Limit Theorems of Integrals with Respect to Vector Random Measures in Complete Paranormed Spaces

http://www.sciencepublishinggroup.com/j/ijsda doi: 10.11648/j.ijsd.20170304.14

ISSN: 2472-3487 (Print); ISSN: 2472-3509 (Online)

Limit Theorems of Integrals with Respect to Vector

Random Measures in Complete Paranormed Spaces

Renying Zeng

School of Mathematical Sciences, Chongqing Normal University, Chongqing, China

Email address:

[email protected]

To cite this article:

Renying Zeng. Limit Theorems of Integrals with Respect to Vector Random Measures in Complete Paranormed Spaces. International Journal of Statistical Distributions and Applications. Vol. 3, No. 4, 2017, pp. 81-86. doi: 10.11648/j.ijsd.20170304.14

Received: September 7, 2017; Accepted: September 26, 2017; Published: November 15, 2017

Abstract:

This paper studies random integral of the form

∫

gdM , where f is a function taking value in a paranormed vector space X, and M is an independent scattered vector random measure. Random integrals of this type are a natural generalization of random series with paranormed space valued coefficients. Some limit theorems of integrals with respect to vector random measures are proved.

Keywords:

Paranormed Vector Space, Random Measure, Random Integral, Limit Theorem, Convergence in Probability

1. Introduction

The idea of random measures first appeared in Bochner's paper [1], and a variety of discussions followed [8, 10, 12, 15, 17, 18, 19, 20]. The aim of this article is to study random

integral of the form

∫

gdM , where g is a function taking value in a paranormed vector space X, and M is an independent scattered vector random measure. Random integrals of this type are a generalization of random series with Banach space valued coefficients. It is well known that the asymptotic behavior of such series depends also on some geometric properties of the Banach space X (or a metrizable space) [3, 4, 6, 8, 16]. On the other hand, the existence of certain bounded linear operators on appropriate function spaces which we call random integrals depends in general on a geometric structure of X [12, 18, 19].

This paper defines and studies random integrals without any restriction in the geometry of X, devotes to the study of independently scattered vector random measures with emphasis on their convergence properties, presents convergence theorems of random integrals of the forms

n

g dM

∫

and

∫

gdM , where g_n and g are taking values in a paranormed vector space X, and M_n and M are R1-valued random measures.

2. Paranormed Spaces

Let K be the field of real numbers or the field of complex numbers, and X be a vector space over K. A paranorm || ||⋅ is a function defined on X, satisfying

|| || 0x ≥ , || || 0x = ⇔ =x 0;

||x+y|| || ||≤ x +||y||;

||− =x|| || ||x ; and

0

lim || _n || 0

n→ α x = , for any sequence {α ⊂n} K;

0

lim || _n|| 0

n→ αx = , for any sequence { }xn ⊂X.

Since ||x− ≤y|| || ||x +||y|| , for ∀x y, ∈X , ||x−y|| defines a metric on X. X equipped with the metric topology is said to be a paranormed space.

A paranormed space is said to be complete if every Cauchy sequence {xn} is convergent in X.

Any Banach space is a complete paranormed space. But the converse is not true.

A seminorm is function q defined on X and satsfying

q(x) ≥ 0;

q(x+y) ≤ q(x) + q(y); and

( ) ( )

q

α

x =

α

q x for any scalar

α

.

Lemma 1.1X is a complete paranormed space if and only if there is a family of continuous siminorms Q = {qn;n = 1, 2, ….} on X, such that

1( ) 2( ) n( )

q x ≤q x ≤ ⋅⋅⋅ ≤q x ≤ ⋅⋅⋅, ∀ ∈x X .

And the paranorm on X can be given by

1

( ) 1

1 ( )

2

n n n

n

q x x

q x

∞ =

=

+

∑

, ∀ ∈x X .

For any sequence{ }x_n ⊂X, and x∈X , the following are equivalent

lim _n

n→∞x =x;

lim _n 0

n

x x

→∞ − = ; or

lim ( _n ) 0

n→∞q x − =x

, ∀ ∈q Q.

3. Random Integrals

Let (T, Σ) be a measurable space, and ( , , )Ω Γ P be a probability space. A function

M: Σ →L₀( , , )Ω Γ P

is said to be an independently scattered random measure if for every pairwise disjoint sets A_n∈ Σ , random variable

( _n)

M A are independent, n =1, 2, …, and

1 ₁

( _{n n}) ( _n)

n

M ∞₌ A ∞ M A

=

=

∑

∪ .

Every independently scattered random measure M can be decomposed into two independent and independently scattered random measure M = Ma + Mc, where Ma is pure

atomic, and Mc is atomless.

Let ( , , )T Σ µ be a finite measure space, and

ν

be an infinitely divisible distribution on R1. The function

M: Σ →L₀( , , )Ω Γ P

is a random measure on ( , , )T Σµ generated by

ν

if M is independently scattered and, for every A∈Σ,

L

* ( )

(M A( ))=

ν

µ A ,

where ν*p is the pth convolution power of

ν

.

Daniell-Kolmogorov’s Consistency Theorem implies the existence of a random measure on every measure space generated by every infinite divisible law.

In what follows we assume that

ν

is symmetric. From [3], we may write

exp( ( )) exp[ ( ) ( )]

E iuM A = −µ A K u , (1)

where

2 2

1

( ) (1 cos ) ( )

2

K u u +∞ uv m dv

−∞

= σ +

_∫

− , (2)

while m is a symmetric

σ

-finite measure on R1 such that 0

}) 0

({ =

M and +∞ min(1,v m dv2) ( )

−∞ < ∞

∫

. We write

2

[ , ]

M≅ σ m if (1) and (2) hold.

In the sequel let X be a paranormed space, X ' be the topological dual space of X.

A function g: T → X is said to be a simple function if there exist pairwise disjoint measurable sets Bj

∈

Σ and xj

∈

X (j= 1,

2, …, k) such that

) , , 1 ( ,

0 )

( s B j k

otherwise if

，

x s

g j ∈ j = ⋅ ⋅⋅

  

=

For everyA∈Σ, we set

1 ( )

k

j j j

A

gdM =

∑

₌ x M B A

∫

∩ ,

the µ- Integralof the simple functiong is defined as

1

( ) k_j j j

T g s dµ =

∑

= x Bµ

∫

Definition 3.1 A function g: T → X is said to be M -Integrable, if there exists a sequence of {gk} of simple

functions from T to X such that lim _k( ) ( )

k→∞g s =g s ，µ-a.e.onT (3)

i.e., there exists E

∈

Σ and µ(E)=0 such that lim _k( ) ( )

k→∞g s =g s for s∈T\A ; and for every A∈Σ , k

A g dM

∫

converges in probability. If g is M-integrable, we put

lim k A gdM = −P k→∞ Ag dM

∫

∫

,

then the symmetry and independence assumptions imply that for every seminorm q∈Q, every simple function g: T → X, and every A∈Σ

{ ( _A k ) } { ( _T k ) }, 0

P q

∫

g dM > ε ≤P q

∫

g dM > ε ε >

Hence condition (3) in the Definition 3.1 is equivalent to

k T g dM

∫

converges in probability. (4)

Let

L

_E(M) denote the linear subspace of 0( , , )

X

L T Σ µ

consisting of all M-integrable functions. Similar to the proof of the Theorem in [5],

L

E(M) is a complete metrizable vector space with the paranorm

|| ||M min{1,|| ||} ( ) min{1,|| ||}

T T

A random variable M≅ σ[ 2, ]m is said to be Poissonian if

2

0

σ = and M R( 1)< ∞.

Proposition 3.1 If M is a poissonian random measure on ( , , )T Σ µ , then

L

_{( )}

E M = 0( , , )

X

L T Σ µ

Proof. Without loss of generality, we may assume that 1

( ) 0

M R > . Foru≥0, set

∫

= min{ | |,1} ( ) )

(u u mdv

H ν

then

( ) ( ) *

0

min{ | ( ) |,1}

( )

min{ | |,1} ( )

! ( ) ( ).

k

A m R k

k

E u M A A

e u v m dv

k H u A

+∞ ∞ −µ = −∞ µ = ≤ µ

∑

∫

Therefore, for any simple function

) , , 1 ( , 0 )

( s B j k

otherwise if

，

x s

g j ∈ j = ⋅ ⋅⋅

  

=

where Bj

∈

Σ pairwise disjoint measurable sets, and for any

seminorm q in Q we have

1

min{ ( ),1}

min{ ( ) | ( ) |,1}

T k

j j

j

E q fdM

E q x M A

=

≤

∫

∑

1 ( ( )) ( )

( ( )) ( ).

k

j j j

T

H q x A

H q f dt

=

≤ µ

= µ

∑

∫

The proof is completed.

Proposition 3.2 Suppose that E M T| ( ) |< ∞. Then, there exists a constant CM such that for every simple function

1

:

g T→R ,

1 2 2

| | _M( )

T T

E

∫

gdM ≤c

∫

g dµ

Proof. Suppose M = Ma + Mc, where Ma is pure atomic,

and Mc is atomless.

Write 1 ( ) n k n B n

g s =

∑

₌ bχ , where Bn

∈

Σ pairwise disjoint

measurable sets, one has

1 2 2

1 2 2 2 1 1 2 2 0 | | ( | |) ( ( )) ( |) , a T a T k

n a n n

T

E gdM

E g dM

b EM B

c g d

= ≤ = = µ

∫

∫

∑

∫

where c₀ is a constant. And there exists a constant c₁ such that 1 2 1 1 1 | | ( | | ( )) | | |. c T k j n n T E gdM

b EM B c g d

= ≤ ≤ µ

∫

∑

∫

Taking 12

0 1 ( )

M

c = + µc c T , then

1 2 2

| | _M( )

T T

E

∫

gdM ≤c

∫

g dµ

Proposition 3.3 If g is M-integrable, x'∈X', then x'(g) is

M-integrable, and

'( ) '( )

A A

x

∫

gdM =

∫

x g dM

Proof. The M-integrability implies that there exists a sequence {gk} of simple functions from T to X such that

lim k( ) ( )

k→∞g s =g s ，µ-a.e.onT

and

lim _k

T gdM = −P k→∞ Tg dM

∫

∫

Noting that

g

_k can be written as

, ( ) 1 ( ) n k n k k _n n B

g s =

∑

₌ xχ

where

,

n k

B

χ is the characteristic functionof the set Bn k, , i.e.,

, , 1 ( ) 0, n k n k s B s otherwise ∈  χ =  Therefore , ( ) 1 '( ( )) '( ) n k n k

k n n B

x g s =

∑

₌ x x χ

Hence

,

( ) 1

lim '( ( )) lim '( ) '( ( ))

n k

n k

k n n B k

k→∞x g s =k→∞

∑

= x x χ =x g s

µ-a.e.onT. And

( ) , 1 ( ) , 1

lim '( )

lim '( ) ( )

'( lim ( ))

'( lim )

'( ).

k T k

n k

n n k n

k

n k

n n k n k k T k T

P x g dM P x x M B

Consequently, x'(g) is M-integrable. And by the uniqueness of the limit one concludes that

'( ) '( )

A A

x

∫

gdM =

∫

x g dM

4. Limit Theorems

This section devotes to the proofs of some limit theorems of integrals with respect to vector random measures.

A sequence { }x_k ⊂X is said to be weakly convergent to

X

x∈ if for every x'∈X', there holds lim '( k) '( )

k→∞x x =x x ,

and denoted by lim _k

k

w x x

→∞

− = .

Definition 4.1 A probability measure µ is said to be a control measure of the random vector measure M, and denoted by M << µ, if µ(A)=0 implies M(A)=0.

Definition 4.2 A sequence {M_k} of random vector measures is said to be weakly converges in probability to a random vector measure M, denoted by lim _k

k

wP M M

→∞

− = , if

for every x'∈X', every E∈Σ, there holds

lim '( _k( ) ( )) 0

k

P x M E M E

→∞

− − =

Definition 4.3 A function g: T → X is said to be weakly M -integrable, if for everyx'∈X',x'(g)is M-integrable.

Theorem 4.1 SupposeM << µ. g,g_k∈

L

E(M) (k = 1, 2,…), such that lim _k( ) ( )

k

w g s g s

→∞

− = µ-a.e. If for any given

t > 0, and if for any 'x ∈X', and any A∈Σ

( ) 0

lim sup {|| '( _A k ) || } 0

A _k P x g dM t

µ →

∫

> =

then

lim _k

A A

k

wP g dM gdM

→∞

−

_∫

=

_∫

Proof. From Egoroff Theorem, for any given ε >0, there exists ( )Aε ∈ Σ, A(ε)⊂A and there exists N > 0, such that

(A A\ ( ))

µ ε < ε, and for all s∈A(ε)

||g s_k( )−g s( ) ||< ε >,k N (5)

Hence ||}, ) ( || || ) ( ' {|| 2 ] || ) ) ( ' ( ' [|| ] || ) ( ' [|| ) ( ) ( ) ( T M g g x P t dM g g x x P t gdM dM g x P k k A A k A ⋅ − ≤ > − = > − ∞

∫

∫

∫

ε ε ε (6)

Combining (5) and (6) one concludes that

( ) ( )

lim [|| '( _k ) || ] 0

A A

k→∞P x

∫

ε g dM−

∫

ε gdM > =t

Therefore, for any given x'∈X'

( ) ( ) ( ) ( ) [|| '( ) || ] [|| '( ) || ] [|| '( ) || ] [|| '( ) || ]. k A A k k A A k A A A A

P x g dM gdM t

P x g dM g dM t

P x g dM gdM t

P x gdM gdM t

ε

ε ε

ε

− >

≤ − >

+ − >

+ − >

∫

∫

∫

∫

∫

∫

∫

∫

( ) ( ) ( ) ( ) [|| '( ) || ] [|| '( ) || ] [|| '( ) || ], k B k A A B

P x g dM t

P x g dM gdM t

P x gdM t

ε

ε ε

ε

= >

+ − >

+ >

∫

∫

∫

∫

where B(ε)=A\A(ε). From the assumption

( )

( )

lim [|| '( ) || ] 0

lim [|| '( ) || ] 0

k B k

B k

P x g dM t

P x gdM t

ε →∞ ε →∞ > = > =

∫

∫

Therefore lim _k A A k

wP g dM gdM

→∞

−

_∫

=

_∫

Similar to the proof of Theorem 4.1 one has the following Theorem 4.2.

Theorem 4.2 SupposeM <<µ. g,g_k∈

L

_E(M) (k = 1, 2,…), such that lim _k( ) ( )

k→∞g s =g s µ-a.e. For given t > 0, if

0 } || ) ( ' {|| sup lim 0 )

(A→ _k P x

∫

A gkdM >t =

µ

Then, for any A∈Σ

lim _k

A A

k

P g dM gdM

→∞

−

_∫

=

_∫

Theorem 4.3 Suppose M<< µ. g g, k∈

L

E(M) (k = 1,

2, …), such that lim k( ) ( )

k→∞g s =g s µ-a.e. If there exists C>0

such that ||g sk( ) ||≤C n, ( =1, 2,...), then g is M-integrable, and for every A∈Σ

lim _k

A A

k

P g dM gdM

→∞

−

_∫

=

_∫

.

Proof. Since M<< µ,

( ) 0

( ) 0

( ) 0

lim sup [|| || ]

lim sup 2 [|| || || 1 || ]

lim sup 2 [ || ( ) || ]

0. k A A _k k A A _k A _k

P g dM t

P g dM t

P C M A t

µ →

∞ µ →

µ →

>

≤ ⋅ >

≤ ⋅ >

=

∫

∫

Therefore, from Theorem 4.2

lim _k

A A

k

P g dM gdM

→∞

Theorem 4.4 Suppose M,M_k(k=1,2,...) are random vector measures, and lim _k

k

P M M

→∞

− = , g∈L _{( )}

E M and g is

a bounded function under the weak topology of X, then g is weakly M-integrable and for every A∈Σ

lim _n

A A

k

wP gdM gdM

→∞

−

_∫

=

_∫

Proof. From Proposition 2.3, g is weakly M-integrable. Noting that lim _k

k

P M M

→∞

− =

( )

[|| '( ) || ]

[|| ' ( ) || ]

2 [|| '( ) || || ( )( ) || ]

[|| '( ) || ].

0.

k

A A

k A

k

A A

P x gdM gdM t P x g dM dM t

P x g M M T t P x gdM gdM t

∞

ε

− >

≤ − >

≤ ⋅ − >

+ − >

=

∫

∫

∫

∫

∫

This completes the proof.

Theorem 4.5 Suppose M,M_k(k=1,2,...) are random vector measures, where Mk≅ σ µ[ 2, n], and E M T| k( ) |< ∞(k

=1, 2, …), g is a bounded continuous function. If {µ_k} weakly converges to µ, and if {M_k}weakly converges to

M, then

lim _n

T T

k

wP gdM gdM

→∞

−

_∫

=

_∫

Proof. For every n > 0, there exists a compact subset

T

B⊂ and a constant c > 0 such that

sup k( )

k

B c

µ < < ∞,sup _k( \ ) 1

k

T B n

µ < ,

and there exists a sequence of simple function

( ) , 1

( ) n l

l _n n B l

g s =

∑

₌ xχ (l = 1, 2, …)

such that

1 ||g sl( ) g s( ) ||

n

− < , s∈B,

and

0

sup || _l( ) || sup || ( ) ||

s s

g s ≤ g s <c (l = 1, 2, …).

Therefore

2

2 2

\

2 2

0

2 2 1

0

|| ||

|| || || ||

( ) (2 ) ( \ ) (2 ) .

l n T

l k l k

B T B

n n

g g d

g g d g g d

n B c T B

n c c n

−

− −

− µ

= − µ + − µ

≤ µ + µ

≤ +

∫

∫

∫

Hence, there exists a constant c₁ >0 such that

1 2 2 1 2

1 0

|| ||

( (2 ) ) ,

l k k

T T

E g dM gdM

c m c− c n−

−

≤ +

∫

∫

By Chebyshev’s Inequality, for every t > 0

lim lim sup {|| l k k || } 0

T T

l→∞ _k P

∫

g dM −

∫

gdM > =t

Similarly,

lim {|| _l || } 0

T T

l→∞P

∫

g dM −

∫

gdM > =t

Noting that

lim sup {|| || }

lim sup {|| || }

k

T T

k

k l k

T T

k

P gdM gdM t

P gdM g dM t

− >

≤ − >

∫

∫

∫

∫

lim sup {|| || }

lim sup {|| || }

l k l

T T

k

l

T T

k

P g dM g dM t

P g dM gdM t

+ − >

+ − >

∫

∫

∫

∫

lim sup {|| || }

lim sup {|| || }.

l k l

T T

k

l

T T

k

P g dM g dM t

P g dM gdM t

= − >

+ − >

∫

∫

∫

∫

Hence

lim {|| _k || } 0

T T

k→∞P

∫

gdM −

∫

gdM > =t

That is to say

lim _T n _T

k

P gdM gdM

→∞

−

_∫

=

_∫

On the other hand, for any An

∈

Σ (n = 1, 2,…) satisfying

lim ( _n) 0

n→∞µ A = there exists a constant c2 > 0 such that

1 2 ₂ 2

|| || ( || || )

n n

k k

A A

E

∫

gdM ≤c

∫

g dµ

Since { }µk weakly converges toµ,

2 2

lim lim sup || || || ||

n k n

A A

n→∞ _k

∫

g dµ ≤

∫

g dµ

Hence

lim lim sup || || 0

n k

A

n→∞ _k E

∫

gdM =

Again, Chebyshev’s Inequality implies that

lim lim sup {|| || } 0

n

k A

n→∞ _k P

∫

gdM > =t

5. Conclusion Remark

Let (T, Σ) be a measurable space. A countably additive set function on Σ and taking values in L0( , )Ω P is said to be a

random measure. Vector random measures arise naturally as a "Banach space generalization" of real-valued random measures. Let X be a Banach space, 0( , )

X

L ΩP the set of all

X-valued random variables. A countably additive set function on Σ taking values in 0( , )

X

L ΩP is said to be an X-valued random measure. Vector random measures can be regarded as a "natural random generalization" of vector non-random measures studied by many authors.

This paper extends the definitions of random measure and random integral to complete paranormed spaces, and devotes to the study of limit theorems of random integrals with respect to vector random measures.

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