On complete convergence for weighted sums of asymptotically linear negatively dependent random field

Statistics and Probability Letters 83 (2013) 2615–2620

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On complete convergence for weighted sums of asymptotically linear negatively dependent random field Mi-Hwa Ko Division of Mathematics and Informational Statistics, Wonkwang University, Jeonbuk, 570-749, Republic of Korea

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Article history: Received 18 April 2013 Received in revised form 28 June 2013 Accepted 14 August 2013 Available online 26 August 2013

In this paper we establish the complete convergence for weighted sums of asymptotically linear negatively quadrant dependent random field, which contains a linear negatively quadrant dependent field and a ρ ∗ -mixing random field. © 2013 Elsevier B.V. All rights reserved.

MSC: 60F05 60F15 Keywords: Complete convergence Linear negatively quadrant dependence ρ ∗ -mixing Weighted sums Asymptotically linear negatively quadrant dependence

1. Introduction and definitions Lehmann (1966) introduced the concept of negatively (positively) quadrant dependence as follows: two random variables X and Y are said to be negatively quadrant dependent (NQD) (resp. positively quadrant dependent (PQD)) if P (X ≤ x, Y ≤ y) − P (X ≤ x)P (Y ≤ y) ≤ 0[resp. ≥ 0] for all real numbers x, y. A field {Xk , k ∈ Zd+ } is said to be linear negatively [resp. positively] quadrant dependent (LNQD) [resp. (LPQD)] if for   any disjoint finite subsets A, B ⊂ Zd+ and any positive real numbers ri , rj , i∈A ri Xi and j∈B rj Xj are NQD [resp. PQD]. (See Newman, 1984.) A field {Xk , k ∈ Zd+ } is said to be negatively associated (NA) [resp.positively associated (PA)] if for any disjoint finite subsets A, B ⊂ Zd+ and for any two increasing functions f : RA → R, g : RB → R, Cov(f (Xi , i ∈ A), g (Xj , j ∈ B)) ≤ 0[resp. ≥ 0]. The concepts of NA and PA were given by Joag-Dev and Proschan (1983) and Esary et al. (1967), respectively. For two nonempty disjoint subsets A, B ⊂ Zd+ , we define dist(A, B) to be inf{∥x − y∥ =

Let σ (A) be the σ -field generated by {Xn , n ∈ A} and σ (B) similarly. Let

ρ ∗ (k) = sup(ρ(A, B); dist(A, B) ≥ k, A, B ⊂ Zd+ ) where ρ(A, B) = sup(

|Cov(f ,g )| 1

1

(Var f ) 2 (Var g ) 2

; f ∈ L2 (σ (A)), g ∈ L2 (σ (B))).

E-mail address: [email protected]. 0167-7152/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2013.08.009

d

k=1

(xk − yk )2 ; x ∈ A, y ∈ B}.

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M.-H. Ko / Statistics and Probability Letters 83 (2013) 2615–2620

A field {Xn , n ∈ Zd+ } is said to be ρ ∗ -mixing if

ρ ∗ (k) → 0 as k → ∞. See Peligrad and Gut (1999). Now we consider these two kinds of dependent (LNQD and ρ ∗ -mixing) random fields simultaneously. Define a measure of dependence of X and Y by

 ρ (X , Y ) = 0 ∨ sup −



Cov(f (X ), g (Y )) 1

1

(V (f (X ))) 2 (V (g (Y ))) 2

where the sup is taken over f , g ∈ C such that E [f (X )]2 < ∞ and E [g (Y )]2 < ∞ and C = {f = f (x1 , . . . , xd ) : f is coordinatewise increasing, d ≥ 1}. For any disjoint subsets A, B ⊂ Zd+ , define

ρ − (A, B) = sup(ρ − (X , Y ), X ∈ F (A), Y ∈ F (B))  where F (A) = { k∈A ak Xk , ak ≥ 0 and ak ̸= 0 for finitely many k′ s}, and F (B) is defined similarly. Definition (Zhang, 2000). A field {Xn , n ∈ Zd+ } is said to be asymptotically linear negatively quadrant dependent (ALNQD) if

ρ − (k) = sup{ρ − (A, B), dist(A, B) ≥ k, A, B ⊂ Zd+ are finite} → 0 as k → ∞.

(1.1)

It is obvious that either an LNQD field or a ρ ∗ -mixing field is an ALNQD field. Let Zd+ (d ≥ 2) denote the positive integer d-dimensional lattice with coordinatewise partial ordering ≤. The notation m ≤ n, where m = (m1 , m2 , . . . , md ) and d n = (n1 , n2 , . . . , nd ) thus means that mk ≤ nk , for k = 1, 2, . . . , d. We also use |n| for k=1 nk and n → ∞ means min1≤k≤d nk → ∞, that is all coordinates tend to infinity. Note that |n| → ∞ is equivalent to max{n1 , n2 , . . . , nd } → ∞ which is weaker than the condition min{n1 , n2 , . . . , nd } → ∞ when d ≥ 2. Let{Xn , n ∈ Zd+ } be a field of random variables, and {an,k , n ∈ Zd+ , k ≤ n} be an array of real numbers. The weighted

sum k≤n an,k Xk can play an important role in various applied and theoretical problems, such as those of the least squares estimators (see Kafles and Bhaskara Rao, 1982) and M estimators (see Rao and Zhao, 1992) in linear models, the nonparametric regression estimators (see Priestley and Chao, 1972), the design regression estimators (see Gu et al., 2007), etc. So the study of the limiting behavior of the weighted sums is very important and significant. The aim of this paper is to give a notion of asymptotically linear negatively quadrant dependence (ALNQD) and some d results concerning complete convergence of weighted sums i≤n an,i Xi , where {an,i , n ∈ Z+ , i ≤ n} is an array of real numbers and {Xn , n ∈ Zd+ } is a field of ALNQD random variables. 2. Main result The following theorem is the main result of this paper. Theorem 2.1. Let {Xn , n ∈ Zd+ } be a random field satisfying (1.1) and {an,i , n ∈ Zd+ , i ≤ n} an array of positive numbers. Let α p > 1, α > 21 and for some q > 2,

  |n|αp−2 P (|an,i Xi | > |n|α ) < ∞, n α(p−q)−2i≤n q  (b) |n| 2 i≤n |an,i |q E (|Xi |q I [|an,i Xi | ≤ |n|α ]) < ∞, n |n|  (c) maxj≤n | i≤j E (an,i Xi I [|an,i Xi | ≤ |n |α ])| = o(|n|α ).

(a)

Then,



|n|αp−2 P {max |Sj | ≥ ϵ|n|α } < ∞ for all ϵ > 0,

(2.1)

j≤ n

n



where Sj = i ≤ j an , i X i . To prove Theorem 2.1 we need the following lemma. Lemma 2.2 (Zhang, 2000). Let {Xk , k ∈ Zd+ } be a centered ALNQD random field. Then for any q > 2 there exists a positive constant Dq = D(q, ρ − (·)) such that

   q   E max  Xj  ≤ Dq |n|q/2 max E |Xj |q . m≤n   j≤ n j≤ m

(2.2)

M.-H. Ko / Statistics and Probability Letters 83 (2013) 2615–2620

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Proof of Theorem 2.1. Let, for i ≤ n Xn′ ,i = Xi I [|an,i Xi | ≤ |n|α ],

(2.3)

Yn,i = an,i Xn,i − E (an,i Xn,i ) ′

′

′

and Sn,k =



Yn,i .

i≤k

Let us notice that if the series n |n|α p−2 is convergent, then (2.1) always holds. Therefore we consider only the case such  that n |n|α p−2 is divergent. Let C be a positive constant which is not the same in each of its appearances. By simple computations, using (2.3) and the assumption (c), we obtain the following estimation



P [max |Sj | > ϵ|n|α ] ≤ P [max |Sj | > ϵ|n|α , |an,1 X1 | ≤ |n|α , . . . , |an,n Xn | ≤ |n|α ] j≤ n

j≤n

+ P [max |Sj | > ϵ|n|α , |an,1 X1 | > |n|α ∪ · · · ∪ |an,n Xn | > |n|α ] j≤n         α  α ≤ P max  an,i Xi I [|an,i Xi | ≤ |n| ] > ϵ|n| + P [|an,i Xi | > |n|α ]  j≤ n  i ≤j i ≤n       ′ α  α ≤ P max |Sn,j | > ϵ|n| − max  E (an,i Xi I [|an,i Xi | ≤ |n| ])  j≤ n j≤ n  i≤j  + P [|an,i Xi | > |n|α ].

(2.4)

i≤n

Thus, by (2.4), the Markov inequality, Cr inequality, (c) and Lemma 2.2, we get

 α

P [max |Sj | > ϵ|n| ] ≤ C

 α

P [max |Sn,j | > ϵ|n| ] + ′

j≤ n

j≤n



α

P [|an,i Xi | > |n| ]

i≤n

         α α + P [|an,i Xi | > |n| ] Yn,i  > ϵ|n| = C P max   j≤ n  i ≤n i≤j    q q α −α q 2 P [|an,i Xi | > |n | ] ≤ C |n| |n| max E |Yn,i | +  

i≤n



i≤n

q

= C |n |−αq |n | 2 max E |an,i Xi I [|an,i Xi | ≤ |n|α ] i≤n

 α

− E (an,i Xi I [|an,i Xi | ≤ |n| ]) | + q



α

P [|an,i Xi | > |n| ]

i≤n





q 2

≤ C |n|−αq |n| max E |an,i Xi |q I [|an,i Xi | ≤ |n|α ] + i≤n

 −α q

≤ C |n|

|n|

q 2



P [|an,i Xi | > |n|α ]

i ≤n

 

q

α

E |an,i Xi | I [|an,i Xi | ≤ |n| ] +

i≤n



α

P [|an,i Xi | > |n| ] .

(2.5)

i≤n

Therefore, from (a), (b) and (2.5), the result (2.1) follows. The proof of Theorem 2.1 completes. 3. Some corollaries Using Theorem 2.1 we obtain the following results. Corollary 3.1. Let {Xn , n ∈ Zd+ } be a field of ALNQD random variables with EXn = 0 and E |Xn |q < ∞ for q > 2 and for all n ∈ Zd+ . Let {an,i , n ∈ Zd+ , i ≤ n} be an array of real numbers. Let α p > 1, α > with 0 < λn ≤ 1 and some q > 2



|n|αp−2 |n|−α(1+λn )+q/2

n

Then, for all ϵ > 0 (2.1) holds.

 i ≤n

|an,i |1+λn E |Xi |1+λn < ∞.

1 . 2

Assume that for some field {λn , n ∈ Zd+ }

(3.1)

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M.-H. Ko / Statistics and Probability Letters 83 (2013) 2615–2620

Proof. First, note that E |Xn |1+λn < ∞ since q > 1 + λn . If n |n|α p−2 is finite then (2.1) always holds. Hence we consider  only the case that n |n|α p−2 is divergent. It follows from (3.1) that



J = |n|−α(1+λn ) |n|q/2



|an,i |1+λn E |Xi |1+λn < 1.

(3.2)

i ≤n

|n|αp−2 = ∞ we obtain    |n|αp−2 |n|−α(1+λn )+q/2 |an,i |1+λn E |Xi |1+λn = |n|αp−2 J

Note that if J ≥ 1 then by the fact that

n



n

n

i≤n

≥



|n|αp−2

n

≥ ∞. By (3.1) and the Markov inequality we obtain



|n|αp−2

n



P (|an,i Xi | > |n|α ) ≤



|n|αp−2 |n|−α(1+λn )



n

i≤n

|an,i |1+λn E |Xi |1+λn

i≤n

<∞

(3.3)

and for q > 2



|n|αp−2 |n|−αq |n|q/2

n



|an,i |q E (|Xi |q I [|an,i Xi | ≤ |n|α ])

i≤n

≤



|n|αp−2−α(1+λn ) |n|q/2

n



|an,i |1+λn E |Xi |1+λn < ∞

(3.4)

i≤n

since q > (1 + λn ). Hence, (3.3) and (3.4) satisfy (a) and (b), respectively. Finally, it follows from (3.2) and the fact EXn = 0 that −α

|n|

          α  −α α  an,i E (Xi I [|an,i Xi | > |n| ]) an,i E (Xi I [|an,i Xi | ≤ |n| ]) = |n| max  max    j≤ n  j≤ n  i ≤j i≤j  ≤ |n|−α max |an,i |E (|Xi |I [|an,i Xi | > |n|α ]) j≤ n

≤ |n|

−α



i≤j

|an,i |E (|Xi |I [|an,i Xi | > |n|α ])

i≤n

≤ |n|−α

 E (|an,i Xi |1+λn I [|an,i Xi | > |n|α ]) |n|αλn

i≤n

≤ |n|−α(1+λn )



|an,i |1+λn E |Xi |1+λn

i≤n

=

J

|n|q/2

→0

as |n| → ∞, which satisfies (c). Hence, the proof is completed. Corollary 3.2. Let {Xn , n ∈ Zd+ } be a field of ALNQD random variables with EXn = 0 and E |Xn |p < ∞ for 1 < p ≤ 2. Let {an,i , n ∈ Zd+ , i ≤ n} be an array of real numbers. If



 q |an,i |p E |Xi |p = O n− 2

for q > 2

(3.5)

i ≤n

then (2.1) holds. Proof. By (3.5) and the Markov inequality we have

|n|αp−2

 i≤n

P (|an,i Xi | > |n|α ) ≤ |n|α p−2

 |an,i |p E |Xi |p i≤n q

≤ C |n|−2− 2 ,

|n|αp

M.-H. Ko / Statistics and Probability Letters 83 (2013) 2615–2620

2619

which satisfies (a). Since p < q

|n|α(p−q)−2 |n|q/2



|an,i |q E (|Xi |q I [|an,i Xi | ≤ |n|α ]) ≤ |n|α(p−q)−2 |n|q/2

i ≤n



|an,i |p E |Xi |p |n|α(q−p)

i≤n

≤ |n|−2 < ∞, which satisfies (b). −α

|n|

          α  −α α  max  an,i E (Xi I [|an,i Xi | ≤ |n| ]) = |n| max  an,i E (Xi I [|an,i Xi | > |n| ])   j ≤n  j≤ n  i≤j i≤j  ≤ |n|−α max |an,i |E (|Xi |I [|an,i Xi | > |n|α ]) j≤ n

≤ |n|−α



i ≤j

|an,i |E (|Xi |I [|an,i Xi | > |n|α ])

i ≤n

−α p

≤ |n|



|an,i |p E |Xi |p

i≤n

≤ |n|−(αp+ 2 ) → 0 as |n| → ∞, q

which satisfies (c). Hence, the proof is completed. Corollary 3.3. Let {Xn , n ∈ Zd+ } be a field of ALNQD random variables with EXn = 0 and E |Xn |p < ∞ for 1 < p ≤ 2 and {Xn } stochastically dominated by a random variable X , such that E |X |p < ∞. Let {an,i , n ∈ Zd+ , i ≤ n} be an array of real numbers. If



 q |an,i |p = O n− 2

for q > 2,

i≤n

then (2.1) holds. Proof. The proof is similar to that of Corollary 3.2. Kuczmaszewska and Lagodowski (2011) proved the following complete convergence for the ρ ∗ -mixing random field as follows. Theorem 3.4. Let {Xn , n ∈ Zd+ } be a ρ ∗ -mixing random field. Let α p > 1, α > (i) (ii) (iii) (iv)

1 2

and for some q ≥ 2,

  α |n|αp−2 n P (|Xi | > |n| ) < ∞, n α(p−q)−2i≤ q |n| E (|Xi | I [|Xi | ≤ |n|α ]) < ∞, n α(p−q)−2 i≤n qd  q (log2 |n|) ( i≤n E (Xi2 I [|Xi | ≤ |n|α ])) 2 < ∞, n |n|  α α maxj≤n | i≤j E (Xi I [|Xi | ≤ |n | ])| = o(|n| ).

Then

       |n|αp−2 P max  Xi  > ϵ|n|α < ∞  j≤ n  i≤j 

 n

for all ϵ > 0. Ko (in press) generalized the above theorem to the weighted sums. Theorem 3.5 (Ko, in press). Let {Xn , n ∈ Zd+ } be a field of ρ ∗ -mixing random variables with EXn = 0. Let {an,i , n ∈ Zd+ , i ≤ n} be an array of weights. If for α p > 1, α > 12 and for some q ≥ 2,

  α |n|αp−2 n P (|an,i Xi | > |n| ) < ∞, n α(p−q)−2i≤ q q (b) |n| |an,i | E (|Xi | I [|an,i Xi | ≤ |n|α ]) < ∞, n α(p−q)−2 i≤n qd  q (c) (log2 |n|) ( i≤n a2n,i E (Xi2 I [|an,i Xi | ≤ |n|α ])) 2 , n |n|  (d) maxj≤n | i≤j an,i E (Xi I [|an,i Xi | ≤ |n |α ])| = o(|n|α ), (a)

then (2.1) holds. Remarks. 1. Theorem 2.1 still holds under either an LNQD random field or a ρ ∗ -mixing random field. 2. Theorems 3.4 and 3.5 do not hold under an ALNQD random field.

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M.-H. Ko / Statistics and Probability Letters 83 (2013) 2615–2620

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