Weak convergence to isotropic complex \(S\alpha S\) random measure

R E S E A R C H

Open Access

Weak convergence to isotropic complex

S

α

S

random measure

Jun Wang

1*

_{, Yunmeng Li}

2

_{and Liheng Sang}

1

*_{Correspondence:} [email protected] 1_{School of Mathematics and} Finance, Chuzhou University, Chuzhou, 239000, People’s Republic of China

Full list of author information is available at the end of the article

Abstract

In this paper, we prove that an isotropic complex symmetric

α

-stable random measure (0 <

α

< 2) can be approximated by a complex process constructed by integrals based on the Poisson process with random intensity.

Keywords: weak convergence; isotropic; complexSαSrandom measure; Poisson processes

1 Introduction

Letα∈(, ),XandXbe real random variables defined on the same probability space.

A complex random variableX=X+iXis called symmetricα-stable (in shortSαS) if the

random vector (X,X) isSαSinR, whereiis an imaginary unit. Furthermore, a complex

SαSrandom variableX=X+iXis isotropic if and only if there exist two independent

and identically distributed (in short for i.i.d.) zero mean normal random variablesG,G

and anα/-stable random variableA, independent of (G,G), totally skewed to the right,

such that (X,X) is sub-Gaussian with underlying vector (G,G), that is to say,

(X,X) d

= (√AG,

√ AG),

where= denotes the finite-dimensional distribution equality. Letd Gbe a complex Gaussian random variable satisfyingG=G+iG, whereG,Gare i.i.d. zero mean normal random

variables. Thus, every complex isotropicSαSrandom variableXwithα<  can be denoted byX=√AG.

Much of present works mainly focus on real-valued random variables; however, in Samorodnitsky and Taqqu [], Cohen [, ], Benassi et al. [], they encounter an important class of real-valued stable processes, which are defined in terms of integral with respect to complexSαSrandom measure. These show that complex random measure is very impor-tant in theory. Next, we introduce a series of processes approximating to different random measures based on the primary works of Stroock []. Finally, we construct a process to ap-proximate to complex-valued random measure.

Let{N(t),t∈[,∞)}be a standard Poisson process. For alln≥, define a processUn=

{Un(t),t∈[,∞)}by

Un(t) =

√ n

t



(–)N(ns)ds, t≥.

Stroock [] proved that whenn→ ∞, the law ofUnconverges weakly in the Banach space

C([,T]) of continuous functions on [,T], to a Wiener measure. Bardina and Jolis [] proved that

Un(s,t) =n t

 s



√

xy(–)N(√nx,√ny)dx dy

converges in law in the space of continuous functions on C([, ]_{), the space of}

con-tinuous functions on [, ]×[, ], asn→ ∞, to the ordinary Brownian sheet, where {N(x,y), (x,y)∈R

+}is a standard Poisson process in the plane. Let{Nα(t),t∈[,∞)}be

a Poisson process with random intensity. Under this condition, whenn→ ∞, Dai and Li [] proved that the processXn={Xn(t)}defined by

Xn(t) =

√ n

t



(–)Nα(ns)_{ds, }≤_t≤_,

converges weakly inC([, ]) to a sub-Gaussian processXα={

√

AW(t),t∈[, ]}, where

{W(t),t∈[, ]}is a standard Brownian motion andAis a random variable with the same

law asA. They also proved under certain conditions that

Xn(s,t) =n t

 s



√

xy(–)Nα(√nx,√ny)_{dx dy, }≤_t≤_,

converges weakly inC([, ]_{) to a two-parameter standard sub-Gaussian process, where}

{Nα(x,y);x,y∈[,∞)}is a two-parameter Poisson process with random intensity.

On the other hand, on approximation to complex Brownian motion, Bardina [] con-sidered the processUθ

n={Unθ(t),t∈[,∞)}defined by

U_nθ(t) =√n

t



eiθN(ns)ds, t≥,

whereiis an imaginary unit. Bardina proved that ifθ∈(,π)∪(π, π), whenn→ ∞, Pθ

nthe image law ofUnθ in the Banach spaceC([,T],C) converges weakly to the law of a

complex Brownian motion inC([,T],C). Whenθ=π,Pθ

n, converges weakly to the law of

√

W(t), where{W(t),t∈[,T]}is a standard Brownian motion.

Inspired by the above works, in this paper, we will prove similar results about an isotropic complexSαSrandom variable. Define

X_nθ(t) =√n

t



eiθNα(nr)_{dr, }≤_t≤_{, ()}

where{Nα(t),t∈[,∞)}is a Poisson process with random intensity _A. In the trivial case,

whenθ= , the processXθ

n(t) is deterministic, and whenntends to infinity,Xnθ(t) goes to

infinity. Whenθ=π, the processXθ

n(t) is real and () becomes

X_nθ(t) =√n

t



(–)Nα(nr)_{dr, }≤_t≤_{, ()}

We will prove under certain conditions, when θ ∈(,π)∪(π, π), that the law of Xθ

n converges weakly inC([, ],C) to the law of isotropic complexSαS random

mea-sure.

The rest of the paper is organized as follows. First we give preliminaries and the main result, then we present some lemmas, including proving the tightness and identification of the limit law, to prove the main result.

2 Preliminaries and the main result

Now we give some preliminary definitions and the main result.

Definition .([]) Let (,F,P) be a probability space. Suppose thatAis a

nonnega-tive random variable on the probability space (,F,P). Let (,F,P) = (×,F×

F,P×P) be the underlying probability space of this paper. LetN={N(t),t≥}be a

counting process on (,F,P) satisfying the following assumptions:

(a) WhenA> is given,Nis a Poisson process on(,F,P)with intensity_A;

(b) WhenA= ,N= a.s.

Then we callNas the Poisson process with random intensity  A.

Throughout this paper, we define the Poisson process{Nα(t),t≥}with random

inten-sity_A, we assume thatAis a strictlyα

-stable random variable with respect to (,F,P),

totally skewed to the right, with Laplace transform given by

Eexp{–λA}=exp–λα_.

Considering the sequencesXθ

ndefined by (), we have the following theorem.

Theorem . Let the process {Xθ

n(t), ≤t≤}, considering Pθn the image law of Xnθ,

in the Banach space C([, ],C).Then, ifθ ∈(,π)∪(π, π), when n tends to infinity, Pθ

nconverges weakly to the law inC([, ],C)of isotropic complex SαS random measure

{X(t) =√AG(t)}, ≤t≤,where{G(t)}is complex Gaussian measure,Ais a random

variable with the same law as A.

The main idea of the proof of the theorem is that, whenNα={Nα(t),t∈[,∞)} is a

Poisson process with random intensityA, the processXθ

n/

√

Aconverges weakly to complex Brownian motion independent ofA. Then, according to the idea of [], we need to check that ifθ∈(,π)∪(π, π), the familyPθ

nis tight and the law of all possible weak limits of

Pθ

nis the law of a complex Brownian motion.

We split the proof of Theorem . into two parts. We first prove the tightness of the processXθ

nand then identify the limit law of the processXnθ.

In this paper,Kdenotes a positive constant independent ofn, it may change value from one expression to another.

Proof of tightness We give an auxiliary processYθ

n={Ynθ(t),t∈[,T]},T> , denoted by

Y_nθ(t) = {A>}

n A

t



For anyn≥, we have

PXθ

n(t) =

√ AYθ

n(t),t∈[, ]

= . ()

SinceP(A= ) = . Denote

Iθ

,n(t) =

{A>}

n A

t



cosθNα(nr)

dr

=ReYθ

n(t)

,

and

I_,θ_n(t) =

{A>}

n A

t



sinθNα(nr)

dr

=ImY_nθ(t).

Lemma . There exists a constant K such that,whenθ ∈(,π)∪(π, π),for any s,t∈ [, ]and n> ,

EI_,θ_n(t) –I_,θ_n(s)+EIθ_,n(t) –I_,θ_n(s)≤K(t–s).

Proof Without loss of generality, we assumes<t. Then

EI_,θ_n(t) –I_,θ_n(s)+EIθ_,n(t) –I_,θ_n(s)

=E

{A>}

n

A

[s,t]



j=

cosθNα(nrj)

drdrdrdr

+ {A>}n



A

[s,t]



j=

sinθNα(nrj)

drdrdrdr

=:B+B,

where

B=

 E

{A>}

n

A

[s,t]



j=

cosθNα(nrj) –Nα(nrj–)



l=

drl

,

B=

 E

_{A>}n



A

[s,t]



j=

cosθNα(nrj) +Nα(nrj–)



l=

drl

.

Using the independence increments of the Poisson process, we obtain

B=

 E

_{A>}n



A

[s,t]



j=

EcosθNα(nrj) –Nα(nrj–) |A  l= drl

≤E

{A>}

n A t s r s

EcosθNα(nr) –Nα(nr)

|Adrdr

×

t

s r

s

EcosθNα(nr) –Nα(nr)

≤E

{A>}

n

A t

s r

s _E

cosθNα(nr) –Nα(nr)

|Adrdr

×

t

s r

s _E

cosθNα(nr) –Nα(nr)

|Adrdr ,

where · denotes the modulus of the complex number. It is easy to obtain

_E

cosθNα(nr) –Nα(nr)

|A≤e–nA(r–r)(–cosθ)

and

_E

cosθNα(nr) –Nα(nr)

|A≤e–An(r–r)(–cosθ)_.

Then

B≤E

_{A>}n



A t

s r

s t

s r

s

e–nA(r–r)(–cosθ)_e–An(r–r)(–cosθ)



l

drl

≤E

_{A>}n



A

A(t–s) ( –cosθ)_n

= 

( –cosθ)(t–s) _.

Next we calculateB. Considering the fact that

cos(x+x)cos(x+x)

=cos(x–x) + (x–x) + (x+x–x)

cos(x+x)

=cos(x–x) + (x–x)

cos(x+x–x)cos(x+x)

–sin(x–x) + (x–x)

sin(x+x–x)cos(x+x)

≤cos(x–x) + (x–x)+sin

(x–x) + (x–x)

≤cos(x–x)+sin(x–x)×cos(x–x)+sin(x–x).

By the independence increments of the Poisson process, we have

B=

 E

_{A>}n



A

[s,t]



j=

cosθNα(nrj) +Nα(nrj–)



l=

drl

≤  E

{A>}

n

A

[s,t]

_E

cosθNα(nr) –Nα(nr)

|A

+EsinθNα(nr) –Nα(nr)

|AEcosθNα(nr) –Nα(nr)

|A

+EsinθNα(nr) –Nα(nr)

|A



j=

≤  E

{A>}

n

A

[s,t]

_Eeiθ(Nα(nr)–Nα(nr))|_AEeiθ(Nα(nr)–Nα(nr))|_A



j=

drj

≤ 

( –cosθ)(t–s) _.

Then we obtain that

EI_,θ_n(t) –I_,θ_n(s)+EIθ_,n(t) –I_,θ_n(s)≤ 

( –cosθ)(t–s)

_=K(t–s)_.

This completes the proof.

Lemma . The set of laws of{Xθ

n}n≥inC([, ],C)is tight.

Proof To prove the tightness, we have to prove that the law corresponding to the real part and the imaginary part of the processXθ

nis tight.

In fact, it is almost sure thatYθ

n() =  for alln≥. Lemma . and Theorem . of []

show that the set of the law of the real part and the imaginary part of the process{Yθ

n}n≥

is tight inC([, ],R) for a fixed constant A. Hence, for anyε> , there exists a compact setS⊂C([, ],R) such that

PIθ

,n∈Sε,n≥

>  –ε/. ()

BecauseAis a finite positive random variable, then there exists a bounded setNε such

that

P(A∈Nε) >  –ε/. ()

Observe that the set ε:={af;a∈Nε,f ∈Sε}is compact inC([, ],R). Then, combining

() and (), for anyn≥, we have

PAI_,θ_n∈ ε

≥PA∈Nε,I_,θ_n∈Sε

≥ –ε. ()

Then, combining () and (), we obtain that, for anyε> , the real part of the processXθ

n

has

PReX_nθ∈ ε

≥ –ε.

Using a similar idea, we obtain that, for anyε> , the imaginary part of the processXθ

nhas

PImX_nθ∈ ε

≥ –ε.

This completes the proof.

Identification of the limit law Let{Pθ

nk} be a subsequence of{P

θ

n}weakly convergent to

some probabilityPθ_{. Ifθ∈(,π)∪(π, π), whenA>  is given, then the canonical process}

real part and the imaginary part of the process are two independent Brownian motions with respect to the probability space (,F,P).

Using Paul Lévy’s theorem, it suffices to prove that underPθ_{, whenA>  is given, the real}

part and the imaginary part of the canonical process are both martingales with respect to the natural filtration{Ft}, with quadratic variationsRe[Z],Re[Z]t=At,Im[Z],Im[Z]t=

At, and covariationRe[Z],Im[Z]t=  with respect to the probability space (,F,P).

Next, we first prove the martingale property and then prove the quadratic variations and covariation; these proofs are similar to the proof in []. Here, we give a sketch of the proof with some lemmas.

Letθ∈(,π)∪(π, π), in order to prove that underPθ_{the real and imaginary parts of}

the canonical processZare martingales with respect to its natural filtration{Ft}, we have

to prove that for anys≤s≤ · · · ≤sk≤s,k≥ and for any bounded continuous function

ϕ:Ck_{→Rsuch that}

EPθ

ϕ(Zs, . . . ,Zsk)

Re[Zt] –Re[Zs]

= , ()

and

EPθ

ϕ(Zs, . . . ,Zsk)

Im[Zt] –Im[Zs]

= . ()

We first consider (). Since{Pθ

n}weakly converges toPθ, combining Lemma ., we have

lim

n→∞EPθn

ϕX_nθ(s), . . . ,Xnθ(sk)

ReX_nθ(t)–ReX_nθ(s)

=EPθ

ϕX_nθ(s), . . . ,Xnθ(sk)

ReX_nθ(t)–ReXθ_n(s).

So it suffices to prove that

EϕX_nθ(s), . . . ,Xnθ(sk)

I_,θ_n(t) –Iθ_,n(s)

converges to zero whenn→ ∞. Now, we just need to prove that

lim

n→∞E

ϕX_nθ(s), . . . ,Xnθ(sk)

I_,θ_n(t) –I_,θ_n(s)= .

In fact

_E

ϕX_nθ(s), . . . ,Xθn(sk)

X_nθ(t) –X_nθ(s)

=EϕX_nθ(s), . . . ,Xnθ(sk)

eiθNα(ns)E

_√

n

t

s

eiθ(Nα(nr)–Nα(ns))_dr

≤KE

_√

n

t

s

Eeiθ(Nα(nr)–Nα(ns))_dr

≤K√n

t

s _E

eiθ(Nα(nr)–Nα(ns))_dr

=K√n

t

s

=K

A n

 –e–An(t–s)(–cosθ)  –cosθ

→, n→ ∞.

Using the same idea of proof (), we can obtain the proof of ().

We give the following auxiliary lemma.

Lemma . Letθ ∈(,π)∪(π, π).Consider the natural filtration{F_tn,θ}of the process Yθ

n.Then,for any s<t and for any realFsn,θ-measurable and bounded random variable H,

if A> is given,we have

(a) lim_n→∞E[n__str

s ei

θ(Nα(nr)–Nα(nr))_dr

dr] =A(t–s)( +i_–cossinθθ);

(b) limn→∞ E[nst r

s[ei

θ(Nα(nr)+Nα(nr))_H]dr

dr] = .

Proof We first prove (a). In fact

E

n

t

s r

s

eiθ(Nα(nr)–Nα(nr))_dr

dr

=n

t

s r

s

e–An(r–r)(–eiθ)_dr

dr

=A(t–s)

 +i sinθ  –cosθ

+o

A n

,

then

lim

n→∞E

n

t

s r

s

eiθ(Nα(nr)–Nα(nr))_dr

dr =A(t–s)

 +i sinθ  –cosθ

.

Now we prove (b). Using the fact that the Poisson process has independent increments, we can obtain

E

n

t

s r

s

eiθ(Nα(nr)+Nα(nr))_Hdr

dr

=n

t

s r

s

Eeiθ(Nα(nr)–Nα(nr))+iθ

Nα(nr)–Nα(ns)+iθNα(ns)_Hdr dr

=n

t

s r

s

Eeiθ(Nα(nr)–Nα(nr))_Eeiθ(Nα(nr)–Nα(ns))_EeiθNα(ns)_Hdr

dr

≤Kn

t

s r

s

e–nA(r–r)(–eiθ)e– n

A(r–s)(–eiθ)dr_dr_

=Kn

t

s r

s

e–n_A(r–r)(–cosθ)_e–n_A(r–s)(–cos(θ))_dr dr

= KAe

ns A(–cosθ) cosθ–cos(θ)

t

s

e–An(–cosθ)rdr_

–KAe

ns A(–cos(θ)) cosθ–cos(θ)

t

s

e–An(–cos(θ))r_dr

= KA

_{( –cosθ)}

n(cosθ–cos(θ))–

KA_{( –e}–n(_At–s)(–cos(θ))₎

n(cosθ–cos(θ))( –cos(θ))

→, n→ ∞.

This completes the proof.

Lemma . Consider{Pθ

n}the laws on C([,T],C)of the processes Xnθ,and assume that

{Pθ

nk}is a subsequence weakly convergent to P

θ_{.Let Z be the canonical process,and let}

{Ft}be its natural filtration.Then,if A is given,under Pθ,whenθ ∈(,π)∪(π, π),we

have the quadratic variationsRe[Z],Re[Z]t=At,Im[Z],Im[Z]t=At,and covariation

Re[Z],Im[Z]t= with respect to the probability space(,F,P).

Proof Let θ ∈(,π)∪(π, π), A is given. We will prove that Re[Z],Re[Z]t =At,

Im[Z],Im[Z]t=At,Re[Z],Im[Z]t= . It is enough to prove that for anys≤s≤ · · · ≤

sk≤s,k≥ and for any bounded continuous functionϕ:Ck→R, such that

lim

n→∞E

ϕXθ_n(s), . . . ,Xnθ(sk)

I_,θ_n(t) –I_,θ_n(s)–A(t–s)= , ()

lim

n→∞E

ϕXθ_n(s), . . . ,Xnθ(sk)

I_,θ_n(t) –I_,θ_n(s)–A(t–s)= , ()

lim

n→∞E

ϕXθ_n(s), . . . ,Xnθ(sk)

I_,θ_n(t) –I_,θ_n(s)I_,θ_n(t) –I_,θ_n(s)= . ()

In fact

E

ϕX_nθ(s), . . . ,Xnθ(sk)

√

n

t

s

cosθNα(nr)

dr



= E

ϕX_nθ(s), . . . ,Xnθ(sk)

n

t

s r

s

cosθNα(nr)

cosθNα(nr)

drdr

=E

ϕX_nθ(s), . . . ,Xnθ(sk)

n

t

s r

s

cosθNα(nr) –Nα(nr)

+cosθNα(nr) +Nα(nr)

drdr

=n

t

s r

s

cosθNα(nr) –Nα(nr)

drdrE

ϕX_nθ(s), . . . ,Xnθ(sk)

+E

t

s r

s

ϕXθ

n(s), . . . ,Xnθ(sk)

cosθNα(nr) +Nα(nr)

drdr

=Re

n

t

s r

s

eθ(Nα(nr)–Nα(nr))_dr

dr

EϕXθ

n(s), . . . ,Xθn(sk)

+Re

E

n

t

s r

s

ϕX_nθ(s), . . . ,Xnθ(sk)

eθ(Nα(nr)+Nα(nr))_dr

dr .

Combining Lemma . and thatPθ

nconverges weakly toPθ, we can obtain that whenn→

Similarly, whenntends to infinity, we obtain the expression

EϕXθ

n(s), . . . ,Xnθ(sk)

Iθ

,n(t) –I,θn(s) 

converges toA(t–s)E[ϕ(Zs, . . . ,Zsk)]. This completes the proof of (). Now, we prove ().

EϕX_nθ(s), . . . ,Xnθ(sk)

I_,θ_n(t) –I_,θ_n(s)I_,θ_n(t) –I_,θ_n(s)

=E

ϕXθ

n(s), . . . ,Xnθ(sk)

n

t

s

cosθNα(nr)

dr

t

s

sinθNα(nr)

dr

=E

n

t

s r

s

ϕX_nθ(s), . . . ,Xnθ(sk)

cosθNα(nr)

sinθNα(nr)

drdr

+E

n

t

s r

s

ϕXθ_n(s), . . . ,Xnθ(sk)

sinθNα(nr)

cosθNα(nr)

drdr

=E

n

t

s r

s

ϕX_nθ(s), . . . ,Xnθ(sk)

sinθNα(nr) +Nα(nr)

drdr

=Im

E

n

t

s r

s

ϕXθ_n(s), . . . ,Xnθ(sk)

eθ(Nα(nr)+Nα(nr))_dr

dr

.

From () of Lemma ., whenntends to infinity, we have

EϕX_nθ(s), . . . ,Xnθ(sk)

I_,θ_n(t) –I_,θ_n(s)I_,θ_n(t) –I_,θ_n(s)→.

This completes the lemma.

Lemma . Ifθ∈(,π)∪(π, π),A> is given.For any T> ,when n→ ∞,the real part and the imaginary part of Xθ

nconverge weakly inC([,T],R),with respect to(,F,P),to

independent zero mean and variance At Brownian motions B(t)and B(t),respectively.

Proof WhenAis given,Nα(t) is the Poisson process with non-random intensity _A with

respect to (,F,P). Thus, for anyT> , by Lemma ., the real part and the imaginary

part ofXθ

nconverge weakly inC([,T],R), with respect to (,F,P), to two independent

zero mean and varianceAtBrownian motionsB(t) andB(t), respectively.

Lemma . Ifθ ∈(,π)∪(π, π).When n→ ∞,the real part and the imaginary part of Xθ

nconverge in law to sub-Gaussian processes{

√

AG(t)},{

√

AG(t)},t∈[, ],

respec-tively,where{G(t)},{G(t)},t∈[, ]are independent copies of a standard Brownian

mo-tion G(t),Ais a random variable with the same law as A.

Proof Combining Lemma ., the proof is similar to the proof of Theorem  of [].

Proof of Theorem. Combining Lemma ., Lemma . and Corollary .. of [], we

3 Conclusions

We prove that an isotropic complex symmetricα-stable random measure ( <α< ) can be approximated by a complex process constructed by integrals based on the Poisson pro-cess with random intensity, which gives a new method to construct complex-valued ran-dom measure.

Acknowledgements

We thank the anonymous referees and the associate editor for their valuable comments and suggestions of this paper.

Funding

This work is partly supported by the Natural Science Foundation of Universities of Anhui Province (KJ2017A426, KJ2016A527, KJ2014A180). Distinguished Young Scholars Foundation of Anhui Province (1608085J06). Natural Science Foundation of China (11271020). Anhui Province Natural Science Foundation (1508085QA14). The Natural Science Foundation of Chuzhou University (2016QD13).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Author details

1_{School of Mathematics and Finance, Chuzhou University, Chuzhou, 239000, People’s Republic of China.}2_{Department of} Mathematics, Anhui Normal University, Wuhu, 241000, People’s Republic of China.

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Received: 22 June 2017 Accepted: 8 September 2017 References

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