H ´AJEK-R´ENYI INEQUALITY FOR DEPENDENT RANDOM VARIABLES IN HILBERT SPACE AND APPLICATIONS
YU MIAO
Abstract. In this paper, we obtain some H´ajek-R´enyi inequalities for sequences of Hilbert valued random variables which are associated, negatively associated andφ-mixing. As applications, we give some almost sure convergence theorems for these dependent sequences in Hilbert space. These results extend and improve some well-known results.
1. Introduction
In the paper [10], H´ajek and R´enyi established an inequality which they for-mulated in the following way: X1, X2,· · · are independent random variables and
Sn=Pni=1Xi, n≥1. For eachk,EXk = 0 andEXk2<∞, while{bk, k ≥1} is a non-increasing sequence of positive numbers. Then, for anyε >0 and any positive integersnandm(n < m),
P
max
n≤k≤mbk|Sk| ≥ε|
≤ 1
ε2 b 2
n n
X
k=1
EXk2+ m
X
k=n+1
b2kEXk2
!
. (1.1)
It is well-known that Kolmogorov’s inequality is the particular casebk = 1, for all kandn= 1 in (1.1).
Afterwards this inequality was extended to real valued martingales (see [4]). Since then, this inequality has been studied by many authors. For the case of R-valued random variables, Sung [22] obtained the H´ajek-R´enyi inequality for the associated sequence. Liu et al. [19] considered the negatively associated random variables. Cohn [5] studied a H´ajek-R´enyi inequality for Markov chain. T´om´acs and L´ıbor [23], Hu et al. [12] showed the inequality for demimartingale. For the case of Banach space, Gan [8] gave the H´ajek-R´enyi inequality for martingale. Furthermore, Gan and Qiu [9] studied a general version of this inequality.
In this paper, we extend the H´ajek-R´enyi inequality of associated, negatively associated and φ-mixing for random sequences to a separable real Hilbert space and derive the strong law of large numbers for these dependent sequences with values in a separable real Hilbert space.
2000 Mathematics Subject Classification. 60F05, 60F17, 60G10.
2. Associated sequences
Lehmann [17] introduced the notion of positive quadrant dependence: Two ran-dom variablesX1andX2are called positively quadrant dependent if
P(X1> x1, X2> x2)≥P(X1> x1)P(X2> x2), for all x1, x2.
This definition was subsequently extended to the multivariate case by Esary et al. in [6]. A finite sequence {Xi,1 ≤ i ≤ n} is said to be associated if for any componentwise nondecreasing functionsf andg onRn
Cov(f(X1, . . . , Xn), g(X1, . . . , Xn))≥0
where the covariance exists. An infinite sequence {Xn, n ≥ 1} is said to be as-sociated if every finite subfamily is asas-sociated. It is easy to see that if {Xn} is a sequence of associated random variables, then the covariance is nonnegative. For gaussian processes, it is well-known that positive association corresponds with posi-tive correlation. Let us recall that the independent random variables are associated and nondecreasing functions of associated random variables are also associated.
The definition of association has found several applications in reliability theory [1]. The basic concept actually appears in [11] in the context of percolation mod-els and it was subsequently applied to the Ising modmod-els of statistical mechanics in [7]; in the statistical mechanics literature (see, e.g., [16]), which developed inde-pendently of reliability theory, associated random variables are said to satisfy the FKG inequalities.
As in Burton et al. [3] we can give definition of association for random vectors with values inRd. Let{X1, . . . , Xn} be a sequence ofRd-valued random vectors.
{X1, . . . , Xn} is said to be associated if
Cov(f(X1, . . . , Xn), g(X1, . . . , Xn))≥0
for any nondecreasing functionsf andg onRmd, such that the covariance exists. Let H be a separable real Hilbert space with the norm k · k generated by an inner product,h·,·iand let{ek, k ≥1}be an orthonormal basis inH. A sequence of random variables {Xn, n ≥ 1} with values in a separable real Hilbert space (H,h·,·i) is said to be associated, if for some orthonormal basis {ek, k ≥ 1} of H and for any d≥1 the d-dimensional sequence (hXi, e1i,· · ·,hXi, edi),i ≥1 is associated.
Lemma 2.1. [20]SupposeX1, . . . , Xn are associated random variables with mean
zero and finite variance. Then
E
max
1≤k≤nSk
2
≤E(S2n).
Lemma 2.2. Let {Xn, n ≥ 1} be an associated sequence of H-valued random
variables with EXn= 0 andEkXnk2<∞,n≥1. Then for anyε >0, we have P
max
1≤i≤nkSik ≥ε
≤ 2EkSnk 2
Proof. Let{ek, k≥1}be an orthonormal basis inH. Then, by Parseval’s identity and Lemma2.1, we have
E max
1≤k≤n
k
X
i=1 Xi
2
=E max
1≤k≤n ∞
X
j=1
* k
X
i=1 Xi, ej
+!2
≤ ∞
X
j=1 E max
1≤k≤n k
X
i=1
hXi, eji
!2
=
∞
X
j=1 Emax
max
1≤k≤n k
X
i=1
hXi, eji
!2
, max
1≤k≤n − k
X
i=1
hXi, eji
!!2
≤ ∞
X
j=1
E max
1≤k≤n k
X
i=1
hXi, eji
!2 +
∞
X
j=1
E max
1≤k≤n − k
X
i=1
hXi, eji
!!2
≤2
∞
X
j=1 E
n
X
i=1
hXi, eji
!2
= 2EkSnk2.
Theorem 2.3. Let {Xn, n ≥ 1} be an associated sequence of H-valued random
variables withEXn= 0andEkXnk2<∞,n≥1. Let{bn, n≥1}be a sequence of
positive nondecreasing real numbers. Then for anyε >0 and any α >1, we have
P
max
1≤k≤n
1 bk
k
X
j=1
Xj
≥ε
≤
2 ε2
α2
1− 1
α2
n
X
j=1
EkXjk2 b2
j + 2
n
X
j=1
EhXj, Sj−1i b2
j
.
Remark 2.1. Here the minimum in the above coefficient can be obtained, i.e.,
inf α>1
α2
1− 1
α2
= 4.
Proof. Without loss of generality, we may assume that bn ≥1 for all n≥1. Let α >1. Fori≥0, define
When Ai 6= ∅, we let ν(i) = max{k;k ∈ Ai}. Let tn be the index of the last nonempty setAi. Ifk∈Ai, thenαi ≤bk≤bν(i)< αi+1. By Lemma2.2, we have
P
max
1≤k≤n
1 bk k X j=1 Xj ≥ε =P
max 0≤i≤tn,Ai6=∅
max k∈Ai 1 bk k X j=1 Xj ≥ε ≤ tn X
i=0,Ai6=∅
P
1
αi1≤maxk≤ν(i)
k X j=1 Xj ≥ε ≤ 2 ε2 tn X
i=0,Ai6=∅
1 α2iE
ν(i) X j=1 Xj 2 = 2 ε2 tn X
i=0,Ai6=∅
1 α2i
∞
X
l=1
E
*ν(i) X
j=1
Xj, el
+2
= 2 ε2
tn X
i=0,Ai6=∅
1 α2i
ν(i) X
j=1
EkXjk2+ 2EhXj, Sj−1i
= 2 ε2
n
X
j=1
EkXjk2+ 2EhXj, Sj−1i
tn X
i=0,Ai6=∅,ν(i)≥j
1 α2i.
Leti0= min{i;Ai 6=∅, ν(i)≥j}, then it is easy to checkbj ≤bν(i0) < α
i0+1. It
follows that
tn X
i=0,Ai6=∅,ν(i)≥j
1 α2i <
1 1− 1
α2
1 α2i0 <
α2
1− 1
α2
1 b2
j .
From the above discussions, the proof of the theorem can be completed.
By Theorem2.3, we can obtain the following H´ajek-R´enyi inequality for associ-ated random variables.
Theorem 2.4. Let {Xn, n ≥ 1} be an associated sequence of H-valued random
variables withEXn = 0and EkXnk2 <∞,n≥1. Let {bn, n≥1} be a sequence
of positive nondecreasing real numbers. Then for any ε >0, any α >1 and any
m < n, we have
P
max
m≤k≤n
1 bk k X j=1 Xj ≥ε ≤ 4 ε2b2
m
EkSmk2+ 4 ε2
α2
1− 1
α2
n
X
j=m+1
EhXj, Sji b2
j
≤ 8
ε2b2
m m
X
i=1
EhXi, Sii+ 16 ε2
α2
1− 1
α2
n
X
j=m+1
EhXj, Sji b2
Proof. Observe that
P max
m≤k≤n
1
bk k
X
j=1 Xj
≥ε
!
≤P
max
m≤k≤n kSmk
bk ≥ ε
2
+P
max
m≤k≤n
kSk−Smk bk ≥
ε
2
=:In+IIn.
For the term In, it is easy to see that
In≤ 4 ε2b2
m
EkSmk2= 4 ε2b2
m E
m
X
i=1
kXik2+ 2
X
i<j
hXi, Xji
≤
8 ε2b2
m m
X
i=1
EhXi, Sii.
(2.1) For the termIIn, noting that
max m≤k≤n
kSk−Smk bk
= max
1≤k≤n−m
Pk
j=1Xm+j
bm+k ,
then, by Theorem2.3, we obtain
IIn ≤ 8 ε2
α2
1− 1
α2
n−m
X
j=1
EkXm+jk2 b2
m+j + 2
n−m
X
j=1
EhXm+j, Sm+j−1−Smi b2
m+j
≤16
ε2
α2
1− 1
α2
n−m
X
j=1
EhXm+j, Sm+j−Smi b2
m+j =16
ε2
α2
1−α12
n
X
j=m+1
EhXj, Sj−Smi b2
j
≤16
ε2
α2
1− 1
α2
n
X
j=m+1
EhXj, Sji b2
j .
(2.2)
Here we use the definition of association. Thus the desired result can be obtained
by (2.1) and (2.2).
Corollary 2.5. Let {bn, n≥1} be a sequence of positive nondecreasing real
num-bers. Let{Xn, n≥1} be an associated sequence ofH-valued random variables with EXn= 0 and satisfyingP∞j=1EhXj, Sji/b2j <∞. If0< r <2, then
E
sup n≥1
(kSnk/bn)r
Proof. By Theorem2.3, we get
E
sup n≥1
(kSnk/bn)r
=
Z ∞ 0
P
sup n≥1
(kSnk/bn)r> t
dt
≤1 +
Z ∞ 1
P
sup n≥1
(kSnk/bn)r> t
dt
≤1 +
Z ∞ 1
2 t2/r
α2
1− 1
α2
∞
X
j=1
EkXjk2 b2
j + 2
∞
X
j=1
EhXj, Sj−1i b2
j
dt
≤1 + 4α
2
1− 1
α2
∞
X
j=1
EhXj, Sji b2
j
Z ∞ 1
1
t2/rdt <∞.
Corollary 2.6. Let{bn, n≥1}be a nondecreasing unbounded sequence of positive
real numbers. Let {Xn, n ≥ 1} be an associated sequence of H-valued random
variables with EXn= 0 and satisfyingP∞j=1EhXj, Sji/b2j <∞. Then Sn/bn→0, a.s.
Proof. For anym < N, by Theorem2.4, it follows that
P
max m≤n≤N
kSnk bn
> ε
≤ 8
ε2b2
m m
X
i=1
EhXi, Sii+ 4 ε2
α2
1− 1
α2
N
X
j=m+1
EhXj, Sji b2
j ,
then, by Kronecker’s lemma, we can obtain
lim m→∞P
∞
[
n=m
kS
nk bn
> ε
!
= lim m→∞P
∞
[
N=m
max m≤n≤N
kSnk bn
> ε
!
= lim
m→∞Nlim→∞P
max m≤n≤N
kSnk bn
> ε
→0
which implies the desired result.
Remark 2.2. In particular, takingbn=n, the result of Ko et al. in [15, Theorem 2.4] is a simple corollary.
Corollary 2.7. Let {an, n≥1} be a sequence of positive real numbers and bn =
Pn
i=1ai, n ≥1. Let {Xn, n≥1} be an associated sequence of H-valued random
variables with EXn= 0 and satisfying ∞
X
j=1
1 b2
j j
X
i=1
aiajEhXj, Xii<∞.
Then
n
X
i=1
aiXi bn
Proof. Noting that {anXn, n ≥ 1} is a sequence of associated random variables and by Corollary2.6, we can complete the proof of the desired result.
Remark 2.3. All the above results extend the works of Sung in [22] fromR-valued random variables to Hilbert space.
3. Negatively associated sequences
A finite sequence{Xi,1≤i≤n}is said to be negatively associated (abbreviated to NA) if for any disjoint subsets A, B ⊂ {1,2, . . . , n} and any coordinatewise nondecreasing functionsf onR|A|andgonR|B|
Cov(f(Xi, i∈A), g(Xi, i∈B))≤0
where the covariance exists. An infinite sequence{Xn, n ≥ 1} is said to be NA if every finite subfamily is NA. This concept was introduced by Joag-Dev and Proschan in [14]. A number of well-known multivariate distributions posses the NA property, such as multinomial distribution, multivariate hypergeometric dis-tribution, negatively correlated normal disdis-tribution, permutation distribution and joint distribution of ranks.
As the definition of associated random variables, we can define the NA se-quences in Hilbert space. A sequence of random variables{Xn, n≥1}with values in a separable real Hilbert space (H,h·,·i) is said to be as NA, if for some or-thonormal basis{ek, k ≥1} of H and for any d≥1 the d-dimensional sequence (hXi, e1i,· · · ,hXi, edi),i≥1 is NA.
Lemma 3.1. [21] Let {Xn, n ≥ 1} be a sequence of NA random variables with
finite second moments and zero means. Then we have
E max
1≤k≤n k
X
i=1
Xi
!2 ≤2
n
X
i=1
EXi2.
Theorem 3.2. Let {Xn, n≥1} be a NA sequence of H-valued random variables
withEXn = 0andEkXnk2<∞,n≥1. Let{bn, n≥1} be a sequence of positive
nondecreasing real numbers. Then for anyε >0, we have
P max
1≤k≤n
1 bk
k
X
i=1
Xi
≥ε
! ≤8
n
X
i=1
EkXik2 ε2b2
Proof. From Lemma3.1, we obtain
E max
1≤k≤n
k X i=1 Xi 2
=E max
1≤k≤n ∞ X j=1 * k X i=1
Xi, ej
+!2 ≤ ∞ X j=1 E max
1≤k≤n
* k X
i=1
Xi, ej
+!2 = ∞ X j=1 E max
1≤k≤n k
X
i=1
hXi, eji
!2 ≤2 ∞ X j=1 n X i=1
E(hXi, eji)2= 2 n
X
i=1
EkXik2.
(3.1)
LetSn=Pnj=1Xj. Without loss of generality, setting b0= 0, we have
Sk= k
X
j=1
bjXj bj = k X j=1 j X i=1
(bi−bi−1)Xj
bj ! = k X i=1
(bi−bi−1) X
i≤j≤k Xj
bj .
Since (1/bk)Pkj=1(bj−bj−1) = 1, then we have
max
1≤k≤n 1 bk
kSkk ≤ max
1≤k≤n1max≤i≤k
X
i≤j≤k Xj bj ≤ max
1≤i≤k≤n
X
j≤k Xj bj −X j<i Xj bj
≤2 max
1≤i≤n
i X j=1 Xj bj .
Since{Xj/bj, j≥1}is still a sequence of NA random variables, then by (3.1), we have
P
max
1≤k≤n 1 bk
kSkk ≥ε
≤P
2 max
1≤i≤n
i X j=1 Xj bj ≥ε ≤8
n
X
i=1
EkXik2 ε2b2
i .
Theorem 3.3. Let {Xn, n≥1} be a NA sequence of H-valued random variables
with EXn= 0 andEkXnk2<∞,n≥1. Let {bn, n≥1} be a sequence of positive
nondecreasing real numbers. Then for anyε >0 and anym < n, we have
P max
m≤k≤n
1 bk k X i=1 Xi ≥ε ! ≤ 32 ε2 1 b2 m m X i=1
EkXik2+ n
X
i=m+1
EkXik2 b2
i
!
Proof. Observe that
P
max
m≤k≤n
1 bk
k
X
j=1
Xj
≥ε
≤P
max m≤k≤n
kSmk bk
≥ ε
2
+P
max m≤k≤n
kSk−Smk bk
≥ ε
2
=:In+IIn.
For the termIn, by the similar proof of Theorem3.2, it is easy to check that
In ≤ 32 ε2b2
m m
X
i=1
EkXik2. (3.2)
For the termIIn, noting that
max m≤k≤n
kSk−Smk bk
= max
1≤k≤n−m
Pk
j=1Xm+j
bm+k ,
then, by Theorem3.2, we obtain
IIn ≤32 ε2
n−m
X
i=1
EkXm+ik2 b2
m+i
= 32 ε2
n
X
i=m+1
EkXik2 b2
i
. (3.3)
Here we use the definition of association. Thus the desired result can be obtained
by (3.2) and (3.3).
Remark 3.1. Let{Xn, n≥ 1} be a NA sequence of R-valued random variables withEXn = 0 andE|Xn|2<∞,n≥1. Let{bn, n≥1} be a sequence of positive nondecreasing real numbers. Then for anyε >0 and any m < n, Liu et al. [19] obtained
P max
m≤k≤n
1 bk
k
X
i=1
Xi
≥ε
! ≤128
ε2
1 b2
m m
X
i=1
EXi2+ n
X
i=m+1
EX2
i b2
i
!
. (3.4)
Hence Theorem3.3improves and extends the result of Liu et al. in [19].
As the proofs of Corollary2.5and 2.6, we have following
Corollary 3.4. Let {bn, n≥1} be a sequence of positive nondecreasing real
num-bers. Let {Xn, n ≥ 1} be a NA sequence of H-valued random variables with EXn= 0 and satisfyingP∞j=1EkXjk/b2j<∞. If0< r <2, then
E
sup n≥1
(kSnk/bn)r
<∞.
Corollary 3.5. Let{bn, n≥1} be a nondecreasing unbounded sequence of positive
real numbers. Let {Xn, n ≥ 1} be a NA sequence of H-valued random variables
Remark 3.2. In particular, takingbn=n, the result of Ko et al. in [15, Corollary 3.5] is a simple corollary.
Corollary 3.6. Let {Xn, n≥1} be a NA sequence of H-valued random variables
with EXn= 0. Then for any 0< t <2 andε >0, we have
P
sup j≥m
kSjk j1/t ≥ε
≤8ε−2 2
2−tm
(t−2)/tsup n
EkXnk2.
Corollary 3.7. Let {Xn, n≥1} be a NA sequence of H-valued random variables
with EXn= 0 andsupnEkXnk2<∞. Then for any0< t <2, we have Esup
n
kS
nk n1/t
r
<∞, for all 0< r <2
and
Sn
n1/t →0, a.s. 4. φ-mixing sequences
Let{Xi;i≥1} be a sequence of random variables and for any 1≤i≤j ≤ ∞ denote Mji as the σ-field generated by {Xk;i ≤k ≤ j}. A sequence of random variables{Xi;i≥1} is said to beφ-mixing, if for anyA∈ Mk1 andB∈ M∞k+j,
|P(B|A)−P(B)| ≤φ(j), φ(j)≥0,
where 1 ≥ φ(1) ≥ φ(2) ≥ · · ·, and limj→∞φ(j) = 0. For more information of φ-mixing (see Billingsley [2]). Intuitively,{X1, X2,· · · , Xn} isφ-mixing ifXi and Xi+j become virtually independent as j becomes large. For example, the waiting time Wi of an M/M/1 delay-in-queue is φ-mixing, becauseWi and Wi+j become virtually independent asj becomes large. In addition,m-dependent sequence im-pliesφ-mixing, while for gaussian processes,φ-mixing corresponds tom-dependence (see [13]).
A sequence of random variables {Xn, n ≥ 1} with values in a separable real Hilbert space (H,h·,·i) is said to be as φ-mixing, if for some orthonormal basis
{ek, k≥1}ofHand for anyd≥1 thed-dimensional sequence (hXi, e1i,· · · ,hXi, edi), i≥1 is φ-mixing.
Lemma 4.1. [18] Let {Xn, n ≥ 1} be a sequence of φ-mixing random variables
with finite second moments, zero means andP
nφ1/2(n)<∞. Then there exists a
positive constant C, such that
E max
1≤k≤n k
X
i=1
Xi
!2 ≤C
n
X
i=1
EXi2.
Acknowledgements
The author is very grateful to the editor and referees for their critical and valu-able comments, which improved the presentation of this article. This work is supported by NSFC (No. 11001077), NCET (No. NCET-11-0945) and HASTIT (No. 2011HASTIT011).
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Yu Miao
College of Mathematics and Information Science, Henan Normal University,
Henan Province, 453007, China yumiao728@yahoo.com.cn
