On complete moment convergence for CAANA random vectors in Hilbert spaces

STAPRO: 8190

Model 3G

pp. 1–7 (col. fig: nil)

Statistics and Probability Letters xx (xxxx) xxx–xxx

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On complete moment convergence for CAANA random vectors in Hilbert spaces Mi-Hwa Ko Division of Mathematics and Informational Statistics, Wonkwang University, Jeonbuk, 54358, Republic of Korea

article

info

Article history: Received 25 September 2017 Received in revised form 22 December 2017 Accepted 28 February 2018 Available online xxxx

a b s t r a c t For a sequence {Xn , n ≥ 1} of coordinatewise asymptotically almost∑ negatively associated ∞ random vectors in Hilbert space, we investigate the convergence of n=1 n11+α E(max1≤k≤n

∑ ∑k log n + α + ∥Sk ∥ − ϵ nα )+ and ∞ = n=1 n1+α E(max1≤k≤n ∥Sk ∥ − ϵ n ) , where Sk = i=1 Xi and a max{a, 0}. This investigation provides the complete moment convergence for the case α r = 1. © 2018 Elsevier B.V. All rights reserved.

MSC: 60F15 60B12 Keywords: Complete moment convergence Coordinatewise asymptotically almost negatively associated random vectors Hilbert spaces

1. Introduction

1

A sequence {Xn , n ≥ 1} of random variables is said to be asymptotically almost negatively associated(AANA) if there exists a nonnegative sequence q(m) → 0 as m → ∞ such that Cov (f (Xm ), g(Xm+1 , Xm+2 , . . . , Xm+k ))

≤ q(m)(Var(f (Xm ))Var(g(Xm+1 , Xm+2 , . . . , Xm+k )))

(1.1) 1 2

for all, m, k ≥ 1 and for all coordinatewise nondecreasing continuous functions f and g whenever the right side of (1.1) is finite. The concept of AANA was introduced by Chandra and Ghosal (1996a, b). Obviously, AANA random variables contain independent random variables (with q(n) = 0 for n ≥ 1) and NA random variables (see Chandra and Ghosal, 1996a). Since the concept of AANA was introduced, various kinds of investigations for AANA random variables have been established. For more details, we can refer to Ko et al. (2005), Yuan and An (2009, 2012), Wang et al. (2010), Tang (2013), Shen and Wu (2014), and so forth. As Ko et al. (2009) extended the concept of negative association to Rd -valued random vectors and H-valued random vectors the concept of AANA can be extended to finite dimensional random vectors and to Hilbert space valued random vectors. Definition 1.1. A sequence {Xn , n ≥ 1} of Rd -valued random vectors is said to be AANA if (1.1) holds. E-mail address: [email protected]. https://doi.org/10.1016/j.spl.2018.02.068 0167-7152/© 2018 Elsevier B.V. All rights reserved.

Please cite this article in press as: Ko M.-H., On complete moment convergence for CAANA random vectors in Hilbert spaces. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.068.

2 3

4 5 6 7 8 9 10 11 12 13 14 15

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1 2

3 4

5 6 7

8 9

10 11 12 13 14 15 16 17 18 19 20 21

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M.-H. Ko / Statistics and Probability Letters xx (xxxx) xxx–xxx

Let H be a real separable Hilbert space with the norm ∥ · ∥ generated by an inner product ⟨·, ·⟩. Let {ej , j ≥ 1} be an orthonormal basis in H and ⟨X , ej ⟩ be denoted by X (j) , where X is an H-valued random vector. Definition 1.2. A sequence {Xn , n ≥ 1} of H-valued random vectors is said to be AANA if for some orthonormal basis {ek , k ≥ 1} in H and for any d ≥ 1 the d-dimensional sequence {(⟨Xi , e1 ⟩, . . . , ⟨Xi , ed ⟩), i ≥ 1} is AANA. As Huan (2015) introduced the concept of H-valued CNA random vectors we can introduce another concept of asymptotically almost negative association for H-valued random vectors which is more general than the concept of H-valued AANA random vectors as follows. Definition 1.3. A sequence {Xn , n ≥ 1} of H-valued random vectors is said to be coordinatewise asymptotically almost (j) negatively associated(CAANA) if for each j ≥ 1, the sequence {Xn , n ≥ 1} of random variables is AANA.

∑∞

A sequence {Xn , n ≥ 1} of random variables is said to converge completely to a constant C if n=1 P(|Xn − C | > ϵ ) < ∞ ∑∞ q 1 for all ϵ > 0 and a sequence {Xn , n ≥ 1} is called complete moment convergence if n=1 E(b− n |Xn | − ϵ )+ < ∞ for all ϵ > 0, where an > 0, bn > 0 and q > 0. The concept of complete convergence was introduced by Hsu and Robbins (1947) and the concept of complete moment convergence was introduced by Chow (1988), respectively. Recently, Huan et al. (2014) extended the Baum–Katz theorem to a sequence of H-valued coordinatewise negatively associated random vectors for the case r > α1 and Huan (2015) studied this problem for the case r = α1 . In this paper we investigate the complete moment convergence of H-valued CAANA random vectors for the case r = α1 . Throughout this paper, the symbol C will denote a generic positive constant which is not necessarily the same one in each appearance. The logarithms are to the base 2, for a ∈ R, log(max{2, a}) will be denoted by log+ a. Let {X , Xn , n ≥ 1} be a sequence of H-valued random vectors. We consider the following inequalities C1 P(|X (j) | > t) ≤

n 1∑

n

(j)

P(|Xk | > t) ≤ C2 P(|X (j) | > t).

(1.2)

k=1

28

If there exists a positive constant C1 (C2 ) such that left-hand side (right-hand side) of (1.2) is satisfied for all j ≥ 1, n ≥ 1 and t ≥ 0, then the sequence {Xn , n ≥ 1} is said to be coordinatewise weakly lower (upper) bounded by X . The sequence {Xn , n ≥ 1} is said to be coordinatewise weakly bounded by X if it is both coordinatewise weakly lower and upper bounded by X . Note that (1.2) is, of course, automatic with X = X1 and C1 = C2 = 1 if {Xn , n ≥ 1} is a sequence of identically distributed random vectors.

29

2. Some lemmas

23 24 25 26 27

30 31 32

33 34

35

Lemma 2.1. Let {Xn , n ≥ 1} be a sequence of AANA random variables with mixing coefficients {q(n), n ≥ 1} and let {fn , n ≥ 1} be a sequence of all nondecreasing continuous functions. Then {fn (Xn ), n ≥ 1} is still a sequence of AANA random variables with mixing coefficients {q(n), n ≥ 1}. Lemma 2.2. ∑∞Let 2{Xn , n ≥ 1} be a sequence of H-valued AANA random vectors with mixing coefficients {q(n), n ≥ 1} and zero means. If n=1 q (n) < ∞, there exists a positive constant C such that E( max ∥ 1≤k≤n

36 37

38

Xi ∥2 ) ≤ C

i=1

n ∑

(2.1)

i=1

E max ∥

k ∑

Xi ∥2 = E max

1≤k≤n

i=1

≤

∞ ∑ j=1

40

E ∥Xi ∥2 .

Proof. From the last part of the proof of Theorem 1 in Chandra and Ghosal (1996a) we obtain the following maximal inequality

1≤k≤n

39

k ∑

=

∞ ∑ j=1

∞ k ∑ ∑ (⟨ Xi , ej ⟩)2 j=1

i=1

E max (⟨ 1≤k≤n

E max ( 1≤k≤n

k ∑

Xi , ej ⟩)2

i=1 k ∑ ⟨Xi , ej ⟩)2 i=1

Please cite this article in press as: Ko M.-H., On complete moment convergence for CAANA random vectors in Hilbert spaces. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.068.

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≤C

∞ ∑ n ∑

3

E(⟨Xi , ej ⟩)2

1

j=1 i=1

=C

n ∑

E ∥Xi ∥2 .

2

i=1

Inspired by Huan (2015) (2.3) and (2.6) can be proved and thus the proof of (2.3) and (2.6) is omitted.

3

(2.4) and (2.7) are quite similar, so we will only show (2.7).

4

Lemma 2.3. Let α be a positive real number such that 21 < α < 1 and let {Xn , n ≥ 1} be a sequence of H-valued CAANA random vectors with mixing coefficients {q(n), n ≥ 1} and zero means. Suppose that {Xn , n ≥ 1} is coordinatewise weakly upper bounded by a random vector X . If ∞ ∑

5 6 7

1

E |X (j) | α < ∞,

(2.2)

8

j=1

then we obtain ∞ ∑ 1 n=1

n

9

P( max ∥

k ∑

1≤k≤n

Xl ∥ > ϵ nα ) < ∞ for all ϵ > 0

(2.3)

10

l=1

and

11

∞ ∑ n=1

1 nα+1

k

∞

∫

P( max ∥

nα

1≤k≤n

∑

Xi ∥ > u)du < ∞ for all u > 0.

(2.4)

Lemma 2.4. Let α be a positive real number such that 12 < α ≤ 1 and let {Xn , n ≥ 1} be a sequence of H-valued CAANA random vectors with mixing coefficients {q(n), n ≥ 1} and zero means. Suppose that {Xn , n ≥ 1} is coordinatewise weakly upper bounded by a random vector X . If ∞ ∑

12

i=1

13 14 15

1

E(|X (j) | α log+ |X (j) |) < ∞,

(2.5)

16

j=1

then we obtain

17

∞ ∑ log n

n

n=1

P( max ∥ 1≤k≤n

k ∑

Xl ∥ > ϵ nα ) < ∞ for all ϵ > 0

(2.6)

18

l=1

and

19

∞ ∑ n=1

log n nα+1

k

∞

∫

P( max ∥

nα

∑

1≤k≤n

Xi ∥ > u)du < ∞ for all u > 0.

(2.7)

Proof of 2.7. For all u > 0 and j ≥ 1 set (j)

(j)

(j)

21

(j)

(j)

Yui = Xi I(|Xi | ≤ u) − uI(Xi < −u) + uI(Xi > u).

(2.8)

Then, we have ∞ ∑ n=1

=

≤

log n nα+1

nα+1

∫ ∞ ∑ log n n=1

+

nα+1

k

∞ nα

P( max ∥ 1≤k≤n

∞ nα

P( max ∥ 1≤k≤n

∑

Xi ∥ > u)du

24

i=1 k ∞ ∑ ∑

(j)

Xi ej ∥ > u)du

25

i=1 j=1

∞ (j)

nα

P( max max|Xk | > u)du

∫ ∞ ∑ log n n=1

22 23

∫

∫ ∞ ∑ log n n=1

20

i=1

nα+1

26

1≤k≤n j≥1

∞ nα

P( max ∥ 1≤k≤n

k ∞ ∑ ∑

(j)

Yui ej ∥ > u)du

i=1 j=1

Please cite this article in press as: Ko M.-H., On complete moment convergence for CAANA random vectors in Hilbert spaces. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.068.

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≤

n=1

2

+

nα+1 ∞ ∑ n=1

3

4 5

6

nα

j=1 i=1

log n nα+1

nα

I1 =

≤C

nα+1

nα

Yui ∥ > u)du

i=1

=C

∞ ∫ ∑

nα

≤C

≤C

≤C

=C

∞

∞ ∫ ∑

P(|X (j) | > u)du (by (1.2)) ∞

∫

xα ∞

xα

1

y α −2 log y

P(|X (j) | > u)dudx ∞

∫

1

∞ ∫ ∑

(2.9)

P(|X (j) | > u)dudy (letting xα = y)

y

∞

P(|X (j) | > u)

u

∫

1

∞ ∫ ∑

∞ ∑

∞ nα

log x

1

(j)

P(|Xi | > u)du

j=1 i=1

∫ ∞ ∑ ∞ ∑ log n

1

y α −2 log ydydu 1

∞

1

u α −1 log uP(|X (j) | > u)du 0

j=1

12

∞ ∑ n ∞∑

∫ ∞ ∑ log n

j=1

11

1≤k≤n

∫∞

j=1

10

P( max ∥

Note that E |Y |p = p 0 yp−1 P(|Y | > y)dy. For I1 , by the Markov inequality, (1.2) and (2.5) we have

j=1

9

k ∑

∞

∫

j=1 n=1

8

(j)

P(|Xi | > u)du

= I1 + I2 .

n=1

7

∞ ∑ n ∞∑

∫ ∞ ∑ log n

1

E(|X (j) | α log+ |X (j) |) < ∞ (by (2.5)).

j=1 13

14

For I2 , we estimate that I2 =

∫ ∞ ∑ log n n=1

15

≤C

nα+1

18 19

20

+C

∞

P( max ∥

nα

nα+1

k ∑

1≤k≤n

Yui − EYui ∥ >

i=1

∞

P( max ∥

nα

Yui ∥ > u)du

i=1

k ∑

1≤k≤n

EYui ∥ >

i=1

u 2

u 2

)du

)du

(j)

(j)

Since {Yui − EYui , i ≥ 1} is CAANA for all j ≥ 1, and so {Yui − EYui } is AANA. Hence by the Markov inequality and Lemma 2.2 we have I21 ≤ C

∫ ∞ ∑ log n

≤C

≤C

≤C

nα+1

nα+1

u−2 E( max ∥ 1≤k≤n

∞

nα

∫ ∞ ∑ log n nα+1

∞

nα

∫ ∞ ∑ log n

n=1

23

k ∑

= I21 + I22 .

n=1

22

1≤k≤n

∫ ∞ ∑ log n

n=1

21

nα

nα+1

n=1 17

P( max ∥

∫ ∞ ∑ log n n=1

16

∞

u−2

u−2

j=1 n=1

i=1

E ∥Yui − EYui ∥2 du (by (2.1))

n ∑

E ∥Yui ∥2 du

i=1

∫ ∞ ∑ ∞ ∑ log n nα+1

(Yui − EYui )∥)2 du

i=1

∞

nα

n ∑

k ∑

∞

nα

u−2

n ∑

(j)

E(Yui )2 du

i=1

Please cite this article in press as: Ko M.-H., On complete moment convergence for CAANA random vectors in Hilbert spaces. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.068.

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≤C

∫ ∞ ∑ ∞ ∑ log n nα

j=1 n=1

+C

∞

P(|X (j) | > u)du

nα

∫ ∞ ∑ ∞ ∑ log n nα

j=1 n=1

5

∞

1

2

u−2 E(|X (j) | I(|X (j) | ≤ u))du

nα

(2.10)

= I211 + I212 .

3

For I211 , by (2.9) we have that I211 < ∞. For I212 , by a standard calculation we observe that I212 = C

∞ ∑

≤C

nα

∞ ∑ ∞ ∑

u−2 E((X (j) )2 I(|X (j) | ≤ u))du

2

E(|X (j) | I(|X (j) | ≤ (m + 1)α )) mα+1

∞ ∑ ∞ m+1 ∑ log m ∑

∞ ∑ ∞ ∑

5

6

m−α−1 E((X (j) )2 I(|X (j) | ≤ (m + 1)α ))

1

m2α

j=1 m=1

≤C

4

(2.11)

7

j=1 m=n

j=1 m=1

≤C

mα

j=1 m=n

∞ ∞ ∞ ∑ log n ∑ ∑ n=1

=C

log n

∞ ∑ ∞ ∫ (m+1)α ∑

nα

n=1

2

m ∑ log n n=1

8

nα

2

E(|X (j) | I((n − 1)α < |X (j) | ≤ nα ))

9

n=1

m log mP((m − 1)α < |X (j) | ≤ mα )

10

j=1 m=1

≤C

∞ ∑

1

E(|X (j) | α log+ |X (j) |) < ∞.

11

j=1 (j)

It remains to prove I22 < ∞. From (1.2), (2.8) and the fact that EXi = 0 for all i ≥ 1 and j ≥ 1 we obtain I22 = C

∞ ∑ n=1

≤C

nα+1

P( max ∥

nα

nα+1

k

∞ 1≤k≤n

∞

P(

nα

∫ ∞ ∑ log n n=1

≤C

nα+1

∫ ∞ ∑ log n n=1

≤C

log n

∫

+C

nα

u−1

u 2

u 2

)du

)du

13

(2.12)

E ∥Yui ∥du

(P(|X (j) | > u))du

∫ ∞ ∑ ∞ ∑ log n nα

≤C

u−1 E(|X (j) |I(|X (j) | ≤ u))du

nα

≤C

nα

n2α

19 20

∞ nα

u−1 E(|X (j) |I(|X (j) | ≤ u))du

∞ ∑ ∞ ∞ ∫ ∑ log n ∑

∞ ∑

17

18

∫ ∞ ∑ ∞ ∑ log n

j=1 n=1

16

∞

By (2.9) we have that I221 < ∞. For I222 , by a standard calculation as in (2.11) we obtain

j=1 n=1

15

∞

= I221 + I222 .

I222 = C

14

i=1

nα

j=1 n=1

i=1

E ∥Yui ∥ >

n ∑

∞ nα

EYui ∥ >

i=1

∫ ∞ ∑ ∞ ∑ log n j=1 n=1

n ∑

∑

12

m=n

(m+1)α mα

E(|X (j) |I(|X (j) | ≤ (m + 1)α ))du

21

22

1

E(|X (j) | α log+ |X (j) |) < ∞,

23

j=1

which yields I22 < ∞, together with I221 < ∞. Hence, the proof is completed. Please cite this article in press as: Ko M.-H., On complete moment convergence for CAANA random vectors in Hilbert spaces. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.068.

24

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2 3

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M.-H. Ko / Statistics and Probability Letters xx (xxxx) xxx–xxx

Remark. (2.3) and (2.6) are the complete convergence for sequences of H-valued CAANA random vectors for the case r =

1

α

.

Lemma 2.5. Let α be a positive real number such that 21 < α < 1 and let {Xn , n ≥ 1} be a sequence of H-valued CAANA random vectors with mixing coefficients {q(n), n ≥ 1} and zero means. If {Xn , n ≥ 1} are identically distributed random vectors with ∞ ∑

1

(j) α

< ∞,

E | X1 |

(2.2′ )

j=1 5

6 7

8

(j)

where X1 = ⟨X1 , ej ⟩, then (2.3) and (2.4) hold. Lemma 2.6. Let α be a positive real number such that 12 < α ≤ 1 and let {Xn , n ≥ 1} be a sequence of H-valued CAANA random vectors with mixing coefficients {q(n), n ≥ 1} and zero means. If {Xn , n ≥ 1} are identically distributed random vectors with ∞ ∑

1

(j) α

(j)

E(|X1 | log+ |X1 |) < ∞,

(2.5′ )

j=1 9

10

11

12 13 14

15

then (2.6) and (2.7) hold. 3. Main results The proofs of Theorems 3.1 and 3.3 are quite similar, so we will only show Theorem 3.3. Theorem 3.1. Let α be a positive real number such that 12 < α < 1. Let {Xn , n ≥ 1} be a sequence of H-valued CAANA random vectors with mixing coefficients {q(n), n ≥ 1} and zero means. If {Xn , n ≥ 1} is coordinatewise weakly upper bounded by a random vector X and satisfying (2.2), then we obtain ∞ ∑ 1 n=1

16

17 18 19

20 21 22

23

n

E( max ∥ α+1 1≤k≤n

Proof. It is sufficient to prove that (2.3) and (2.4) hold. To prove (2.3): See Example 3.2 in Huan (2015). To prove (2.4):

∫ ∞ ∑ 1

≤C

nα+1

≤C

=C

=C

29 30 31

32

nα

nα+1

nα

Xi ∥ > u)du

i=1

∞

u−2 E( max ∥ 1≤k≤n

∞

u−2

nα

u−2 (

j=1

Xi ∥)2 du

i=1

E ∥Xi ∥2 du

∞ ∑ 1 j=1

∞ ∞ ∑ 1 ∑ 1

(

n ∑

k ∑

i=1

∞ nα

n2α

k ∑

1≤k≤n

nα

nα+1

∫ ∞ ∑ 1

n=1 28

P( max ∥

∫ ∞ ∑ 1

n=1

27

∞

∫ ∞ ∑ 1

n=1

26

(3.1)

i=1

Example 3.2. Consider the space l2 consisting of square summable real sequences x = {xk , k ≥ 1} with norm ∥x∥ = ∑∞ 1 ( k=1 x2k ) 2 . Let α be a real number ( 12 < α < 1) and let {X , Xn , n ≥ 1} be a sequence of l2 -valued i.i.d. random vectors with P(X (j) = ±j−α ) = 12 for all j ≥ 1. It is well known that the space l2 is of type 2. Then for every ϵ > 0, (3.1) holds.

n=1

25

Xi ∥ − ϵ nα )+ < ∞,

where a+ = max{a, 0}.

n=1

24

k ∑

j2α

j2α

)du

)<∞

so that (2.4) holds. Thus the proof is complete. Theorem 3.3. Let α be a positive real number such that 12 < α ≤ 1. Let {Xn , n ≥ 1} be a sequence of H-valued CAANA random vectors with zero means. If {Xn , n ≥ 1} is coordinatewise weakly upper bounded by a random vector X and satisfying (2.5) then we obtain ∞ ∑ log n n=1

n

E( max ∥ α+1 1≤k≤n

k ∑

Xi ∥ − ϵ nα )+ < ∞.

(3.2)

i=1

Please cite this article in press as: Ko M.-H., On complete moment convergence for CAANA random vectors in Hilbert spaces. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.068.

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7

Proof. From (2.6) and (2.7) we obtain ∞ ∑ n=1

=

1≤k≤n

n1+α

P(( max ∥

+

P( max ∥

n

P( max ∥

n=1

1≤k≤n

nα

1≤k≤n

n1+α

k ∑

k ∑

P( max ∥ 1≤k≤n k ∑

3

Xi ∥ − ϵ nα > u)du

4

Xi ∥ − ϵ nα > u)du

5

k ∑

Xi ∥ − ϵ nα > u)du

6

i=1

Xi ∥ > ϵ nα )

7

i=1

∞ nα

Xi ∥ − ϵ nα )+ > u)du

i=1

∞

P( max ∥

∫ ∞ ∑ log n

k ∑

i=1

nα

n1+α

∞ ∑ log n

+

1≤k≤n

0

2

i=1

∞

∫ ∞ ∑ log n n=1

n=1

1≤k≤n

0

n1+α

X i ∥ − ϵ nα ) +

i=1

0

n1+α

∑

∞

∫ ∞ ∑ log n n=1

≤

E( max ∥

∫ ∞ ∑ log n n=1

=

nα+1

∫ ∞ ∑ log n n=1

=

log n

1

k

P( max ∥ 1≤k≤n

k ∑

Xi ∥ > u)du

8

i=1

< ∞ (by (2.6) and (2.7)). Remark. Let α be a real number such that 21 < α ≤ 1. We consider the sequence {X , Xn , n ≥ 1} in Example 3.2. By using same arguments as in Example 3.2, we can show that (3.2) holds.

9

10 11

Corollary 3.4. Let α be a positive real number such that 21 < α < 1. Let {Xn , n ≥ 1} be a sequence of H-valued CAANA random vectors with coefficients {q(n), n ≥ 1} and zero means. If {Xn , n ≥ 1} are identically distributed and satisfying (2.2′ ), then (3.1) holds.

13

Corollary 3.5. Let α be a positive real number such that 12 < α < 1. Let {Xn , n ≥ 1} be a sequence of H-valued CAANA random vectors with coefficients {q(n), n ≥ 1} and zero means. If {Xn , n ≥ 1} are identically distributed and satisfying (2.5′ ), then (3.2) holds.

16

Acknowledgment

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This paper was supported by Wonkwang University in 2018. References Chandra, T.K., Ghosal, S., 1996a. Extensions of strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables. Acta Math. Hungar. 71, 327–336. Chandra, T.K., Ghosal, S., 1996b. The strong law of large numbers for weighted averages under dependence assumptions. J. Theoret. Probab. 9, 797–809. Chow, Y.S., 1988. On the rate of moment complete convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin. 16, 177–201. Hsu, P.L., Robbins, H., 1947. Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33 (2), 25–31. Huan, N.V., 2015. On the complete convergence for sequences of random vectors in Hilbert spaces. Acta Math. Hungar. 147 (1), 205–219. Huan, N.V., Quang, N.V., Thuan, N.T., 2014. Baum-Katz type theorems for coordinatewise negatively associated random vectors in Hilbert spaces. Acta Math. Hungar. 144 (1), 132–419. Ko, M.H., Kim, T.S., Han, K.H., 2009. A note on the almost sure convergence for dependent random variables in a Hilbert space. J. Theoret. Probab. 22, 506–513. Ko, M.H., Kim, T.S., Lin, Z.Y., 2005. The Hàjeck-Rènyi inequality for the AANA random variables and its applications. Taiwanese J. Math. 9 (1), 111–122. Shen, A.T., Wu, R.C., 2014. Strong convergence for sequences of asymptotically almost negatively associated random variables. Stochastics 86 (2), 291–303. Tang, X.F., 2013. Some strong laws of large numbers for weighted sums fo asymptotically almost negatively associated random variables. J. Inequal. Appl. http://dx.doi.org/10.1186/1029-242x-2013-4. Wang, X.J., Hu, S.H., Yang, W.Z., 2010. Convergence properties for asymptoticaly almost negatively associated sequence. Discrete Dyn. Nat. Soc. 15. Article ID 21830. Yuan, D.M., An, J., 2009. Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications. Sci. China Ser. A Math. 52, 1887–1904. Yuan, D.M., An, J., 2012. Laws of large numbers for Cesàro alpha-integrable random variables under dependence condition AANA or AQSI. Sin. Engl. Ser. 28 (6), 1103–1118.

Please cite this article in press as: Ko M.-H., On complete moment convergence for CAANA random vectors in Hilbert spaces. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.068.

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