DSpace at VNU: Strong laws of large numbers for random fields in martingale type p Banach spaces

Statistics and Probability Letters 80 (2010) 756–763

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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 Dung a,∗ , Nguyen Duy Tien b
a

Faculty of Mathematics, Danang University of Education, 459 Ton Duc Thang, Lien Chieu, Danang, Viet Nam

b

Faculty of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam

article

abstract

info

Article history:
Received 17 June 2009
Received in revised form 10 January 2010
Accepted 11 January 2010
Available online 29 January 2010

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 of Gut and Stadtmüller (2009) and some other ones.
© 2010 Elsevier B.V. All rights reserved.

MSC:
60B11
60B12
60F15
60G42

1. Introduction
Let Nd be the positive integer d-dimensional lattice points, where d is a positive integer. For m = (m1 , . . . , md ) and n =
α
(n1 , . . . , nd ) ∈ Nd , notation m ≺ n means that mi ≤ ni , 1 ≤ i ≤ d, |nα | is used for di=1 ni i , [m, n) = di=1 [mi , ni ) is a

d-dimensional rectangle and i=1 (mi < ni ) means that there is at least one of m1 < n1 , m2 < n2 ,. . ., md < nd holds. We
write 1 = (1, . . . , 1) ∈ Nd .
Consider a random field {Vn , 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 space X with norm · . In the current work, we establish strong laws
of large numbers (SLLN) for |nα |−1 maxk≺n Sk . This can be done by studying convergence of sums of type
d

|n|−1 P {max Sk > |nα |} for every > 0.
n

k≺n

Many authors have investigated the Marcinkiewicz type strong laws of large numbers for random fields {Xn , n ∈ Nd }
of random variables. For example, Fazekas and Tómács (1998) studied strong laws of large numbers |n|−1/r Sn (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 (Sn − ESn ) (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: (L.V. Dung), (N.D. Tien).

0167-7152/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.spl.2010.01.007


L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763

757

2. 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 -algebra G of F , the conditional expectation E (V |G) is defined analogously to that in the random
variable case and enjoys similar properties.
A real separable Banach space X is said to be martingale type p (1 ≤ p ≤ 2) if there exists a finite positive constant C
such that for all martingales {Sn ; n ≥ 1} with values in X,
∞
p

sup E Sn

≤C

n ≥1

p

E Sn − Sn−1

.

n =1

It can be shown using classical methods from martingale theory that if X is of martingale type p, then for all 1 ≤ r < ∞
there exists a finite constant C such that
r
p

∞
r

E sup Sn

≤ CE

Sn − Sn−1

n ≥1

.

p

n =1

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 the Hoffmann-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 to Pisier (1975, 1986).
To prove the main result we need the following lemma which was proved by Dung 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 {Vk , k ≺ n} be a collection of |n| random elements in a real separable martingale
type p Banach space with E (Vk |Fk ) = 0 for all k ≺ n, where Fk is the σ -field generated by the family of random elements
{Vl : di=1 (li < ki )}, F1 = {∅, Ω }. Then
p

E max Sk

≤C

k≺n

E Vk

p

,

k≺n

where Sk =

i ≺k

Vi . In the case of p = 1, the hypothesis that E (Vk |Fk ) = 0 for all k ≺ n is superfluous.

Lemma 2.2. Let 1 < p ≤ 2. Let α1 , . . . , αd be positive constants satisfying 1/p < min{α1 . . . , αd } < 1, let q be the number of
integers s such that αs = min{α1 . . . , αd }. If E V r (log+ V )q−1 < ∞ then we have
∞

1

(i)
n

| nα |

|nα |

n

≥ t }dt < ∞,

|nα |p

1

(ii)

P{ V

|nα |p

P{ V

p

≥ t }dt < ∞.

0

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 of Stadtmüller and Thalmaier (2009) that
g (j) =

1∼C
αq+1 /α1
α /α
1≤n1 ...nq .nq+1
...nd d 1 ≤j

j(log j)q−1

as j → ∞.

(q − 1)!

Denote ∆g (j) = g (j) − g (j − 1) we get
1
n

|nα |

∞

∞
|nα |

P{ V

≥ t }dt ≤
k=1

∞

≤
k=1

∞

≤
k=1

1
kα1
1

kα1
1
kα1

∞

∞

∆g (k)

kα1

∞

∆g (k)
i=k

P{ V

≥ t }dt =
k=1

(i+1)α1
i α1

P{ V

iα1 −1 P { V
i=k

kα1

≥ iα1 }dt

∞

∆g (k)

1

≥ iα1 }

∞

∆g (k)
i=k

(i+1)α1
iα1

P{ V

≥ t }dt


758

L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763

∞

∞

1

≤
k=1

∞

i =k

k=1

∞

kα1

∞

< (j + 1)α1 }

jα1 P {jα1 ≤ V

∆g (k)
j =k

jα1 P {jα1 ≤ V

< (j + 1)α1 }

jα1 P {jα1 ≤ V

< (j + 1)α1 }

jα1 P {jα1 ≤ V

< (j + 1)α1 }

j =1

j

j =1

∞

=C
j =1

1

∆g (k)

kα1
k=1
j

∞

=C

iα1 −1

i=1

∞

≤C

j

< (j + 1)α1 }

j =k

1

k=1

< (j + 1)α1 }

j =i

P {jα1 ≤ V

∆g (k)

kα1

≤C

P {jα1 ≤ V

∞

1

≤

∞

i α 1 −1

∆g (k)

kα1

1

∆g (k)

kα1
k=1
j−1

k=1

1
kα1

1

−

(k + 1)α1

−

(k + 1)α1

g (k)

∞

P {jα1 ≤ V

+C

< (j + 1)α1 }g (j)

j =1

∞

jα1 P {jα1 ≤ V

≤C

< (j + 1)α1 }

j =1

j−1

k=1

1
kα1

1

k(log k)q−1

∞

P {jα1 ≤ V

< (j + 1)α1 }j(log j)q−1

jα1 P {jα1 ≤ V

< (j + 1)α1 }(log j)q−1

+C
j =1

∞

≤C

j =1

j −1

1

k=1

kα1

−

1

(k + 1)α1

∞

P {jα1 ≤ V

< (j + 1)α1 }j(log j)q−1

jα1 P {jα1 ≤ V

< (j + 1)α1 }(log j)q−1

+C
j =1

j

∞

≤C
j =1

1

kα1
k=1

∞

P {jα1 ≤ V

+C

< (j + 1)α1 }j(log j)q−1

j =1

∞

P {jα1 ≤ V

≤C

< (j + 1)α1 }j(log j)q−1 < ∞.

j =1

Now we prove (ii).
|nα |p

1
n

|nα |p

P{ V

p

0

1

1

≥ t }dt =

| nα | p

n

1

≤C
n

|nα |p

P{ V

p

0

n

n

|nα |p

|nα |p

P{ V
1

Noting that the first term on the right-hand side is finite, it remains to prove that
|nα |p

1
n

|nα |p

Denote d(k) =
∞
j=k

d(j)
jpα1

P{ V

p

≥ t }dt .

1
n1 ...nq =k

∼C

1, we have by Lemma 3.1 of Gut (2001) that

(log k)q−1
kpα1 −1

.

P{ V
1

|nα |p

1

+C

|nα |p

1

≥ t }dt +

p

≥ t }dt .

p

≥ t }dt

k


L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763

759

Hence, we have

n

αq+1

|nα |p

1

|nα |p

∞

P{ V

p

d(k)

≥ t }dt ≤
k,nq+1 ,...,nd =1

1

α

[kα1 nq+1 ...nd d ]

1
pαq+1

E( V

pαd

kpα1 .nq+1 . . . nd

p

I (j ≤ V

< j + 1))

j=1

(where [x] denotes the greatest integer not exceeding x)
αq+1

∞

d(k)

≤C
k,nq+1 ,...,nd =1

α

[kα1 nq+1 ...nd d ]

1
pαq+1

jp P (j ≤ V

pαd

kpα1 .nq+1 . . . nd

αq+1

∞

d(k)

≤C
k,nq+1 ,...,nd =1

d(k)
k,nq+1 ,...,nd =1

pαq+1

[jp − (j − 1)p ]P ( V ≥ j)

pαd

kpα1 .nq+1 . . . nd

j =1
αq+1
α
[kα1 nq+1 ...nd d ]

1
pαq+1

pjp−1 P { V

pαd

kpα1 .nq+1 . . . nd

=C

d(k)

∞

1
pαq+1

pαd

nq+1 ,...,nd =1 nq+1 . . . nd

≤C

pαq+1
nq+1 ,...,nd =1 nq+1

...

∞

+

pα

nd d

...

≤C

+

∞

≤C

. . . nd

which is finite if E ( V

r

kpα1

≥ j}
k=

αq+1
α 1/α1
j/nq+1 ...nd d

d(k)
kpα1

α

∞

pj(1/α1 −1) P { V

≥ j}

d(k)

k=1

kpα1
∞

∞
pα
nd d
αq+1
α
j=[nq+1 ...nd d ]+1

pjp−1 P { V

≥ j}
k=

αq+1
α 1/α1
j/nq+1 ...nd d

d(k)
kpα1

∞

1

βq+1
nq+1 ,...,nd =1 nq+1

d(k)

∞

pjp−1 P { V

j =1

1

...

∞
k=1

[nq+1 ...nd d ]
βd

pαq+1

nq+1 ,...,nd =1 nq+1

≥ j}

pα
nd d
αq+1
α
i=[nq+1 ...nd d ]+1

1

βq+1
nq+1 ,...,nd =1 nq+1
∞

pj(1/α1 −1) j(p−1/α1 ) P { V

α

∞

αq+1

∞

≥ j}

j =1

1

pαq+1
nq+1 ,...,nd =1 nq+1

pjp−1 P { V
j =1

[nq+1 ...nd d ]

1

α

[kα1 nq+1 ...nd d ]

kpα1
k=1
αq+1

∞

≥ j}

j =1
αq+1

∞

α

[kα1 nq+1 ...nd d ]

1

∞

≤C

< j + 1)

j =1

βd

. . . nd

jr −1 (log i)q−1 P { V

≥ j}

j =1

log+ V )q−1 < ∞ and since βl = αl /α1 > 1 for q + 1 ≤ l ≤ d.

The random field {Vn , n ∈ Nd } is said to be weakly mean dominated by the random element V if, for some 0 < C < ∞,
1

|n|

P { Vk ≥ x} ≤ CP { V

≥ x}

k≺n

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 {Vn ; n ∈ Nd }
be an array of random elements in a real separable Banach space X, Fk is the σ -field generated by the family of random
d
elements {Vl : i=1 (li < ki )}, F1 = {∅, Ω }.
The first theorem is a general a.s. convergence one.
Theorem 3.1. Let α1 , . . . , αd be positive constants. Let {Vn , n ∈ Nd } be a random field of random elements. If
1
n

|n|

P {max Sk > |nα |} < ∞
k≺n

for every

> 0,

(3.1)


760

L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763

then

|2nα | < ∞ for every

P max Sl >
l≺2n

n

>0

(3.2)

and, a fortiori, the SLLN
1

| nα |

max |Sk | → 0 a.s. as |n| → ∞

(3.3)

k≺n

obtains.
Conversely, (3.2) implies that (3.1) holds.

> 0, denote 2n = (2n1 , . . . , 2nd ) and 2nα = (2n1 α1 , . . . , 2nd αd ). We have the inequalities

Proof ((3.1) ⇒ (3.2)). Fix

P max Sl > |2nα |

≤

l≺2n

n

min

k∈[2n ,2n+1 )

n

P max Sl >
l≺k

1

≤
k∈[2n ,2n+1 )

n

|2n |
2d

≤
k∈[2n ,2n+1 )

n

1

≤ 2d
n

| n|

|k|

2α1 +···,αd

P max Sl >
l ≺k

P max Sl >
l≺k

P max Sl >
l ≺n

|kα |

2α1 +···+αd

2α1 +···+αd

2α1 +···+αd

|kα |

|kα |

|nα | < ∞. (by (3.1))

This implies by the Borel–Cantelli lemma that
1

|2nα |

max Sk → 0

a.s. as |n| → ∞.

k≺2n

(3.4)

Now for k ∈ [2n , 2n+1 ) we have
1

1

max Sl ≤

|kα | l≺k

|kα |

max
l≺2n+1

Sl ≤

1

|

2nα

max

| l≺2n+1

Sl =

2α1 +···+αd

|2(n+1)α |

max
l≺2n+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

1
n

|n|

P max Sl > |nα |
l ≺n

P max Sl >

≤
n

l≺2n

2α1 +···+αd

Sl

(3.5)

> 0,

|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 let X be a separable Banach space. Then the following two statements are equivalent:
(i) The Banach X is of martingale type p.
(ii) For every random field {Vn ; n ∈ Nd } in X with E (Vn |Fn ) = 0 for all n ∈ Nd and for every α = (α1 , . . . , αd ) with αi > 0 for

all 1 ≤ i ≤ d, the condition
E Vn
n

|nα |p

p

<∞

implies that, for every
1
n

| n|

> 0,

P {max Sk > |nα |} < ∞
k≺n

and, a fortiori, the SLLN
1

|nα |
obtains.

max Sk → 0
k≺n

a.s. as |n| → ∞


L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763

761

Proof. In order to prove [(i) ⇒ (ii)] we show that
P {max Sk > |2nα |} < ∞

> 0.

for every

k≺2n

n

Applying Markov’s inequality and Lemma 2.2 we have that
1

P {max Sk > |2nα |} ≤

p

k≺2n

n

n

|2nα |p

E (max Sk

1

≤C
n

2nα p

|

|

p

k≺2n

E Vk
k≺2n

p

)
E Vk

≤C
k

|kα |p

p

< ∞.

Now we prove [(ii) ⇒ (i)]. Assume that (ii) holds. Let {Wn1 , Gn1 ; n1 ≥ 1} be an arbitrary sequence of martingale difference
in X such that
∞

E Wn1

p

p
n1

n 1 =1

<∞

Set
Vn1 ,...,nd = Wn1

if n2 = · · · = nd = 1 otherwise Vn1 ,...,nd = 0.

Then {Vn1 ,...,nd } is the random field in X satisfies E (Vn1 ,...,nd |Fn1 ,...,nd ) = 0 for all (n1 , . . . , nd ) ∈ Nd , and
∞

E Vn1 ,...,nd p
=
(n1 . . . nd )p
n1 ,...,nd =1

∞

p

E Wn1

n1 =1

p
n1

< ∞.

By (ii),
1
n1 . . . nd

Vi1 ,...,id → 0

a.s. as (n1 . . . nd ) → ∞.

i1 ≤n1
id ≤nd

Taking n2 = · · · = nd = 1 and letting n1 → ∞ we obtain

1

n1

n1 j = 1

Wj → 0

a.s. as n1 → ∞.

Then by Theorem 2.2 of Hoffmann-Jørgensen and Pisier (1976), X is 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. Let X be a martingale type p Banach space with 1 < p ≤ 2. Let α1 , . . . , αd be positive constants satisfying
1/p < min{α1 . . . , αd } < 1, let q be the number of integers s such that αs = min{α1 . . . , αd } and let {Vn , n ∈ Nd } be a
random field satisfying E (Vn |Fn ) = 0 for all n ∈ Nd . Suppose that {Vn , n ∈ Nd } is weakly mean dominated by V such that
E V r (log+ V )q−1 < ∞ with r = min{α1...,α } . Then
1

1
n

|n|

d

P {max Sk > |nα |} < ∞

(3.6)

k≺n

and, a fortiori, the SLLN
1

|nα |

max Sk → 0 a.s. as |n| → ∞

(3.7)

k≺n

obtains.
Proof. For each n ∈ Nd , set
Vk = Vk I ( Vk ≤ |nα |), Vk = Vk I ( Vk > |nα |),
Yk = Vk − E (Vk |Fk ), Yk = Vk − E (Vk |Fk ),

Sn =
k≺n Yk , Sn =
k≺n Yk .
Since E (Vk |Fk ) = 0, it follows that Vk = Yk + Yk . Moreover, if Gk and Gk are the σ -fields generated by the family of

random elements {Yl : i=1 (li < ki )} and {Yl : i=1 (li < ki )}, respectively, then Gk ⊂ Fk and Gk ⊂ Fk for all k ≺ n, which
imply that E (Yk |Gk ) = E (Yk |Gk ) = 0 for all k ≺ n.
d

d

762

L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763

> 0,

We now begin the proof. For every
1
n

|n|

1

P {max Sk > 2 |nα |} ≤
k≺n

|n|

n

1

P {max Sk > |nα |} +
k≺n

|n|

n

> |nα |}.

P {max Sk
k≺n

(3.8)

First, we show that
1
n

|n|

P {max Sk
k≺n

> |nα |} < ∞.

Applying Markov’s inequality and Lemma 2.2, we obtain
1
n

|n|

P {max Sk
k≺n

1

> |nα |} ≤

|n||n|α

n

k≺n

1

1

≤C

|nα | |n|

n

1

≤C

|nα |

n

|nα |

1

|n|

1

=C

|n|

n

≤C

k≺n

0

n

n

1

1

| nα |

|n|

1

|n|

|nα |

∞
k≺n

|nα |

P { Vk ≥ t }dt

P { Vk ≥ t }dt
k≺n

| nα |

|nα |

P{ V

≥ t }dt

∞

1

|nα |

n

|nα |

≥ t }dt

∞

1

≥ |nα |} + C

∞

1
n

k≺n

n

≤C

P { Vk
0

k≺n

P { Vk ≥ |nα |}dt + C

P { Vk ≥ |nα |} + C

P{ V

∞

|nα | |n|

n

k≺n

1

1

=C

k≺n

E Yk

|nα | |n|

n

E Vk

1

1

E (max Sk ) ≤ C

|nα |

≥ t }dt < ∞ (by Lemma 2.2).

P{ V

By (3.8), in order to complete the proof, we next show that
1
n

|n|

P {max Sk > |nα |} < ∞.
k≺n

Again applying Markov’s inequality, we find that
1
n

|n|

1

P {max Sk > |nα |} ≤
k≺n

1

n

|n| |nα |p

n

|n| |nα |p

1

=

1

1

≤C
n

k≺n

k≺n

1

n

|nα |p

n

|nα |p

k≺n

k≺n
|nα |p

p

∞

1

|nα |p |n|

E Yk
k≺n

P { Vk
k≺n

p

≥ t }dt

0

P { Vk

p

≥ t }dt

P { Vk

p

≥ t }dt

0

1

|n|

0

k≺n

|nα |p

1

≤C

1

=C

|nα |p |n|

|nα |p

|nα |p |n|
1

p

n

1

n

=C

n

E Vk

1

1

E (max Sk > |nα |p ) ≤ C

|nα |p |n|
1

≤C

E (max Sk > |nα |)p

P{ V

p

≥ t }dt < ∞ (by Lemma 2.2).

0

The proof is completed.
Remark. Note that in the case of q < d, positive constants α1 , . . . , αd are not upper bounded by 1, which is weaker than
condition (2.1) of Theorem 2.1 of Gut and Stadtmüller (2009).
Theorem 3.4. Let α1 , . . . , 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 {Vn , n ∈ Nd } is weakly mean dominated by V such that E V (log+ V )q−1 < ∞. Then
(3.3) holds and then, the SLLN (3.4) obtains.
Proof. The proof is similar to that of Theorem 3.2 with p = 1 and we use Tn =
Sn and Sn , respectively.

k≺n

Vk and Tn =

k≺n

Vk are instead of


L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763

763

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