛿 > 0
(2) 𝑋𝑛𝑖
and some sequence {𝑢𝑛 } of constants such that 𝑢𝑛 → 0.
The proof is complete.
Proof of Theorem 7. The proof of (11) is similar to that of
󸀠
:= (𝑋𝑖 −
(12); we only prove (12). We denote 𝑋𝑛𝑖
󸀠󸀠
󸀠󸀠󸀠
𝜇)𝐼{|𝑋𝑖 −𝜇|≤(log 𝑛)𝛿1 } , 𝑋𝑛𝑖 := (𝑋𝑖 − 𝜇)𝐼{|𝑋𝑖 −𝜇|≤(log 𝑛)1/𝑟 } , and 𝑋𝑛𝑖
:=
(𝑋𝑖 − 𝜇)𝐼{(log 𝑛)𝛿1 ≤|𝑋𝑖 −𝜇|≤(log 𝑛)1/𝑟 } for 1 ≤ 𝑖 ≤ 𝑛 and 𝑛 ≥ 1,
󸀠
where 𝛿1 = 1/(1 + 𝛼(𝑟 − 1)). Denote 𝑎𝑛𝑖
:= 𝑎𝑛𝑖 𝐼{|𝑎𝑛𝑖 |≤𝑛1/𝛼 (log 𝑛)𝛿2 }
󸀠󸀠
󸀠
and 𝑎𝑛𝑖 := 𝑎𝑛𝑖 − 𝑎𝑛𝑖 for 1 ≤ 𝑖 ≤ 𝑛 and 𝑛 ≥ 1, where
𝛿2 = (𝑟 − 1)/(1 + 𝛼(𝑟 − 1)). Then,
󸀠
𝑛
󸀠
∑𝑛𝑖=1 𝑎𝑛𝑖 (𝑋𝑖 − 𝜇) ∑𝑖=1 𝑎𝑛𝑖 𝑋𝑛𝑖
∑𝑛 𝑎󸀠󸀠 𝑋󸀠
=
+ 𝑖=1 𝑛𝑖 𝑛𝑖
𝑏𝑛
𝑏𝑛
𝑏𝑛
+
Then,
𝑛 → ∞ 𝑖=1
(13)
Proof. From the inequality 𝑒𝑥 ≤ 1 + 𝑥 + (1/2)𝑥2 𝑒|𝑥| for all
𝑥 ∈ 𝑅, we have
̂ [𝑋2 𝑒𝑡|𝑎𝑛𝑖 𝑋𝑛𝑖 | ]
̂ [𝑒𝑡𝑎𝑛𝑖 𝑋𝑛𝑖 ] ≤ 1 + 1 𝑡2 𝑎2 E
E
𝑛𝑖
2 𝑛𝑖
(14)
for any 𝑡 > 0. Let 𝜀 > 0 be given. We set 𝑡 = 2 log 𝑛/𝜀 and
obtain by (1) and (2) in Lemma 8 and (8) that
󵄨󵄨 −1 󸀠 󸀠 󵄨󵄨
󵄨󵄨𝑏𝑛 𝑎𝑛𝑖 𝑋𝑛𝑖 󵄨󵄨 ≤
󵄨
󵄨
𝑛
󸀠2
󸀠2
𝑋𝑛𝑖
≤
∑𝑏𝑛−2 𝑎𝑛𝑖
𝑖=1
≤
2
𝛽
2𝑢𝑛 log 𝑛 𝑎𝑛𝑖
̂ [󵄨󵄨󵄨𝑋𝑖 󵄨󵄨󵄨𝛿 𝑒(2/𝜀)𝐶|𝑋𝑖 | ]
E
𝑛
󵄨
󵄨
2
2
𝜀
∑𝑖=1 𝑎𝑛𝑖
2
𝛽
2𝐶 𝑢 log 𝑛 𝑎𝑛𝑖
̂ [𝑒𝐶(𝜀)|𝑋𝑖 | ]
E
≤ 1 + 1 𝑛2
𝑛
2
𝜀
∑𝑖=1 𝑎𝑛𝑖
󵄨 󵄨𝛼
𝑛(2−𝛼)/𝛼 ∑𝑛𝑖=1 󵄨󵄨󵄨𝑎𝑛𝑖 󵄨󵄨󵄨
(2−𝛼)𝛿2
𝑏𝑛2 (log 𝑛)
(2+𝛼(𝑟−1))/(1+𝛼(𝑟−1))
(log 𝑛)
(19)
𝑋𝑖2
𝐵𝑛 ≤
]
󵄨 󵄨𝛼
× ∑󵄨󵄨󵄨𝑎𝑛𝑖 󵄨󵄨󵄨 = 𝐴𝛼𝛼,𝑛
= 𝑛−3/2 .
a.s. 𝐶.
Namely,
𝐶 (lim sup𝐵𝑛 ≤ 𝐴𝛼𝛼,𝑛 ) = 1.
(16)
(21)
𝑖=1
𝑛→∞
2
log 𝑛 𝑎𝑛𝑖
1 𝑛
exp
{
}
∏
2
𝑛2 𝑖=1
2 ∑𝑛𝑖=1 𝑎𝑛𝑖
(20)
(1+(𝛼−1)(𝑟−1))/(1+𝛼(𝑟−1))
(log) 1 𝑛 󵄨󵄨 󸀠󸀠 󵄨󵄨 (log 𝑛)
∑ 󵄨󵄨𝑎 󵄨󵄨 ≤
𝑏𝑛 𝑖=1 󵄨 𝑛𝑖 󵄨
𝑏𝑛 𝑛(𝛼−1)/𝛼
𝑛
𝑖=1
≤
a.s. 𝐶.
For 𝐵𝑛 , we observe that
for all large 𝑛. For the large 𝑛, it follows by the Markov
inequality and (15) that
̂ [𝑒
𝐶 (∑𝑎𝑛𝑖 𝑋𝑛𝑖 > 𝜀) ≤ 𝑒−𝑡𝜀 E
󸀠2
𝑋𝑛𝑖
𝐴𝛼𝛼,𝑛
𝛿
𝑡 ∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑛𝑖
a.s. 𝐶,
𝜀
𝐶 (lim sup𝐴 𝑛 > ) = 0 ∀𝜀 > 0.
2
𝑛→∞
(15)
2
log 𝑛 𝑎𝑛𝑖
}
𝑛
2
2 ∑𝑖=1 𝑎𝑛𝑖
𝑛
󵄨 󵄨
󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨𝑋𝑖 󵄨󵄨󵄨
𝑋
=
󵄨
󵄨
𝑖
󵄨
󵄨
𝛿
log 𝑛
𝑏𝑛 (log 𝑛) 2
𝑛1/𝛼
Hence, by Lemma 8, we have
2
log 𝑛 𝑎𝑛𝑖
≤1+
2
2 ∑𝑛𝑖=1 𝑎𝑛𝑖
≤ exp {
(18)
For 𝐴 𝑛 , we will apply Lemma 8 to the random variable
󸀠
󸀠
𝑋𝑛𝑖
and weight 𝑏𝑛−1 𝑎𝑛𝑖
. Note that
2
̂ [𝑒𝑡𝑎𝑛𝑖 𝑋𝑛𝑖 ] ≤ 1 + 2log 𝑛 𝑎2 E
̂ 2 𝑡|𝑎𝑛𝑖 𝑋𝑛𝑖 | ]
E
𝑛𝑖 [𝑋𝑛𝑖 𝑒
𝜀2
≤1+
󸀠󸀠󸀠
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑛𝑖
∑𝑛 𝑎 𝑋󸀠󸀠
+ 𝑖=1 𝑛𝑖 𝑛𝑖
𝑏𝑛
𝑏𝑛
:= 𝐴 𝑛 + 𝐵𝑛 + 𝐶𝑛 + 𝐷𝑛 .
𝑛
𝐶 (lim sup∑𝑎𝑛𝑖 𝑋𝑛𝑖 > 𝜀) = 0 ∀𝜀 > 0.
(17)
(22)
By replacing 𝑋𝑖 by 𝜃𝑋𝑖 (𝜃 > 0), we have
lim sup𝐵𝑛 ≤
𝑛→∞
𝐴𝛼𝛼,𝑛
𝜃
a.s. 𝐶.
(23)
4
Journal of Applied Mathematics
Letting 𝜃 → ∞, we have
which implies
𝐶 (lim sup𝐵𝑛 ≤ 0) = 1.
(24)
𝑛→∞
󵄨󵄨 𝑎 𝑋󸀠󸀠󸀠 󵄨󵄨 𝐴
󵄨󵄨 𝑛𝑖 𝑛𝑖 󵄨󵄨
𝛼,𝑛 󵄨󵄨 󵄨󵄨𝑟
󵄨󵄨
󵄨
󵄨󵄨 𝑏𝑛 󵄨󵄨󵄨 ≤ log 𝑛 󵄨󵄨𝑋𝑖 󵄨󵄨
󵄨
󵄨
≤
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
> 𝜇) = 0.
𝑏𝑛
(31)
𝐶 (lim sup
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
≤ 𝜇) = 1.
𝑏𝑛
(32)
𝑛→∞
For 𝐶𝑛 , note that
󸀠󸀠󸀠2 𝑛
2
𝑋𝑛𝑖
∑𝑖=1 𝑎𝑛𝑖
2
𝑏𝑛
𝐶 (lim sup
𝐴2𝛼,𝑛
2
log 𝑛
Also,
a.s. 𝐶,
󵄨󵄨 󵄨󵄨2𝑟
󵄨󵄨𝑋𝑖 󵄨󵄨
(25)
a.s. 𝐶.
𝑛→∞
Similarly, considering the sequence {−𝑋𝑖 , 𝑖 ≥ 1}, from
(29), we have
Then, by Lemma 8, we have
𝜀
𝐶 (lim sup𝐶𝑛 > ) = 0 ∀𝜀 > 0.
2
𝑛→∞
(26)
For 𝐷𝑛 , assumption (8) implies ∑𝑛𝑖=1 𝐶(|𝑋𝑛 | > log1/𝑟 ) <
󸀠󸀠
| is bounded a.s. 𝐶. It
∞. Hence, by Lemma 6, ∑𝑛𝑖=1 |𝑋𝑛𝑖
follows that
󵄨 󸀠󸀠 󵄨󵄨
󵄨 󵄨
󵄨󵄨
∑𝑛𝑖=1 󵄨󵄨󵄨𝑎𝑛𝑖 󵄨󵄨󵄨 ∑𝑛𝑖=1 󵄨󵄨󵄨󵄨𝑋𝑛𝑖
󵄨
𝐷𝑛 ≤
𝑏𝑛
≤
𝐴 𝛼,𝑛
(1+(𝛼−1)(𝑟−1))/(1+𝛼(𝑟−1))
(log 𝑛)
(27)
𝑛
󵄨 󸀠󸀠 󵄨󵄨
󵄨󵄨 󳨀→ 0 a.s. 𝐶
× ∑ 󵄨󵄨󵄨󵄨𝑋𝑛𝑖
󵄨
𝐶 (lim sup
𝑛→∞
∑𝑛𝑖=1 𝑎𝑛𝑖 (−𝑋𝑖 )
̂ [−𝑋1 ]) = 1.
≤E
𝑏𝑛
̂
Note that 𝜇 = −E[−𝑋
1 ]. So
𝐶 (lim sup
𝑛→∞
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
≥ 𝜇) = 1.
𝑏𝑛
(34)
Therefore, the proof of Theorem 7 is complete.
The following theorem shows that if the norming constant
𝑏𝑛 is stronger than that of Theorem 7, then condition (8) in
Theorem 7 can be replaced by a weaker condition.
Theorem 9. Let {𝑋𝑖 , 𝑖 ≥ 1} be a sequence of i.i.d. random
̂ 1 ], 𝜇 = −E[−𝑋
̂
variables in H satisfying 𝜇 = E[𝑋
1 ] and for
some ℎ, 𝑟 > 0
𝑖=1
as 𝑛 → ∞.
Thus,
𝑟
𝐶 (lim sup𝐷𝑛 ≤ 0) = 1.
𝑛→∞
(28)
𝑛→∞
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
≥ 𝜇 + 𝜀)
𝑏𝑛
≤ 𝐶 (lim sup
𝑛→∞
𝐶 (𝜇 ≤ lim sup
∑𝑛𝑖=1 𝑎𝑛𝑖
(𝑋𝑖 − 𝜇)
𝑏𝑛
≥ 𝜇 (1 − lim sup
𝑛→∞
∑𝑛𝑖=1 𝑎𝑛𝑖
) + 𝜀)
𝑏𝑛
𝑛→∞
(29)
𝑛→∞
𝑛→∞
∑𝑛𝑖=1 𝑎𝑛𝑖
) + 𝜀) ,
𝑏𝑛
𝑛→∞
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
≥ 𝜇 + 𝜀) = 0 ∀𝜀 > 0,
𝑏𝑛
∑𝑛 𝑎 𝑋
≤ lim sup 𝑖=1 𝑛𝑖 𝑖 ≤ 𝜇) = 1,
𝑏𝑛
𝑛→∞
𝐶 (𝜇 ≤ lim sup
and the condition 𝜇(1 − lim sup𝑛 → ∞ ∑𝑛𝑖=1 𝑎𝑛𝑖 /𝑏𝑛 ) ≥ 0: from
(20), (24), (26), and (28), we conclude that
𝐶 (lim sup
(35)
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
𝑏𝑛
where 𝑏𝑛 = 𝑛1/𝛼 (log 𝑛)1/𝑟+𝛽 .
Moreover, for 𝑟 > 1 and 𝛽
lim sup𝑛 → ∞ (∑𝑛𝑖=1 𝑎𝑛𝑖 /𝑏𝑛 )) ≥ 0, we have
= 𝐶 (lim sup (𝐴 𝑛 + 𝐵𝑛 + 𝐶𝑛 + 𝐷𝑛 )
≥ 𝜇 (1 − lim sup
̂ [𝑒(ℎ|𝑋𝑖 | ) ] < ∞.
E
Let {𝑎𝑛𝑖 , 1 ≤ 𝑖 ≤ 𝑛, 𝑛 ≥ 1} be an array of constants satisfying
(9) in Theorem 7. Then, for 0 < 𝑟 ≤ 1 and 𝛽 > 0, if 𝜇(1 −
lim sup𝑛 → ∞ (∑𝑛𝑖=1 𝑎𝑛𝑖 /𝑏𝑛 )) ≥ 0, we have
On the other hand, ∀𝜀 > 0; and we note that
𝐶 (lim sup
(33)
(30)
𝑛→∞
>
0, if 𝜇(1 −
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
𝑏𝑛
∑𝑛 𝑎 𝑋
≤ lim sup 𝑖=1 𝑛𝑖 𝑖 ≤ 𝜇) = 1,
𝑏𝑛
𝑛→∞
where 𝑏𝑛 = 𝑛1/𝛼 (log 𝑛)(1+(𝛼−1)(𝑟−1))/(1+𝛼(𝑟−1))+𝛽 .
(36)
(37)
Journal of Applied Mathematics
5
Proof. We can prove that Lemma 8 is also true except that (8)
and (1) of Lemma 8 are replaced by (35) and the following
condition.
|𝑎𝑛𝑖 𝑋𝑛𝑖 | ≤ V𝑛 |𝑋𝑖 |𝛽 / log 𝑛 a.s. 𝐶, for some 0 < 𝛽 ≤ 𝑟 and
some sequence {V𝑛 } of constants such that V𝑛 → 0.
󸀠
= 𝑋𝑖 𝐼{|𝑋𝑖 |≤(ℎ−1 log 𝑛)1/𝑟 } and
For the case 0 < 𝑟 ≤ 1, we let 𝑋𝑛𝑖
󸀠󸀠
󸀠
𝑋𝑛𝑖 = 𝑋𝑖 − 𝑋𝑛𝑖 for 1 ≤ 𝑖 ≤ 𝑛 and 𝑛 ≥ 1. For the case 𝑟 > 1, we
󸀠
󸀠󸀠
let 𝑋𝑛𝑖
= 𝑋𝑖 𝐼{|𝑋𝑖 |≤(log 𝑛)1/(1+𝛼(𝑟−1)) } , 𝑋𝑛𝑖
= 𝑋𝑖 𝐼{|𝑋𝑖 |>ℎ−1 log1/𝑟 𝑛} , and
󸀠󸀠󸀠
𝑋𝑛𝑖 = 𝑋𝑖 𝐼{(log 𝑛)1/(1+𝛼(𝑟−1)) <|𝑋𝑖 |≤(ℎ−1 log 𝑛)1/𝑟 } . The rest of the proof
is similar to that of Theorem 7 and is omitted.
Remark 10. If 𝜇 = 𝜇 = 0 in Theorem 9, then, for 0 < 𝑟 ≤ 1
and 𝛽 > 0, we have
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
󳨀→ 0 a.s.,
𝑏𝑛
(38)
where 𝑏𝑛 = 𝑛1/𝛼 (log 𝑛)1/𝑟+𝛽 . Moreover, if 𝑟 > 1, we have
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
󳨀→ 0 a.s.,
𝑏𝑛
(39)
where 𝑏𝑛 = 𝑛1/𝛼 (log 𝑛)(1+(𝛼−1)(𝑟−1))/(1+𝛼(𝑟−1))+𝛽 . Since if 𝛼 > 1
and 𝑟 ≥ 1, then 𝑟(𝛼 − 1)/𝛼(1 + 𝑟) > 0 and (1 + (𝛼 − 1)(𝑟 −
1))/(1 + 𝛼(𝑟 − 1)) < 1/𝑟 + 𝑟(𝛼 − 1)/𝛼(1 + 𝑟), we also have
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
󳨀→ 0 a.s.,
𝑏𝑛
(40)
where 𝑏𝑛 = 𝑛1/𝛼 (log 𝑛)1/𝑟+𝑟(𝛼−1)/𝛼(1+𝑟) . The result is similar to
Theorem 2.2 of Bai and Cheng [7].
Remark 11. If 𝜇 = 𝜇 = 0 in Theorem 7, we can get the
classical strong law of large numbers for weighted sums of
i.i.d. random variables as follows:
∑𝑛𝑖=1 𝑎𝑛𝑖 𝑋𝑖
= 0) = 1.
𝑛→∞
𝑏𝑛
𝑃 ( lim
(41)
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
Acknowledgments
The authors thank Professor Zengjing Chen for his helpful discussion and suggestions and the reviewers for their
valuable comments and suggestions to improve the presentation of this paper. This work is partially supported
by the National Science Foundation of China (11301160),
the Scientific Research Foundation of Yunnan Province
(2013FZ116), the Scientific Research Foundation of Yunnan
Province Education Committee (2011C120), the Reserve Talents Foundation of Honghe University (2014HB0204), and
the Curricula Construction Foundations of Honghe University (ZYDT1308, ZDKC1111).
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