# Limiting Behavior of the Partial Sum for Negatively Superadditive Dependent Random Vectors in Hilbert Space

Authors: Mi-Hwa Ko
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, We study the complete convergence andLp- convergence for the maximum of the partial sum of negatively superadditive dependent random vectors in Hilbert space. The results extend the corresponding ones of Ko (Ko, 2020) to H-valued negatively superadditive dependent random vectors.


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## Body

## 1. Introduction

Alam and Saxena [1] introduced the concept of negative association as follows: a finite family of random variables Xi,1≤i≤n is said to be negatively associated (NA) for every pair of disjoint subsets of A and B of 1,2,…,nCovfXi,i∈AgXj,j∈B≤0, whenever f and g are coordinate-wise nondecreasing functions and the covariance exists. An infinite family is NA if every finite subfamily is NA.A functionϕ:ℝn⟶ℝ is said to be superadditive if ϕx∨y+ϕx∧y≥ϕx+ϕy for all x,y∈ℝn, where ∨ is for componentwise maximum and ∧ is for componentwise minimum. This notion was introduced by Kemperman [2].Based on the above superadditive function, the concept of negatively superadditive dependent (NSD) random variables was introduced by Hu [3] as follows: a random vector X=X1,X2,…,Xn is said to be negatively superadditive dependent (NSD) if EϕX1,X2,…,Xn≤EϕX1∗,X2∗,…,Xn∗, where X1∗,X2∗,…,Xn∗ are independent such that Xi∗ and Xi have the same distribution for each i and ϕ is a superadditive function such that the above expectations exist. A sequence Xn,n≥1 of random variables is said to be NSD if for any n≥1, X1,X2,…,Xn is NSD. Hu [3] gave an example for illustrating that negatively superadditive dependence (NSD) does not imply negative association (NA), and Christofides and Vaggelatou [4] indicated that NA implies NSD. So, the NSD structure is an extension of the NA structure. Sometimes, the NSD structure is more useful than the NA structure and NSD random variables have wide applications in reliability theory and multivariate statistical analysis. For this reason, studying the limit theorems for NSD random variables is much significant.LetH be a real separable Hilbert space with the norm ⋅ generated by an inner product <⋅,⋅>. Let ej,j≥1 be an orthonormal basis in H, X be an H-valued random vector, and <X,ej> be denoted by Xj.Ko et al. [5] and Huan et al. [6] introduced the concept of negatively associated random vectors taking values in H, Ko [7] introduced the concept of asymptotically negatively associated H-valued random vectors, and Dung et al. [8] introduced the concept of pairwise NQD H-valued random vectors.As the concept ofH-valued NA random vectors was introduced by Ko et al. [5], Son et al. [9] presented the concept of H-valued negatively superadditive dependent (NSD) random vectors as follows: a sequence Xn,n≥1 of H-valued random vectors is said to be NSD if for any d≥1, the sequence Xn1,Xn2,…,Xnd,n≥1 of ℝd-valued random vectors is negatively superadditive dependent (NSD).LetX,Xn,n≥1 be a sequence of H-valued random vectors. We will use the following inequalities:(1)C1PX>t≤1n∑k=1nPXk>t≤C2PX>t.If there exists a positive constantC1C2 such that the left-hand side (right-hand side) of (1) is satisfied for all n≥1 and t≥0, then the sequence Xn,n≥1 is said to be weakly lower (upper) bounded by X. The sequence Xn,n≥1 is said to be weakly bounded by X if it is both lower and upper bounded by X.In this paper, we show the complete convergence results andL2-convergence of the maximum of the partial sums for NSD random vectors in Hilbert space. We also consider residual Cesàro alpha-integrability and strongly residual Cesàro alpha-integrability for NSD random vectors in Hilbert space.Throughout the paper, the symbolC denotes a generic constant which is not necessarily the same in each occurrence, and  denotes the L2-norm. Moreover, ≪ represents the Vinogradov symbol O and I⋅ is the indicator function.


## 2. Some Lemmas

The following lemmas will be useful to prove the main results.Lemma 1.
(see [3]). If X1,X2,…,Xn is negatively superadditive dependent (NSD) and f1,f2,…,fn are all nondecreasing functions, then f1X1,f2X2,…,fnXn is also NSD.Lemma 2.
(see [10]). Let Xn,n≥1 be a sequence of NSD random variables with EXn=0 and EXn2<∞ for all n≥1. Then, for all n≥1, there is a positive constant C such that(2)Emax1≤k≤n∑i=1kXi2≤C∑i=1nEXi2.
We extend Lemma2 to a sequence of Hilbert valued random vectors as follows.Lemma 3.
LetXn,n≥1 be a sequence of H-valued NSD random vectors with EXn=0 and EXn2<∞ for every n≥1, where Xn=Xn1,Xn2,…,Xnj for any j≥1 and Xnj=<Xn,ej>. If Xnj,n≥1 is a sequence of NSD random variables with EXnj=0 and EXnj2<∞ for each j≥1, then there is a positive constant C such that(3)Emax1≤k≤n∑i=1kXi2≤C∑i=1nEXi2.Proof.
Inspired by the proof of Lemma 1.7 of Huan et al. [6] and from Lemma 2, we have that(4)Emax1≤k≤n∑i=1kXi2=Emax1≤k≤n∑j=1∞<∑i=1kXi,ej>2≤E∑j=1∞max1≤k≤n<∑i=1kXi,ej>2=∑j=1∞Emax1≤k≤n∑i=1kXij2≤C∑j=1∞∑i=1nEXij2=C∑i=1nEXi2.
It is obvious that ifXn,n≥1 is an NSD sequence of H-valued random vectors, where Xn=Xn1,Xn2,…,Xnj for any j≥1, then Xnj is a sequence of NSD random variables for each j≥1. However, the reverse is not true in general.Lemma 4.
(see [11]). Let an,n≥1 and bn,n≥1 be sequences of non-negative numbers. If supn≥1n−1∑i=1nai<∞and∑n=1∞bn<∞, then ∑i=1naibi≤supm≥1m−1∑i=1mai∑i=1nbi for every n≥1.Lemma 5.
(see [12]). Let Xn,n≥1 be a sequence of H-valued random vectors, weakly upper bounded by a random vector X. Let r>0 for some A>0.(5)Xi′=XiIXi≤A,Xi″=XiIXi>A,X′=XIXi≤A,X″=XIX>A.
And(i)
IfEXr<∞, then 1/n∑i=1nEXir≤CEXr(ii)
1/n∑i=1nEXi′r≤CEX′r+ArPX>A(iii)
1/n∑i=1nEXi″r≤CEX″r


## 3. Main Results

A sequenceXn,n≥1 of random vectors is said to converge completely to a constant C if for any ε>0, ∑n=1∞PXn−C>ε<∞.In this case, we writeXn⟶C completely. This notion was given by Hsu and Robbins [13]. Note that complete convergence implies the almost sure convergence in view of the Borel–Cantelli lemma.Based on Lemma3, we will extend the complete convergence results of the maximum of the partial sum of NSD random variables to the case of H-valued random vectors.Theorem 1.
LetXn,n≥1 be a sequence of H-valued NSD random vectors. If a sequence Xn,n≥1 satisfies(6)supn≥11n∑k=1nEXk2<∞,then for any δ>1/2,(7)n−δmax1≤i≤nSi−ESi⟶0completely,where Si=∑j=1iXj.Proof.
For eachn≥1, let m=mn be the integer such that 2m−1<n≤2m. Then, we obtain that(8)n−δmax1≤i≤nSi−ESi≤n−δmax1≤i≤2mSi−ESi≤2m−1−−δmax1≤i≤2mSi−ESi=2δ⋅2−mδmax1≤i≤2mSi−ESi.
Hence, it is enough to prove(9)2−mδmax1≤i≤2mSi−ESi⟶0completely.
By Lemma3, the Hölder inequality, Lemma 4, and (6), we have that(10)∑m=0∞E2−mδmax1≤i≤2mSi−ESi2≪∑m=0m=02−2mδ∑i=12mEXi2≪∑i=1∞EXi2∑m:2m≥i2−2mδ≤∑i=1∞i−2δEXi2≤supn≥11n∑i=1nEXi2∑n=1∞n−2δ≪∑n=1∞n−2δ<∞,which yields (9). Hence, the desired result (7) follows.
Based on Lemma3, we will extend some L2-convergence of NSD random variables to the case of H-valued random vectors.Theorem 2.
LetXn,n≥1 be a sequence of H-valued NSD random vectors satisfying (6); then, for any δ>1/2,(11)n−δmax1≤i≤nSi−ESi⟶0inL2.Proof.
LetXn=Xn1,Xn2,…,Xnj for any j≥1. Note that Xnj for any j≥1 is a sequence of NSD random variables. Then, Xnj−EXnjn≥1 is a sequence of H-valued NSD random variables by Lemma 1. By Lemma 3, Hölder’s inequality, and (6), we obtain(12)En−δmax1≤i≤nSi−ESi2=n−2δEmax1≤i≤n∑j=1iXj−EXj2≪n−2δ∑i=1nEXi−EXi2≪n−2δ∑i=1nEXi2≪n−2δ+1supn≥11n∑i=1nEXi2,which yields (11) for any δ>1/2. The proof of theorem is completed.
We considerL2-convergence of weakly upper bounded H-valued NSD random vectors.Theorem 3.
LetXn,n≥1 be a sequence of H-valued NSD random vectors which is weakly upper bounded by a random vector X with EX2<∞. Then, for any δ>1/2, (11) holds.Proof.
By Lemma3, Hölder’s inequality, EX2<∞, the proof of Theorem 2, and Lemma 5 (i), we obtain(13)En−δmax1≤i≤nSi−ESi2≤n−2δ∑i=1nEXi2≤Cn−2δ+1EX2⟶0asn⟶∞.
Hence, (11) holds.
We will extend two special kinds of uniform integrability which were introduced by Chandra and Goswami [14] to H-valued random vectors.Definition 1 (see [15]).
Forα∈0,∞, a sequence Xn,n≥1 of random vectors in Hilbert space is said to be residually Cesàro alpha-integrable (RCI (α)) if(14)supn≥11n∑i=1nEXi<∞,limn⟶∞1n∑i=1nEXi−iαIXi>iα=0.
Clearly,Xn is RCI α for any α>0 if Xn,n≥1 is identically distributed with EX1<∞ and Xn2,n≥1 is RCI (α) for any α>0 if Xn,n≥1 is stochastically dominated by a non-negative random vector X with EX2<∞ (see [15]).Theorem 4.
Let a random vectorXn=Xn1,Xn2,…,Xnj for any j≥1 be non-negative where every component Xnj of Xn is non-negative random variable for each j≥1. Let Xn,n≥1 be a sequence of H-valued NSD non-negative random vectors. If Xn2,n≥1 is RCI (α) for some α∈0,∞, then for any δ>1/2,(15)n−δmax1≤i≤nSi−ESi⟶0inL2.Proof.
Letj≥1Xn=Xn1,Xn2,…,Xnj such that Xnj≥0 for all j≥1, where Xnj=<Xn,ej>. For each j≥1, let(16)Ynj=XnjI0≤Xnj≤nα+nαIXnj>nα,(17)Znj=Xnj−Ynj=Xnj−nαIXnj>nα.
Define, for eachn≥1Zn=Xn−Yn,Sn1=∑i=1nYi and Sn2=∑i=1nZi. Then, Sn=Sn1+Sn2=∑i=1nYi+∑i=1nZi.
Note thatXnj is a sequence of NSD random variables and that(18)Yn=minXn,nα,Zn=Xn−nαIXn>nα,(19)Zn2≤Xn2−nαIXn2>nα.
Obviously, for eachj≥1 and n≥1, Ynj and Znj are monotone transforms of the random variable Xnj by (16) and (17), respectively. Thus, j≥1Ynj and Znj are NSD sequences of H-valued random variables by Lemma 1. Ynj−EYnj and Znj−EZnj are also NSD sequences of zero mean random variables. By Lemma 3 and the first condition of the RCI (α) property of (14) of the sequence Xn2,n≥1, we obtain(20)n−2δEmax1≤i≤nSi1−ESi12≪n−2δ∑i=1nEYi−EYi2≤n−2δ∑i=1nEYi2≤n−2δ∑i=1nEXi2≤n−2δ+1supn≥1n−1∑i=1nEXi2⟶0asn⟶∞.
By Lemma3, the Hölder inequality and relation (19), and the second condition of the RCI (α) property (14) of the sequence Xn2,n≥1, we obtain(21)n−2δEmax1≤i≤nSi2−ESi22≪n−2δ∑i=1nEZi−EZi2≪n−2δ∑i=1nEZi2≤n−2δ+1⋅1n∑i=1nEXi2−iαIXi2>iα⟶0asn⟶∞.
Thus, by (20) and (21), we have(22)n−2δEmax1≤i≤nSi−ESi2≤n−2δEmax1≤i≤nSi1−ESi12+n−2δEmax1≤i≤nSi2−ESi22⟶0asn⟶∞,which yields (15). The proof of theorem is completed.Definition 2.
(see [15]). For α∈0,∞, a sequence Xn,n≥1 of H-valued random vectors is said to be strongly residually Cesàro alpha-integrable (SRCI (α)) if(23)supn≥11n∑i=1nEXi<∞,∑n=1∞1nEXn−nαIXn>nα<∞.
Note thatXn2,n≥1 is SRCI α for any α>0, provided that Xn,n≥1 is stochastically dominated by a non-negative random vector X with EX2+δ<∞ for δ>0.Theorem 5.
Define aH-valued non-negative random vector Xn as in Theorem 4 Let Xn,n≥1 be a sequence of H-valued NSD non-negative random vectors. If Xn2,n≥1 is SRCI α for some α∈0,∞, then for any δ>1/2,(24)n−δmax1≤i≤nSi−ESi⟶0completely.Proof.
For eachn≥1, let m=mn be the integer such that 2m−1<n≤2m. Then, we have that(25)n−δmax1≤i≤nSi−ESi≤n−δmax1≤i≤2mSi−ESi≤2m−1−δmax1≤i≤2mSi−ESi≤2δ⋅2−mδmax1≤i≤2mSi−ESi.
It is sufficient to prove(26)2−mδmax1≤i≤2mSi−ESi⟶0completely.
DefineYn,Zn,Sn1, and Sn2 as in the proof of Theorem 4. To prove (26), we will first show that 2−mδmax1≤i≤2mSi1−ESi1⟶0 completely. In other words, we will prove(27)2−mδmax1≤i≤2m∑k=1iYk−EYk⟶0completely.
By Lemma3 and the Hölder inequality, we have that(28)∑m=0∞E2−mδmax1≤i≤2m∑k=1iYk−EYk2≪∑m=0∞2−2mδ∑i=1i=1EYi2≤∑m=0∞2−2mδ∑i=12mEXi2.
In view of the first condition of the SRCIα property (23) of the sequence Xn2,n≥1 and Lemma 4, we conclude that(29)∑m=0∞E2−mδmax1≤i≤2m∑k=1iYk−EYk2≪∑i=1∞EXi2∑m:2m≥i2−2mδ≤∑i=1∞i−2δEXi2≤supn≥11n∑i=1nEXi2∑i=1∞i−2δ<∞,which yields (27). Next, we prove that(30)2−mδmax1≤i≤2mSi2−ESi2⟶0completely.
Namely, we will prove that(31)2−mδmax1≤i≤2m∑k=1iZk−EZk⟶0completely.
By Lemma3, the Hölder inequality, and (19) and the second condition of the SRCI α property (23) of the sequence Xn2,n≥1, we have(32)∑m=0∞E2−mδmax1≤i≤2m∑k=1iZk−EZk2≪∑m=0∞2−2mδ∑i=12mEZi2=∑i=1∞EZi2∑m:2m≥i2−2mδ≤∑i=1∞i−2δEZi2≤∑i=1∞i−1EZi2≤∑i=1∞i−1EXi2−iαIXi2>iα<∞,which yields (31). Thus, by (27) and (31), the desired result (26) follows. The proof of Theorem 5 is completed.


## 4. Conclusions

In this article, we obtain the maximal moment inequality for a sequence ofH-valued NSD random vectors by extending the maximal moment inequality for a sequence of NSD random variables in Wang et al. [10] (see Lemma 3). Using this maximal moment inequality, we investigate the complete convergence results (see Theorem 1), L2-convergence (see Theorems 2 and 3), and residual Cesàro alpha-integrability and strongly residual Cesàro alpha-integrability (see Theorems 4 and 5) for H-valued NSD random vectors.



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