Contents lists available atScienceDirectStatistics and Probability Letters journal homepage:www.elsevier.com/locate/stapro Strong laws of large numbers for random fields in martingale ty
Trang 1Contents lists available atScienceDirect
Statistics and Probability Letters journal homepage:www.elsevier.com/locate/stapro
Strong laws of large numbers for random fields in martingale type p
Banach spaces
Le Van Dunga,∗, Nguyen Duy Tienb
aFaculty of Mathematics, Danang University of Education, 459 Ton Duc Thang, Lien Chieu, Danang, Viet Nam
bFaculty of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam
a r t i c l e i n f o
Article history:
Received 17 June 2009
Received in revised form 10 January 2010
Accepted 11 January 2010
Available online 29 January 2010
MSC:
60B11
60B12
60F15
60G42
a b s t r a c t
We extend Marcinkiewicz–Zygmund strong laws for random fields{Vn;n ∈ Nd}with
values in martingale type p Banach spaces Our results are more general and stronger than
the result ofGut and Stadtmüller(2009) and some other ones
© 2010 Elsevier B.V All rights reserved
1 Introduction
Let Nd be the positive integer d-dimensional lattice points, where d is a positive integer For m= (m1, ,m d)and n= (n1, ,n d) ∈ Nd , notation m≺ n means that m i ≤ n i,1 ≤i ≤d, |nα|is used forQd
i= 1nαi
i ,[m,n) = Qd
i= 1[m i,n i)is a
d-dimensional rectangle andWd
i= 1(m i <n i)means that there is at least one of m1 <n1, m2 <n2, ., m d<n dholds We
write 1= (1, ,1) ∈Nd
Consider a random field{V n,n∈ Nd}of random elements defined on a probability space(Ω,F,P)taking values in a
real separable martingale type p(1≤p≤2)Banach spaceXwith normk · k In the current work, we establish strong laws
of large numbers (SLLN) for|nα|− 1maxk≺nkS kk This can be done by studying convergence of sums of type
X
n
|n|−1P{max
k≺n
kS kk > |nα|} for every >0.
Many authors have investigated the Marcinkiewicz type strong laws of large numbers for random fields{X n,n ∈ Nd}
of random variables For example,Fazekas and Tómács(1998) studied strong laws of large numbers|n|−1/r S n(for some
0 < r < 1) for pairwise independent random variables,Czerebak-Mrozowicz et al.(2002) studied Marcinkiewicz type strong laws of large number|n|− 1 /p(S n−ES n)(for some 1<p<2) for pairwise independent random fields Recently,Gut and Stadtmüller(2009) studied Marcinkiewicz–Zygmund laws of large numbers for random fields of i.i.d random variables
In this paper, we not only extend these results to random fields in martingale type p Banach spaces but also bring more
general and stronger ones
Throughout this paper, the symbol C will denote a generic constant (0<C< ∞) which is not necessarily the same one
in each appearance
∗Corresponding author.
E-mail addresses:lvdunght@gmail.com (L.V Dung), nduytien2006@yahoo.com (N.D Tien).
0167-7152/$ – see front matter © 2010 Elsevier B.V All rights reserved.
Trang 22 Preliminaries
Technical definitions relevant to the current work will be discussed in this section
Scalora(1961) introduced the idea of the conditional expectation of a random element in a Banach space For a random
element V and sub-σ-algebraGofF, the conditional expectation E(V|G)is defined analogously to that in the random variable case and enjoys similar properties
A real separable Banach spaceXis said to be martingale type p(1 ≤p≤2)if there exists a finite positive constant C
such that for all martingales{S n;n≥1}with values inX,
sup
n≥ 1
EkS nkp≤C
∞ X
n= 1
EkS n−S n− 1kp.
It can be shown using classical methods from martingale theory that ifXis of martingale type p, then for all 1≤r< ∞
there exists a finite constant C such that
E sup
n≥ 1
kS nkr ≤CE
∞ X
n= 1
kS n−S n− 1kp
!r p
. Clearly every real separable Banach space is of martingale type 1 and the real line (the same as any Hilbert space) is of
martingale type 2 If a real separable Banach space of martingale type p for some 1<p≤2 then it is of martingale type r for all r∈ [1,p)
It follows from theHoffmann-Jørgensen and Pisier(1976) characterization of Rademacher type p Banach spaces that if a Banach space is of martingale type p, then it is of Rademacher type p But the notion of martingale type p is only superficially similar to that of Rademacher type p and has a geometric characterization in terms of smoothness For proofs and more
details, the reader may refer toPisier (1975,1986)
To prove the main result we need the following lemma which was proved byDung et al.(2009) in the case d=2 If d is
arbitrary positive integer, then the proof is similar and so is omitted
Lemma 2.1 Let 1 ≤ p ≤ 2 and let {V k,k ≺ n}be a collection of |n|random elements in a real separable martingale type p Banach space with E(V k|Fk) = 0 for all k ≺ n, whereFk is theσ-field generated by the family of random elements
{V l: Wd
i= 1(l i<k i)},F1= {∅ ,Ω} Then
E max
k≺n
kS kkp≤CX
k≺n
EkV kkp,
where S k= P
i≺k V i In the case of p=1, the hypothesis that E(V k|Fk) =0 for all k≺n is superfluous.
Lemma 2.2 Let 1<p≤2 Letα1, , αd be positive constants satisfying 1/p<min{ α1 , αd} <1, let q be the number of
integers s such thatαs=min{ α1 , αd} If E kVkr(log+kVk )q− 1
< ∞then we have
(i) X
n
1
|nα|
Z ∞
|nα|
P{kVk ≥t}dt< ∞,
(ii) X
n
1
|nα|p
Z |nα|p
0
P{kVkp≥t}dt< ∞.
Proof Without loss of generality, we may assume min{ α1, , αd} = α1= · · · = αq< αq+ 1≤ αd
We first prove(i) We have by Lemma 3 ofStadtmüller and Thalmaier(2009) that
1 ≤n1 n q.nαq+ 1 /α 1
q+ 1 nαd/α 1
d ≤j
1 ∼ C j(log j)q− 1
(q−1)! as j→ ∞ . Denote∆g(j) =g(j) −g(j−1)we get
X
n
1
|nα|
Z ∞
|nα|
P{kVk ≥t}dt ≤
∞ X
k= 1
1
kα 1∆g(k) Z
∞
kα1
P{kVk ≥t}dt=
∞ X
k= 1
1
kα 1∆g(k)
∞ X
i=k
Z (i+ 1 )α1
iα1
P{kVk ≥t}dt
≤
∞ X
k= 1
1
kα 1∆g(k)
∞ X
i=k
Z (i+ 1 ) α 1
iα1
P{kVk ≥iα 1}dt
≤
∞
kα 1∆g(k)
∞ X
iα 1 − 1P{kVk ≥iα 1}
Trang 3∞ X
k= 1
1
kα 1∆g(k)
∞ X
i=k
iα 1 − 1
∞ X
j=i
P{jα 1 ≤ kVk < (j+1)α 1}
≤
∞ X
k= 1
1
kα 1∆g(k)
∞ X
j=k
P{jα 1 ≤ kVk < (j+1)α 1}
j
X
i= 1
iα 1 − 1
≤ C
∞ X
k= 1
1
kα 1∆g(k)
∞ X
j=k
jα 1P{jα 1 ≤ kVk < (j+1)α 1}
≤ C
∞ X
j= 1
jα 1P{jα 1≤ kVk < (j+1)α 1}
j
X
k= 1
1
kα 1∆g(k)
= C
∞ X
j= 1
jα 1P{jα 1≤ kVk < (j+1)α 1}
j
X
k= 1
1
kα 1∆g(k)
= C
∞ X
j= 1
jα 1P{jα 1≤ kVk < (j+1)α 1}
j− 1
X
k= 1
1
kα 1
(k+1)α 1
g(k)
+C
∞ X
j= 1
P{jα 1 ≤ kVk < (j+1)α 1}g(j)
≤ C
∞ X
j= 1
jα 1P{jα 1≤ kVk < (j+1)α 1}
j− 1
X
k= 1
1
kα 1
(k+1)α 1
k(log k)q− 1
+C
∞ X
j= 1
P{jα 1 ≤ kVk < (j+1)α 1}j(log j)q− 1
≤ C
∞ X
j= 1
jα 1P{jα 1≤ kVk < (j+1)α 1} (log j)q− 1
j− 1
X
k= 1
1
kα 1
(k+1)α 1
k
+C
∞ X
j= 1
P{jα 1 ≤ kVk < (j+1)α 1}j(log j)q− 1
≤ C
∞ X
j= 1
jα 1P{jα 1≤ kVk < (j+1)α 1} (log j)q− 1
j
X
k= 1
1
kα 1
+C
∞ X
j= 1
P{jα 1 ≤ kVk < (j+1)α 1}j(log j)q− 1
≤ C
∞ X
j= 1
P{jα 1≤ kVk < (j+1)α 1}j(log j)q− 1< ∞.
Now we prove (ii)
X
n
1
|nα|p
Z |nα|p
0
P{kVkp≥t}dt = X
n
1
|nα|p
Z 1
0
P{kVkp≥t}dt+ X
n
1
|nα|p
Z |nα|p
1
P{kVkp≥t}dt
n
1
|nα|p+CX
n
1
|nα|p
Z |nα|p
1
P{kVkp≥t}dt. Noting that the first term on the right-hand side is finite, it remains to prove that
X
n
1
|nα|p
Z |nα|p
1
P{kVkp≥t}dt.
Denote d(k) = Pn1 n q=k1, we have by Lemma 3.1 ofGut(2001) that
∞
Xd(j)
j pα 1 ∼ C(log k)q− 1
k pα 1 − 1 .
Trang 4Hence, we have
X
n
1
|nα|p
Z |nα|p
1
P{kVkp≥t}dt≤
∞ X
k,n q+ 1 , ,n d= 1
k pα 1.n pαq+ 1
q+ 1 .n pαd
d
[kα1nαq+ 1
q+ 1 nαd] X
j= 1
E(kVkp I(j≤ kVk <j+1)) (where[x]denotes the greatest integer not exceeding x)
≤C
∞ X
k,n q+ 1 , ,n d= 1
k pα 1.n pαq+ 1
q+ 1 .n pαd
d
[kα1nαq+ 1
q+ 1 nαd] X
j= 1
j p P(j≤ kVk <j+1)
≤C
∞ X
k,n q+ 1 , ,n d= 1
k pα 1.n pαq+ 1
q+ 1 .n pαd
d
[kα1nαq+ 1
q+ 1 nαd] X
j= 1
[j p− (j−1)p]P(kVk ≥j)
≤C
∞ X
k,n q+ 1 , ,n d= 1
k pα 1.n pαq+ 1
q+ 1 .n pαd
d
[kα1nαq+ 1
q+ 1 nαd] X
j= 1
pj p−1P{kVk ≥j}
=C
∞
X
n q+ 1 , ,n d= 1
1
n pαq+ 1
q+ 1 .n pαd
d
∞ X
k= 1
d(k)
k pα 1
[kα1nαq+ 1
q+ 1 nαd] X
j= 1
pj p−1P{kVk ≥j}
≤C
∞
X
n q+ 1 , ,n d= 1
1
n pαq+ 1
q+ 1 .n pαd
d
[nαq+ 1
q+ 1 nαd] X
j= 1
pj( 1 /α 1 − 1 )j(p− 1 /α 1 )P{kVk ≥j} X∞
k= 1
d(k)
k pα 1
+
∞ X
n q+ 1 , ,n d= 1
1
n pαq+ 1
q+ 1 .n pαd
d
∞ X
i=[nαq+ 1
q+ 1 nαd]+ 1
pj p−1P{kVk ≥j}
∞ X
k=
j/nαq+ 1
q+ 1 nαd 1 /α 1
d(k)
k pα 1
≤C
∞
X
n q+ 1 , ,n d= 1
1
nβq+ 1
q+ 1 .nβd
d
[nαq+ 1
q+ 1 nαd] X
j= 1
pj( 1 /α 1 − 1 )P{kVk ≥j} X∞
k= 1
d(k)
k pα 1
+
∞ X
n q+ 1 , ,n d= 1
1
n pαq+ 1
q+ 1 .n pαd
d
∞ X
j=[nαq+ 1
q+ 1 nαd]+ 1
pj p−1P{kVk ≥j}
∞ X
k=
j/nαq+ 1
q+ 1 nαd 1 /α1
d(k)
k pα 1
≤C
∞
X
n q+ 1 , ,n d= 1
1
nβq+ 1
q+ 1 .nβd
d
∞ X
j= 1
j r−1(log i)q− 1P{kVk ≥j}
which is finite if E(kVkrlog+kVk )q− 1< ∞and sinceβl= αl/α1>1 for q+1≤l≤d.
The random field{V n,n∈Nd}is said to be weakly mean dominated by the random element V if, for some 0<C< ∞, 1
|n|
X
k≺n
P{kV kk ≥x} ≤CP{kVk ≥x}
for all n∈Nd and x>0
3 Main results
With the preliminaries accounted for, the main results may now be established In the following, we let{V n;n ∈ Nd}
be an array of random elements in a real separable Banach spaceX,Fkis theσ-field generated by the family of random elements{V l: Wd
i= 1(l i<k i)},F1= {∅ ,Ω}
The first theorem is a general a.s convergence one
Theorem 3.1 Letα1, , αd be positive constants Let{V n,n∈Nd}be a random field of random elements If
|n|P{max
k≺n
Trang 5X
n
P
max
and, a fortiori, the SLLN
1
|nα|maxk≺n
obtains.
Conversely,(3.2)implies that(3.1)holds.
Proof ((3.1)⇒(3.2)) Fix >0, denote 2n= (2n1, ,2n d)and 2nα= (2n1 α 1, ,2n dαd) We have the inequalities X
n
P
max
l≺ 2n kS lk > |2nα| ≤ X
n
min
k∈[ 2n, 2n+ 1 )P
max
l≺k
kS lk >
2α 1 +··· ,αd
|kα|
n
X
k∈[ 2n, 2n+ 1 )
1
|2n|P
max
l≺k
kS lk >
2α 1 +···+ αd
|kα|
n
X
k∈[ 2n, 2n+ 1 )
2d
|k|P
max
l≺k
kS lk >
2α 1 +···+ αd
|kα|
≤2dX
n
1
|n|P
max
l≺n
kS lk >
2α 1 +···+ αd|nα| < ∞. (by(3.1)) This implies by the Borel–Cantelli lemma that
1
Now for k∈ [2n,2n+1)we have
1
|kα|maxl≺k kS lk ≤ 1
|kα|l≺max2n+ 1
kS lk ≤ 1
|2nα|l≺max2n+ 1
kS lk = 2α 1 +···+ αd
|2(n+ 1 )α| l≺max2n+ 1
and so the conclusion(3.3)follows from(3.4)and(3.5)
((3.2)⇒(3.1)) Suppose that(3.2)holds, we easily to prove that for every >0,
X
n
1
|n|P
max
l≺n
kS lk > |nα| ≤ X
n P
max
l≺ 2n kS lk >
2α 1 +···+ αd
|2nα| ,
which implies that(3.1)holds The proof is completed
The following theorem characterizes the martingale type p Banach spaces.
Theorem 3.2 Let 1≤p≤2 and letXbe a separable Banach space Then the following two statements are equivalent:
(i) The BanachXis of martingale type p.
(ii) For every random field{V n;n∈Nd}inXwith E(V n|Fn) =0 for all n∈Nd and for everyα = (α1, , αd)withαi>0 for
all 1≤i≤d, the condition
X
n
EkV nkp
|nα|p < ∞
implies that, for every >0,
X
n
1
|n|P{max
k≺n
kS kk > |nα|} < ∞
and, a fortiori, the SLLN
1
|nα|maxk≺n
kS kk →0 a.s as|n| → ∞
obtains.
Trang 6Proof In order to prove[ (i) ⇒ (ii)]we show that
X
n
P{max
k≺ 2nkS kk > |2nα|} < ∞ for every >0.
Applying Markov’s inequality andLemma 2.2we have that
X
n
P{max
k≺ 2nkS kk > |2nα|} ≤ X
n
1
p|2nα|p E(max
k≺ 2nkS kkp)
n
1
|2nα|p
X
k≺ 2n
EkV kkp≤CX
k
EkV kkp
|kα|p < ∞.
Now we prove[(ii)⇒(i)] Assume that (ii) holds Let{W n1,Gn1;n1≥1}be an arbitrary sequence of martingale difference
inXsuch that
∞
X
n1 = 1
EkW n1kp
n p1 < ∞
Set
V n1, ,n d =W n1 if n2= · · · =n d=1 otherwise V n1, ,n d=0.
Then{V n1, ,n d}is the random field inXsatisfies E(V n1, ,n d|Fn1 , ,n d) =0 for all(n1, ,n d) ∈Nd, and
∞
X
n1 , ,n d= 1
EkV n1, ,n dkp
(n1 .n d)p =
∞ X
n1 = 1
EkW n1kp
n p1 < ∞.
By (ii),
1
n1 .n d
X
i1≤n1
id≤nd
V i1, ,i d →0 a.s as(n1 .n d) → ∞.
Taking n2= · · · =n d=1 and letting n1→ ∞we obtain
1
n1
n1
X
j= 1
Then by Theorem 2.2 ofHoffmann-Jørgensen and Pisier(1976),Xis of martingale type p.
In the next two theorems, we obtain the Marcinkiewicz–Zygmund type laws of large numbers for random fields of random elements
Theorem 3.3 LetXbe a martingale type p Banach space with 1 < p ≤ 2 Letα1, , αd be positive constants satisfying
1/p < min{ α1 , αd} < 1, let q be the number of integers s such thatαs = min{ α1 , αd}and let{V n,n ∈ Nd}be a random field satisfying E(V n|Fn) = 0 for all n ∈ Nd Suppose that {V n,n ∈ Nd}is weakly mean dominated by V such that
E kVkr(log+kVk )q− 1
< ∞with r= 1
min { α 1 ,αd} Then
X
n
1
|n|P{max
k≺n
and, a fortiori, the SLLN
1
obtains.
Proof For each n∈Nd, set
V k0=V k I(kV kk ≤ |nα| ),V k00=V k I(kV kk > |nα| ),
Y k0=V k0−E(V k0|Fk),Y k00 =V k00−E(V k00|Fk),
S0
n= P
k≺n Y0
k,S00
n= P
k≺n Y00
k
Since E(V k|Fk) = 0, it follows that V k = Y0
k+Y00
k Moreover, ifG0
kandG00
kare theσ-fields generated by the family of random elements{Y0
l : Wd
i= 1(l i<k i)}and{Y00
l : Wd
i= 1(l i<k i)}, respectively, thenG0
k⊂FkandG00
k⊂Fk for all k≺n, which
imply that E(Y0|G0) =E(Y00|G00) =0 for all k≺n.
Trang 7We now begin the proof For every >0,
X
n
1
|n|P{max
k≺n
kS kk >2|nα|} ≤ X
n
1
|n|P{max
k≺n
kS k0k > |nα|} + X
n
1
|n|P{max
k≺n
kS00kk > |nα|} (3.8) First, we show that
X
n
1
|n|P{max
k≺n
kS00kk > |nα|} < ∞.
Applying Markov’s inequality andLemma 2.2, we obtain
X
n
1
|n|P{max
k≺n
kS00kk > |nα|} ≤ X
n
1
|n||n|αE(max
k≺n
kS00kk ) ≤CX
n
1
|nα|
1
|n| X
k≺n
EkY k00k
n
1
|nα|
1
|n| X
k≺n
EkV k00k =CX
n
1
|nα|
1
|n| X
k≺n
Z ∞
0
P{kV k00k ≥t}dt
n
1
|nα|
1
|n| X
k≺n
Z |nα|
0
P{kV kk ≥ |nα|}dt+CX
n
1
|nα|
1
|n| X
k≺n
Z ∞
|nα|
P{kV kk ≥t}dt
n
1
|n| X
k≺n
P{kV kk ≥ |nα|} +CX
n
1
|nα|
Z ∞
|nα|
1
|n| X
k≺n
P{kV kk ≥t}dt
n
P{kVk ≥ |nα|} +CX
n
1
|nα|
Z ∞
|nα|
P{kVk ≥t}dt
n
1
|nα|
Z ∞
|nα|
P{kVk ≥t}dt< ∞ (byLemma 2.2).
By(3.8), in order to complete the proof, we next show that
X
n
1
|n|P{max
k≺n
kS0kk > |nα|} < ∞.
Again applying Markov’s inequality, we find that
X
n
1
|n|P{max
k≺n
kS0kk > |nα|} ≤ X
n
1
|n|
1
|nα|p E(max
k≺n
kS k0k > |nα| )p
n
1
|n|
1
|nα|p E(max
k≺n
kS k0k > |nα|p) ≤CX
n
1
|nα|p
1
|n| X
k≺n
EkY k0kp
n
1
|nα|p
1
|n| X
k≺n
EkV k0kp=CX
n
1
|nα|p
1
|n| X
k≺n
Z ∞
0
P{kV k0kp≥t}dt
n
1
|nα|p
1
|n| X
k≺n
Z |nα|p
0
P{kV kkp≥t}dt
n
1
|nα|p
Z |nα|p
0
1
|n| X
k≺n
P{kV kkp≥t}dt
n
1
|nα|p
Z |nα|p
0
P{kVkp≥t}dt< ∞ (byLemma 2.2).
Remark Note that in the case of q< d, positive constantsα1, , αdare not upper bounded by 1, which is weaker than condition (2.1) of Theorem 2.1 ofGut and Stadtmüller(2009)
Theorem 3.4 Letα1, , αd be positive constants satisfying min{ α1 , αd} > 1, let q be the number of integers s such that
αs=min{ α1 , αd} Suppose that{V n,n∈Nd}is weakly mean dominated by V such that E kVk (log+kVk )q− 1
< ∞ Then
(3.3)holds and then, the SLLN(3.4)obtains.
Proof The proof is similar to that ofTheorem 3.2with p=1 and we use T n0 = P
k≺n V k0and T n00 = P
k≺n V k00are instead of
S0
n and S00
n, respectively
Trang 8Czerebak-Mrozowicz, E.B., Klesov, O.I., Rychlik, Z., 2002 Marcinkiewicz-type strong laws of large numbers for pairwise independent random fields Probab Math Statist 22 (Fasc 1), 127–139.
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