The author studies the laws of large numbers for weighted sums of negatively superadditive-dependent random variables. The obtained results in this paper extend and improve the corresponding theorems of Yang et al. [Commun. Stat. Theor. M., 48 (2019), 3044-3054]. Moreover, the author obtains a new theorem of mean convergence for weighted sums of negatively superadditive-dependent random variables, which was not considered in Yang et al. (2019).
Citation: Yongfeng Wu. Limit theorems for negatively superadditive-dependent random variables with infinite or finite means[J]. AIMS Mathematics, 2023, 8(11): 25311-25324. doi: 10.3934/math.20231291
[1] | Mingzhou Xu . Complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations. AIMS Mathematics, 2023, 8(8): 19442-19460. doi: 10.3934/math.2023992 |
[2] | Yuyan Wei, Xili Tan, Peiyu Sun, Shuang Guo . Weak and strong law of large numbers for weakly negatively dependent random variables under sublinear expectations. AIMS Mathematics, 2025, 10(3): 7540-7558. doi: 10.3934/math.2025347 |
[3] | Mingzhou Xu, Xuhang Kong . Note on complete convergence and complete moment convergence for negatively dependent random variables under sub-linear expectations. AIMS Mathematics, 2023, 8(4): 8504-8521. doi: 10.3934/math.2023428 |
[4] | Mingzhou Xu . Complete convergence of moving average processes produced by negatively dependent random variables under sub-linear expectations. AIMS Mathematics, 2023, 8(7): 17067-17080. doi: 10.3934/math.2023871 |
[5] | Haiye Liang, Feng Sun . Exponential inequalities and a strong law of large numbers for END random variables under sub-linear expectations. AIMS Mathematics, 2023, 8(7): 15585-15599. doi: 10.3934/math.2023795 |
[6] | He Dong, Xili Tan, Yong Zhang . Complete convergence and complete integration convergence for weighted sums of arrays of rowwise $ m $-END under sub-linear expectations space. AIMS Mathematics, 2023, 8(3): 6705-6724. doi: 10.3934/math.2023340 |
[7] | Shuyan Li, Qunying Wu . Complete integration convergence for arrays of rowwise extended negatively dependent random variables under the sub-linear expectations. AIMS Mathematics, 2021, 6(11): 12166-12181. doi: 10.3934/math.2021706 |
[8] | Lunyi Liu, Qunying Wu . Complete integral convergence for weighted sums of negatively dependent random variables under sub-linear expectations. AIMS Mathematics, 2023, 8(9): 22319-22337. doi: 10.3934/math.20231138 |
[9] | Baozhen Wang, Qunying Wu . Almost sure convergence for a class of dependent random variables under sub-linear expectations. AIMS Mathematics, 2024, 9(7): 17259-17275. doi: 10.3934/math.2024838 |
[10] | Mingzhou Xu . On the complete moment convergence of moving average processes generated by negatively dependent random variables under sub-linear expectations. AIMS Mathematics, 2024, 9(2): 3369-3385. doi: 10.3934/math.2024165 |
The author studies the laws of large numbers for weighted sums of negatively superadditive-dependent random variables. The obtained results in this paper extend and improve the corresponding theorems of Yang et al. [Commun. Stat. Theor. M., 48 (2019), 3044-3054]. Moreover, the author obtains a new theorem of mean convergence for weighted sums of negatively superadditive-dependent random variables, which was not considered in Yang et al. (2019).
Notation: For two sequences of positive constants {an,n≥1} and {bn,n≥1}, symbols an∼bn, an=O(bn) and an=o(bn) stand for liman/bn=1, liman/bn∈(0,∞) and liman/bn=0, respectively. For simplicity, we shall write P⟶, a.s.⟶ and Lp⟶ to express the convergence in probability, the almost certain convergence and p-mean convergence, respectively.
The following concept of superadditive function was introduced in [1].
Definition 1.1. A function ϕ:Rn→R is called superadditive if ϕ(x∨y)+ϕ(x∧y)≥ϕ(x)+ϕ(y) for all x,y∈Rn, where ∨ is for componentwise maximum and ∧ is for componentwise minimum.
Hu [2] introduced the concept of negatively superadditive-dependent (NSD) based on the above concept of superadditive function.
Definition 1.2. A random vector X=(X1,X2,⋯,Xn) is said to be NSD if
Eϕ(X1,X2,⋯,Xn)≤Eϕ(X∗1,X∗2,⋯,X∗n), |
where X∗1,X∗2,⋯,X∗n are independent such that X∗i and Xi have the same distribution for each i and ϕ is a superadditive function such that the expectations in the above equation exists. A sequence {Xn,n≥1} of random variables is said to be NSD if for each n≥1, (X1,X2,⋯,Xn) is NSD.
Hu [2] established some basic properties and three structural theorems of NSD random variables. An interesting example was also presented in [2], which illustrated that NSD is not necessarily negatively associated (NA, [3]). Christofides and Vaggelatou [4] showed that NA is NSD. Eghbal et al. [5] derived two maximal inequalities and strong law of large numbers of quadratic forms of NSD random variables. Shen et al. [6] studied almost sure convergence and strong stability for weighted sums of NSD random variables. Wang et al. [7] studied complete convergence for arrays of rowwise NSD random variables, with applications to nonparametric regression. For more research of the limit theory for NSD random variables, the author can refer the reader to [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].
NA random variable has been studied many times and attracted extensive attention, so it is very significant to investigate the limit theorems of this wider NSD class, which is highly desirable and of considerable significance in theory and application.
A random variable X is called to be a two-tailed Pareto distribution whose density is
f(x)={qx2ifx≤−1,0if−1<x<1,px2ifx≥1, | (1.1) |
where p+q=1.
Let {Xn,n≥1} be independent Pareto-Zipf random variables satisfying P(Xn=0)=1−1/n,
P(Xn≤x)=1−1x+nforallx>0, | (1.2) |
and fXn(x)=1(x+n)2I(x>0).
Obviously, if the random variable Xn satisfies Eq (1.1) or (1.2), then E|Xn|=∞, n≥1. Alder [25] considered independent and identically distributed (i.i.d.) random variables satisfying Eq (1.1) and studied the strong law of large numbers. Alder [26] obtained the weak law of large numbers for Pareto-Zipf random variables. For more research on laws of large numbers for i.i.d. random variables with infinite mean, the author can refer to works of Adler [27,28] and Matsumoto and Nakata [29,30,31].
Yang et al. [24] investigated the law of large numbers for NSD random variables satisfying Pareto-type distributions with infinite means, and obtained the following theorems which extend and improve the corresponding ones in [25,26]:
Theorem 1.1. Let {Xn,n≥1} be a nonnegative sequence of NSD random variables whose distributions are defined by P(Xn=0)=1−1/cn for n≥1 and the tail probability
P(Xn>x)=1x+cnforallx>0andn≥1, | (1.3) |
where {cn} is a nondecreasing constant sequence with cn≥1 and
Cn=n∑j=11cj→∞. | (1.4) |
Then we have
∑nj=1c−1jXjCnlogCnP⟶1. | (1.5) |
Theorem 1.2. Let {Xn,n≥1} be a sequence of NSD random variables with the same distributions from a two-tailed Pareto distribution defined by Eq (1.1). Then for all β>0 we have
1logβnn∑j=1logβ−2jjXja.s.⟶p−qβ. | (1.6) |
In the current work, the author studies the weak and strong laws of large numbers for NSD random variables. The obtained results in this article extend and improve Theorems 1.1 and 1.2. Meanwhile, the author investigates p-mean convergence for NSD random variables under some appropriate conditions, which was not considered in [24].
Throughout this paper, the symbol C denotes a positive constant which may differ from one place to another. The symbol I(A) denotes the indicator function of the event A.
To prove our main results, we first present some technical lemmas.
Lemma 2.1. ([2]) If (X1,X2,⋯,Xn) is NSD and f1,f2,⋯,fn are all non decreasing, then (f1(X1),f2(X2),⋯,fn(Xn)) is also NSD.
As we know, moment inequalities are very important tools in establishing the limit theorems for sequences of random variables. Shen et al. [6] presented the following Marcinkiewicz-Zygmund inequality with exponent 2.
Lemma 2.2. ([6]) Let {Xn,n≥1} be a sequence of NSD random variables with EXn=0 and EX2n<∞ for n≥1. Then
E(max1≤k≤n(k∑i=1Xi)2)≤2n∑i=1EX2i,n≥1. |
By means of similar methods in Shao [32], Wang et al. [7] established the following Rosenthal-type maximal inequality, which is very useful in establishing the convergence properties for NSD random variables:
Lemma 2.3. ([7]) Let p>1. Let {Xn,n≥1} be a sequence of NSD random variables with E|Xi|p<∞ for each i≥1. Then for all n≥1,
E(max1≤k≤n|k∑i=1Xi|p)≤23−pn∑i=1E|Xi|pfor1<p≤2 |
and
E(max1≤k≤n|k∑i=1Xi|p)≤2(15plnp)p[n∑i=1E|Xi|p+(EX2i)p/2]forp>2. |
Lemma 2.4. ([6]) Let {Xn,n≥1} be a sequence of NSD random variables. If
∞∑n=1Var(Xn)<∞, |
then ∑∞n=1(Xn−EXn) almost certainly converges.
Now we state our main results and the proofs will be presented in next section.
Theorem 2.1. Let {Xn,n≥1} be a nonnegative sequence of NSD random variables whose distributions are defined by P(Xn=0)=1−1/cn for n≥1 and the tail probability
P(Xn>x)=1x+cnforallx>0andn≥1, | (2.1) |
where {cn,n≥1} is a nondecreasing constant sequence with cn≥1 and
Cn=n∑j=11cj→∞. | (2.2) |
Let {Dn,n≥1} be a sequence of constants satisfying Dn→∞ and Cn=o(Dn). Then we have
1Dnmax1≤k≤n|k∑j=1c−1j(Xj−EXnj)|P⟶0, | (2.3) |
where Xnj=XjI(Xj≤Dncj)+DncjI(Xj>Dncj), 1≤j≤n.
Take Dn=CnlogCn, then we can obtain the following corollary which extends Theorem 1.1.
Corollary 2.1. Let {Xn,n≥1} be a nonnegative sequence of NSD random variables whose distributions are defined by P(Xn=0)=1−1/cn for n≥1 and the tail probability Eq (2.1), where {cn,n≥1} is a nondecreasing constant sequence satisfying cn≥1 and Eq (2.2). Then
1CnlogCnmax1≤k≤n|k∑j=1c−1j(Xj−EXnj)|P⟶0. | (2.4) |
Remark 2.1. Yang et al. [24] proved that
1CnlogCnn∑j=1c−1jEXjI(Xj≤cjCnlogCn)→1 |
and
1CnlogCnn∑j=1c−1jE(cjCnlogCnI(Xj>cjCnlogCn))=n∑j=1P(Xj>cjCnlogCn)→0, |
which yields
1CnlogCnn∑j=1c−1jEXnj→1. |
Then we can find that Theorem 1.1 is a special case of Corollary 2.1 for k=n. Therefore, Theorem 2.1 and Corollary 2.1 extend and improve Theorem 1.1.
Theorem 2.2. Let {Xn,n≥1} be a sequence of identically distributed NSD random variables. Let {dn,n≥1} be a sequence of positive constants satisfying dn↑∞, and {cn,n≥1} be a sequence of positive constants such that φ(n)≡cndn satisfies φ(n)→∞ as n→∞,
∞∑m=n1φ2(m)=O(nφ−2(n)) | (2.5) |
and
∞∑n=1P(|X1|>φ(n))<∞. | (2.6) |
Then
1dnn∑j=1c−1j(Xj−E~Xj)⟶0a.s., | (2.7) |
where ~Xj=−φ(j)I(Xj<−φ(j))+XjI(|Xj|≤φ(j))+φ(j)I(Xj>φ(j)), 1≤j≤n.
Remark 2.2. We will show that Theorem 1.2 is a special case of Theorem 2.2. In fact, if we assume that {Xn,n≥1} is a sequence of NSD random variables with the same distributions from a two-tailed Pareto distribution defined by Eq (1.1), and take cn=nlog2−βn and dn=logβn (β>0), then φ(n)=cndn=nlog2n. We can verify that φ(n)=nlog2n satisfies the conditions stated in Theorem 2.2.
First, it is clear that φ(n)=nlog2n satisfies φ(n)→∞ as n→∞.
Second, we have by standard calculations that
∞∑m=n1φ2(m)∼∫∞n1x2log4xdx=O(n−1log−4(n))=O(nφ−2(n)), |
which shows that Eq (2.5) is verified.
Next, we have by Eq (1.1) and φ(n)=nlog2n that
∞∑n=1P(|X1|>φ(n))=∞∑n=1P(|X1|>nlog2n)=∞∑n=1(∫−nlog2n−∞qx−2dx+∫∞nlog2npx−2dx)=∞∑n=1p+qnlog2n=∞∑n=11nlog2n<∞ |
and then Eq (2.6) is verified.
Finally, we also obtain by Eq (1.1) and φ(j)=jlog2j that
1dnn∑j=1c−1jE~Xj=1dnn∑j=1(−djP(Xj<−φ(j))+c−1jEXjI(|Xj|<φ(j))+djP(Xj>φ(j)))=p−qlogβnn∑j=1logβ−2jj+p−qlogβnn∑j=1logβ−1jj=:J1+J2. |
By similar argument as in the proof of H→0 in [24], we can obtain J1→0. By similar argument as in the proof of Eq (3.5) in [24], we can prove J2→p−qβ. Then we obtain by Eq (2.7) that
1logβnn∑j=1logβ−2jjXja.s.⟶p−qβ. |
To sum up, Theorem 1.2 is a special case of Theorem 2.2 and then Theorem 2.2 extends Theorem 1.2.
Next, we present a new theorem of p-mean convergence for NSD random variables under some appropriate conditions, which was not considered in [24,25,26].
Theorem 2.3. Let {Xn,n≥1} be a sequence of NSD random variables satisfying
limx→∞supj≥1xαP(|Xj|>x)<∞,α∈(1,2). | (2.8) |
Let {dn,n≥1} be a sequence of positive constants satisfying dn↑∞, and {cn,n≥1} be a sequence of positive constants such that cj≥1 and
n∑j=1c−αj=o(dαn). | (2.9) |
Then for p∈(1,α),
1dnmax1≤k≤n|k∑j=1c−1j(Xj−E^Xnj)|Lp⟶0, | (2.10) |
where ^Xnj=−dncjI(Xj<−dncj)+XjI(|Xj|≤dncj)+dncjI(Xj>dncj), 1≤j≤n.
Proof of Theorem 2.1. We first observe that for every ε>0,
P(1Dnmax1≤k≤n|k∑j=1c−1j(Xj−EXnj)|>2ε)≤P(max1≤k≤n|k∑j=1c−1j(Xj−Xnj)|>Dnε)+P(max1≤k≤n|k∑j=1c−1j(Xnj−EXnj)|>Dnε)=:H1+H2. |
To prove Eq (2.3), we need only to show that Hi→0 as n→∞, i=1,2. For H1, we have by the definition of Xnj, Cn=o(Dn), Eqs (2.1) and (2.2) that
H1≤P(n⋃j=1(Xj≠Xnj))≤n∑j=1P(Xj>Dncj)=n∑j=11Dncj+cj=1Dn+1n∑j=1c−1j=CnDn+1→0. |
For fixed n≥1, Xnj is the nondecreasing function of Xj. Hence, it follows by Lemma 2.1 that {Xnj,1≤j≤n} is a sequence of NSD random variables. Hence we have by Markov's inequality and Lemma 2.3 with 1<p≤2,
H2≤CDpnE(max1≤k≤n|k∑j=1c−1j(Xnj−EXnj)|)p≤CDpnn∑j=1c−pjE|Xnj|p=CDpnn∑j=1c−pjE|Xj|pI(Xj≤Dncj)+Cn∑j=1P(Xj>Dncj)=CDpnn∑j=1c−pj∫(Dncj)p0P(|Xj|pI(Xj≤Dncj)≥t)dt+Cn∑j=11Dncj+cj=CDpnn∑j=1c−pj∫(Dncj)p0P(|Xj|p≥t)dt+CCnDn+1=CDpnn∑j=1c−pj∫(Dncj)p01t1/p+cjdt+CCnDn+1(by(2.1))≤CDpnn∑j=1c−pj∫(Dncj)p01t1/pdt+CCnDn+1=CCnDn+CCnDn+1→0. |
The proof is completed.
Proof of Theorem 2.2. Obviously, to prove Eq (2.7), we need only to show
1dnn∑j=1c−1j(Xj−~Xj)⟶0a.s. | (3.1) |
and
1dnn∑j=1c−1j(~Xj−E~Xj)⟶0a.s.. | (3.2) |
By Eq (2.6), dn↑∞ and the Borel-Cantelli lemma, we obtain
1dnn∑j=1c−1j|Xj|I(|Xj|>φ(j))⟶0a.s.. |
Noting that
|Xj+φ(j)|I(Xj<−φ(j))+|Xj−φ(j)|I(Xj>φ(j))≤|Xj|I(|Xj|>φ(j)). |
Then
|1dnn∑j=1c−1j(Xj−~Xj)|=|1dnn∑j=1c−1jXjI(|Xj|>φ(j))+(Xj+φ(j))I(Xj<−φ(j))+(Xj−φ(j))I(Xj>φ(j))|≤2dnn∑j=1c−1j|Xj|I(|Xj|>φ(j))⟶0a.s., |
which yields Eq (3.1).
It follows by the definition of ~Xj that
∞∑j=11φ2(j)E(~Xj−E~Xj)2≤C∞∑j=11φ2(j)EX2jI(|Xj|≤φ(j))+C∞∑j=1P(|Xj|>φ(j))=:I1+I2. |
We obtain directly by Eq (2.6) that I2<∞. Let F(x) be the distribution of X1, then
I1=C∞∑j=11φ2(j)EX21I(|X1|≤φ(j))=C∞∑j=11φ2(j)∫∞−∞x2I(|X1|≤φ(j))dF(x) |
=C∫∞−∞x2∑j:φ(j)≥|x|1φ2(j)dF(x). | (3.3) |
Define N(|x|)=♯{j:φ(j)<|x|} and j∗=inf{j:φ(j)≥|x|}. Hence we can obtain N(|x|)≥j∗−1 and
∑j:φ(j)≥|x|1φ2(j)≤∞∑j=j∗1φ2(j)≤Cj∗φ2(j∗)(by Eq (2.5))≤Cj∗x2 |
≤CN(|x|)+1x2. | (3.4) |
It follows by Eqs (2.6), (3.3) and (3.4) that
I1≤C∫∞−∞(N(|x|)+1)dF(x)=CEN(|X1|)+C=CE[∞∑j=1I(|X1|>φ(j))]+C=C∞∑j=1P(|X1|>φ(j))+C<∞. |
Now we obtain by I1<∞ and I2<∞ that
∞∑j=11φ2(j)E(~Xj−E~Xj)2<∞. | (3.5) |
Consequently, by Lemma 2.4 and Eq (3.5), we get
∞∑j=11φ(j)(~Xj−E~Xj)convergesa.s., |
which implies Eq (3.2) by Kronecker's lemma, together with the condition dn↑∞.
The proof is completed.
Proof of Theorem 2.3. Noting that
E{1dnmax1≤k≤n|k∑j=1c−1j(Xj−E^Xnj)|}p≤1dpnE{max1≤k≤n|k∑j=1c−1j(^Xnj−E^Xnj)|}p+1dpnE{max1≤k≤n|k∑j=1c−1j(Xj−^Xnj)|}p≤1dpn{E(max1≤k≤n|k∑j=1c−1j(^Xnj−E^Xnj)|)2}p/2+1dpnE{max1≤k≤n|k∑j=1c−1j(Xj−^Xnj)|}p=:J1+J2. |
To prove Eq (2.10), it is sufficient to prove J1→0 and J2→0. By Lemma 2.1 and the fact that ^Xnj is the nondecreasing function of Xj, {^Xnj,1≤j≤n} is also a sequence of NSD random variables.
We have by Lemma 2.2 that
J2/p1=1d2nE{max1≤k≤n|k∑j=1c−1j(^Xnj−E^Xnj)|}2≤Cd2nn∑j=1c−2jE(^Xnj−E^Xnj)2≤Cd2nn∑j=1c−2jEX2jI(|Xj|≤dncj)+Cn∑j=1P(|Xj|>dncj)=:J3+J4. |
By dn↑∞, Eqs (2.8) and (2.9), we have
J4≤C1dαnn∑j=1c−αj→0asn→∞. | (3.6) |
Now we will show J3→0. Observing
J3=Cd2nn∑j=1c−2j∫(dncj)20P(X2jI(|Xj|≤dncj)≥t)dt≤Cd2nn∑j=1c−2j∫(dncj)20P(X2j≥t)dt. |
Let t=u2, then
J3≤Cd2nn∑j=1c−2j∫dncj0uP(|Xj|≥u)du. |
From Eq (2.8), we know that, there exists M>0 and N0∈N such that
P(|Xj|≥u)≤Mu−αforu>N0. | (3.7) |
Since dn↑∞ and cj≥1, while n is sufficiently large, we can obtain dncj>N0. Hence
J3≤Cd2nn∑j=1c−2j∫N00uP(|Xj|≥u)du+CMd2nn∑j=1c−2j∫dncjN0u1−αdu=:J′3+J″3. |
By α<2, cj≥1 and Eq (2.9), we have
J′3≤Cd2nn∑j=1c−2j∫N00udu≤Cd2nn∑j=1c−2j≤Cd2−αn1dαnn∑j=1c−αj→0asn→∞ |
and
J″3≤Cd2nn∑j=1c−2j[(dncj)2−α−N2−α0]≤Cdαnn∑j=1c−αj→0asn→∞. |
Finally, we need only to show J2→0 as n→∞. Let
Znj=Xj−^Xnj=(Xj+dncj)I(Xj<−dncj)+(Xj−dncj)I(Xj>dncj). |
We first prove that
EZnj→0asn→∞. | (3.8) |
Observing
|EZnj|≤E|Znj|≤E|Xj|I(|Xj|>dncj)=(∫dncj0+∫∞dncj)P(|Xj|I(|Xj|>dncj)≥t)dt=∫dncj0P(|Xj|>dncj)dt+∫∞dncjP(|Xj|≥t)dt=dncjP(|Xj|>dncj)+∫∞dncjP(|Xj|≥t)dt=:J5+J6. |
By Eq (3.7) and α>1, we have
J5≤M(dncj)α−1→0asn→∞ |
and
J6≤M∫∞dncjt−αdt≤CM(dncj)α−1→0asn→∞, |
which yields Eq (3.8). Therefore, we obtain by Lemma 2.3 that
J2≤1dpnE{max1≤k≤n|k∑j=1c−1j(Znj−EZnj)|}p≤Cdpnn∑j=1c−pjE|Znj|p≤Cdpnn∑j=1c−pjE|Xj|pI(|Xj|>dncj).(bythedefinitionofZnj) |
By similar arguments as in the proof of Eq (3.8), we can obtain
E|Xj|pI(|Xj|>dncj)=(dncj)pP(|Xj|>dncj)+∫∞(dncj)pP(|Xj|p≥t)dt. |
Then
J2≤Cn∑j=1P(|Xj|>dncj)+Cdpnn∑j=1c−pj∫∞(dncj)pP(|Xj|p≥t)dt=:J′2+J″2. |
By similar arguments as the proof of J4→0, we obtain J′2→0. We also have by Eq (3.7), p<α and Eq (2.9) that
J″2≤Cdpnn∑j=1c−pj∫∞(dncj)pt−α/pdt≤Cdαnn∑j=1c−αj→0asn→∞. |
The proof is completed.
In this work the author investigated the limit theorems for negatively superadditive-dependent random variables, and obtained some new results on the law of large numbers and mean convergence under some appropriate conditions. As a future work, we propose to consider some other strong convergence for sequence of negatively superadditive-dependent random variables.
The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.
This work is supported by the Social Sciences Planning Project of Anhui Province (AHSKY2018D98).
The author declares that he has no conflict of interest.
[1] |
J. H. B. Kemperman, On the FKG-inequalities for measures on a partially ordered space, Indagat. Math., 80 (1977), 313–331. https://doi.org/10.1016/1385-7258(77)90027-0 doi: 10.1016/1385-7258(77)90027-0
![]() |
[2] | T. Z. Hu, Negatively superadditive dependence of random variables with applications, Chinese J. Appl. Probab. Statist., 16 (2000), 133–144. |
[3] |
K. J. Dev, F. Proschan, Negative association of random variables with applications, Ann. Statist., 11 (1983), 286–295. https://doi.org/10.1214/aos/1176346079 doi: 10.1214/aos/1176346079
![]() |
[4] |
T. C. Christofides, E. Vaggelatou, A connection between supermodular ordering and positive/negative association, J. Multivariate Anal., 88 (2004), 138–151. https://doi.org/10.1016/s0047-259x(03)00064-2 doi: 10.1016/s0047-259x(03)00064-2
![]() |
[5] |
N. Eghbal, M. Amini, A. Bozorgnia, Some maximal inequalities for quadratic forms of negative superadditive dependence random variables, Statist. Probabil. Lett., 80 (2010), 587–591. https://doi.org/10.1016/j.spl.2009.12.014 doi: 10.1016/j.spl.2009.12.014
![]() |
[6] |
Y. Shen, X. J. Wang, W. Z. Yang, S. H. Hu, Almost sure convergence theorem and strong stability for weighted sums of NSD random variables, Acta Math. Sin., 29 (2013), 743–756. https://doi.org/10.1007/s10114-012-1723-6 doi: 10.1007/s10114-012-1723-6
![]() |
[7] |
X. J. Wang, X. Deng, L. L. Zheng, S. H. Hu, Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications, Statistics, 48 (2014), 834–850. https://doi.org/10.1080/02331888.2013.800066 doi: 10.1080/02331888.2013.800066
![]() |
[8] |
N. Eghbal, M. Amini, A. Bozorgnia, On the Kolmogorov inequalities for quadratic forms of dependent uniformly bounded random variables, Scienceasia, 81 (2011), 1112–1120. https://doi.org/10.1016/j.spl.2011.03.005 doi: 10.1016/j.spl.2011.03.005
![]() |
[9] |
X. J. Wang, A. T. Shen, Z. Y. Chen, S. H. Hu, Complete convergence for weighted sums of NSD random variables and its application in the EV regression model, Test, 24 (2015), 166–184. https://doi.org/10.1007/s11749-014-0402-6 doi: 10.1007/s11749-014-0402-6
![]() |
[10] |
Y. C. Yu, H. C. Hu, L. Liu, S. Y. Huang, M-test in linear models with negatively superadditive dependent errors, J. Inequal. Appl., 235 (2017), 235. https://doi.org/10.1186/s13660-017-1509-6 doi: 10.1186/s13660-017-1509-6
![]() |
[11] |
A. T. Shen, X. H. Wang, Kaplan-Meier estimator and hazard estimator for censored negatively superadditive dependent data, Statistics, 50 (2016), 377–388. https://doi.org/10.1080/02331888.2015.1038269 doi: 10.1080/02331888.2015.1038269
![]() |
[12] |
H. Naderi, F. Boukhari, P. Matula, A note on the weak law of large numbers for weighted negatively superadditive dependent random variables, Commun. Stat. Theor. M., 51 (2022), 7465–7475. https://doi.org/10.1080/03610926.2021.1873377 doi: 10.1080/03610926.2021.1873377
![]() |
[13] |
S. C. Ta, C. M. Tran, D. V. Le, On the almost sure convergence for sums of negatively superadditive dependent random vectors in Hilbert spaces and its application, Commun. Stat. Theor. M., 49 (2020), 2770–2786. https://doi.org/10.1080/03610926.2019.1584304 doi: 10.1080/03610926.2019.1584304
![]() |
[14] |
X. H. Wang, S. H. Hu, On the strong consistency of M-estimates in linear models for negatively superadditive dependent errors, Aust. Nz. J. Stat., 57 (2015), 259–274. https://doi.org/10.1111/anzs.12117 doi: 10.1111/anzs.12117
![]() |
[15] |
Y. Wu, X. J. Wang, S. H. Hu, Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications, Appl. Math. Ser. B, 31 (2016), 439–457. https://doi.org/10.1007/s11766-016-3406-z doi: 10.1007/s11766-016-3406-z
![]() |
[16] |
X. H. Wang, X. Q. Li, S. H. Hu, On the complete convergence of weighted sums for an array of rowwise negatively superadditive dependent random variables, Scienceasia, 42 (2016), 66–74. https://doi.org/10.2306/scienceasia1513-1874.2016.42.066 doi: 10.2306/scienceasia1513-1874.2016.42.066
![]() |
[17] |
X. Deng, X. J. Wang, Y. Wu, Y. Ding, Complete moment convergence and complete convergence for weighted sums of NSD random variables, Racsam. Rev. R. Acad. A., 110 (2016), 97–120. https://doi.org/10.1007/s13398-015-0225-7 doi: 10.1007/s13398-015-0225-7
![]() |
[18] |
A. T. Shen, X. H. Wang, H. Y. Zhu, Convergence properties for weighted sums of NSD random variables, Commun. Stat. Theor. M., 45 (2016), 2402–2412. https://doi.org/10.1080/03610926.2014.881492 doi: 10.1080/03610926.2014.881492
![]() |
[19] |
A. Kheyri, M. Amini, H. Jabbari, A. Bozorgnia, Kernel density estimation under negative superadditive dependence and its application for real data, J. Stat. Comput. Sim., 89 (2019), 2373–2392. https://doi.org/10.1080/00949655.2019.1619738 doi: 10.1080/00949655.2019.1619738
![]() |
[20] |
B. Meng, Q. Y. Wu, D. C. Wang, On the strong convergence for weighted sums of negatively superadditive dependent random variables, J. Inequal. Appl., 269 (2017), 269. https://doi.org/10.1080/03610918.2022.2093371 doi: 10.1080/03610918.2022.2093371
![]() |
[21] |
K. Bertin, S. Torres, L. Viitasaari, Least-square estimators in linear regression models under negatively superadditive dependent random observations, Statistics, 55 (2021), 1018–1034. https://doi.org/10.1080/02331888.2021.1993854 doi: 10.1080/02331888.2021.1993854
![]() |
[22] |
M. Q. Chen, K. Chen, Z. J. Wang, Z. L. Lu, X. J. Wang, Complete moment convergence for partial sums of arrays of rowwise negatively superadditive dependent random variables, Commun. Stat. Theor. M., 49 (2020), 1158–1173. https://doi.org/10.1080/03610926.2018.1554136 doi: 10.1080/03610926.2018.1554136
![]() |
[23] |
Y. C. Yu, X. S. Liu, L. Liu, W. S. Liu, On adaptivity of wavelet thresholding estimators with negatively super-additive dependent noise, Math. Slovaca, 69 (2019), 1485–1500. https://doi.org/10.1515/ms-2017-0324 doi: 10.1515/ms-2017-0324
![]() |
[24] |
W. Z. Yang, L. Yang, D. Wei, S. H. Hu, The laws of large numbers for Pareto-type random variables with infinite means, Commun. Stat. Theor. M., 48 (2019), 3044–3054. https://doi.org/10.1080/03610926.2018.1473602 doi: 10.1080/03610926.2018.1473602
![]() |
[25] | A. Adler, Laws of large numbers for two tailed Pareto random variables, Probab. Math. Stat., 28 (2008), 121–128. |
[26] | A. Adler, An exact weak law of large numbers, Bull. Inst. Math. Acad., 7 (2012), 417–422. |
[27] | A. Adler, Exact weak laws and one side strong laws, Bull. Inst. Math. Acad., 12 (2017), 103–124. |
[28] |
A. Adler, One sided strong laws for random variables with infnite mean, Open Math., 15 (2017), 828–832. https://doi.org/10.1515/math-2017-0070 doi: 10.1515/math-2017-0070
![]() |
[29] |
K. Matsumoto, T. Nakata, Limit theorems for a generalized Feller game, J. Appl. Probab., 50 (2013), 54–63. https://doi.org/10.1239/jap/1363784424 doi: 10.1239/jap/1363784424
![]() |
[30] |
T. Nakata, Limit theorems for nonnegative independent random variables with truncation, Acta Math. Hung., 145 (2015), 1–16. https://doi.org/10.1007/s10474-014-0474-5 doi: 10.1007/s10474-014-0474-5
![]() |
[31] |
T. Nakata, Weak laws of large numbers for weighted independent random variables with infnite mean, Statist. Probabil. Lett., 109 (2016), 124–129. https://doi.org/10.1016/j.spl.2015.11.017 doi: 10.1016/j.spl.2015.11.017
![]() |
[32] |
Q. M. Shao, A comparison theorem on moment inequalities between negatively associated and independent random variables, J. Theor. Probab., 13 (2000), 343–355. https://doi.org/10.1023/A:1007849609234 doi: 10.1023/A:1007849609234
![]() |