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Series and spectral representations of random stablemappingsDang Hung Thang aa Department of Mathematics , University of Hanoi , 90 Nguyen Trai Dong Da, Hanoi,VietnamPublished online: 04 Apr 2007.

To cite this article: Dang Hung Thang (1998) Series and spectral representations of random stable mappings, Stochastics andStochastic Reports, 64:1-2, 33-49

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SERIES AND SPECTRAL REPRESENTATIONS OF RANDOM STABLE MAPPINGS*

DANG HUNG THANG

Department of Mathematics, University of Hanoi, 90 Nguyen Trai Dong Da, Hanoi. Vietnam

(Received 18 August 1997, In @a/ form 17 Novembev 1997)

In this paper, series and spectral representations of symmetric stable random mappings taking values in a separable Banach space are obtained. The structure of the linear space of real-valued random variables generated by the random mapping under consideration plays an important role in representation theorems.

Keywords: Random mappings; random operators; symmetric stable random mappings; stable random series; stable random measures; random integral; series and spectral representation

AMS 1991 Subject Classification: Primary: 60H25, 60Bll; Secondary: 60E07, 60G15

1. INTRODUCTION

Let (X, d ) be a complete separable metric space and Y be a separable Banach space. By definition, a deterministic mapping from X into Y is a rule that assigns to each element x E X a unique element @x E Y, which is called the image of @ under x. Due to errors in the measurements and inherent randomness of the environment the image @x is not known exactly. Therefore, instead of considering @x as an element of Y we have to think of it as a random variable with values in Y.

By a random mapping from X into Y we mean rule @ that assign to each element x E X a unique Y-valued random variable @.x. If X is an interval of

*Supported in part by the National Basic Research Program in Natural Science (Vietnam) and Deutscher Akademischer Austauchdienst (Germany). E-mail: dhthang &it-hu.ac.vn

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34 D. H. THANG

R (a domain in R ~ , resp.) then @ is nothing but a Y-valued stochastic process on X (a Y-valued random field on X , resp.). Hence the concept of random mappings is a natural generalization of the familiar concept of stochastic processes and random fields. It is also one of the basis concepts in the theory of Random Dynamical Systems in an infinite dimensional space see [I] and references therein. If X is a Banach space, a random mapping @ is said to be a random operator if the mapping x w @x is linear and continuous in probability. Some aspects of random operators in Banach spaces were discussed in [14 - 16, 181.

It is well-known that a second order stochastic process can be expressed as the sum of orthogonal random variables (the Karhunen-Loeve expansion) and a stationary Gaussian stochastic process can be written as a integral of while noise (the spectral representation). For non-Gaussian processes, similar spectral representations were obtained by several authors (Kuelbs [6] for symmetric stable processes, Rajput, Rama-Murthy [ l l ] for symmetric semi-stable processes and recently Rajput, Rosinski [12] for infinitely divisible processes). Such representations have proved exceedingly useful in the statistical and probabilistic analysis of processes under consideration. A beautiful account on the random series of functions including random Taylor series, random Fourier series and its applications can be found in Kahane's book [7] (see also [8]).

The purpose of the paper is to establish series and spectral representa- tions for random stable mappings from X into Y. We show that there is a connection between the structure of the space [ a ] generated by the random stable mapping @ and the representation theorems. Based on a result due to Bretagnolle et al. [3] and Kanter [5] which said that the space generated by a real separable symmetric stable process can be embedded into some L,(S, A, p ) we prove that a random symmetric stable mapping admits an equivalent random mapping which is represented by a integral with respect to a random stable measure. In the case the space [a] is isometric to some L, (S . A, p ) then @ is written itself in the form of a integral with respect to a random stable measure. Moreover if [@I is isometric to lp then @ can be expanded into a random series of mappings with real stable independent random coefficients. In particular, a Gaussian symmetric random mapping is always written in the form of a random series as well as represented by a integral with respect to a white noise. Finally, the series representation of the random mapping is used to study the path properties of the random mapping under investigation.

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RANDOM STABLE MAPPINGS

2. SERIES REPRESENTATION OF RANDOM MAPPINGS

First let us give some necessary definitions. Let (S, A, p) be a finite measure space, E be a separable Banach space with the dual space E * and B(E) be its Bore1 a-algebra. For each p > 0 the space L;(S, A, p) denotes the set of (A, B(E))-measurable functions from S into E with 1 1 f 11, = (E I IUIJP ) '~P < cm. As usual we identify two functions which are equal p-almost surely. The space L ~ ( S , A, p ) equipped the norm 1 1 f l i p becomes a Banach space for p 2 1 and a F-space (complete metric linear space for 0 < p < 1) . We shall write Lp (S, A, p) instead of L: (s, A, p ) .

Let (R, F, P) be a complete probability space. A measurable mapping u from (0, F) into (E, B(E)) is called a E-valued random variable. The set of all real-valued random variables and the set of all E-valued random variables are denoted by Lo(0) and Lf(0) , resp. We do not distinguish two E-valued random variables which are equal almost surely. The set Lf(R) equipped with the topology of convergence in probability becomes a F- space. The characteristic function (ch. f.) of a probability measure p on E denoted by j2 is a mapping from E x into C given by

The ch. f. of a E-valued random variable u is defined as the ch. f. of its law and is also denoted by fi. If the sequence (u,) of L t (R) converges in probability to u then we writte P - limn u, = u. Frequently we write L;(R) instead of L,"'(O, F, P) , and N stand for the set of positive integers.

Throughout this paper, (X, d ) is a separable metric space and Y is a separable Banach space with the dual space Y*.

DEFINITION 2.1 A family @ = {@x), ,~ of Y-valued random variables indexed by the parameter set X i s called a random mapping from X into Y.

We now give several definitions of regularity for random mappings.

1. Let @ and QJ be two random mappings from X into Y. The random mapping CJ is said to be a modijication of @ if for all x E X we have

Noting that the exceptional set can depend on x.

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Two random mappings @ and 9 are said to be equivalent if it for each k E N and for each finite sequence (xi, y;)f=, of X x Y* the R k-valued

k random variables { (@xi , y l ) ) i= , and {(*xi, y~) ) :=, have the same law. For each fixed sample w E R, the mapping x+ @x(w) is called a sample path (or a sample mapping) of @. The mapping @ is called continuous if the sample path is continuous for almost all w.

@ is stochastically continuous if for each sequence ( x , ) of X converging to x E X we have P-lim @x, = a x . @ is sample-continuous if @ has a continuous modifications 9.

DEFINITION 2.3 Let X be a separable Banach space. A random mapping cP from X into Y is called a random operator if

Note that the exceptional set can depend on A,, X z , x l , x2.

Example 2.4 (Random series of mappings). Let ( fa),"=, be a sequence of deterministic measurable mappings from X into Y and (an):=, be a sequence of real-valued random variables. Assume that for each x E X the series

converges in probability. Then the correspondence

produces a random mapping from X into Y.

DEFINITION 2.5 A random mapping @ from X into Y is called a symmetric Gaussian random mapping if for each k E N and for each finite sequence (x i , y;)f=, of X x Y " the R k-valued random variable { (@xi , Y ~ ) ) L 1 is symmetric and Gaussian.

From definition it follows that if @ is a Gaussian symmetric random mapping then @x is a Y-valued Gaussian symmetric random variable for each x E X.

DEFINITION 2.6 Let (a,) is a standard Gaussian sequence (i.e., a sequence of real Gaussian i.i.d. random variables with the distribution

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RANDOM STABLE MAPPINGS 37

N(0,1)), ( f , ) is a sequence of deterministic mappings from X into Y. The series of the form

is called a Gaussian random series of mappings whenever this series converges a.s. in Y for each x E X.

THEOREM 2.7 Let @ be a stochastically continuous symmetric Gaussian random from X into Y. Then @ can be represented as a Gaussian random series of the continuous mappings.

Moreover, if @ is a symmetric Gaussian rmzdonz operator then @ can be represented as a Gaussian ranclonz series of the linear continuous operators.

Proof Let [@I denote the closed subspace of L2(R) spanned by random variables { (Qx, y*), x E X, y * ~ Y *). Then [@I is a separable Hilbert space and every element of [@I is a symmetric Gaussian random variable. Let (a,) is an orthonormal basis of [@I. Since the sequence (a,) is orthogonal, symmetric and Gaussian it is a standard Gaussian sequence. Now for each n E N define a mapping f,: X+ Y by

Here the Bochner integral (1) exists because

Fix x E X. For each y* E Y*, (ax, y*) is expanded in the basis (a,) in the form

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38 D. H. THANG

where the series converges in L2(R) so it is convergent it probability. Since the sequence (a,f,x) is a sequence of symmetric independent Y-valued random variables by the Ito-Nisio theorem we conclude that

Next we show that f, is continuous. Fix n E N. Let (xk) is a sequence in X converging to x. Since is stochastically continuous it follows that P - limk a x k = a x . From ( 1 ) it is easy to see that

By a fact that for a sequence of Gaussian symmetric Y-valued random variables (u,) all the convergences in L;(R) are equivalent [4], it follows that axk converges to a, in L~'(R). Hence limk,, f,xk = fnx.

Finally, if @ is a random operator then by (1) the linearity o f f , is obvious.

Next we shall be interested in possible extensions of Theorem 2.6 to the case of symmetric stable random mappings.

DEFINITION 2.8 A random mapping Q, from X into Y is called a symmetric p-stable random mapping (0 < p 5 2) if for each k E N and for each finite sequence (x i , y ; )L l of X x Y* the R k-valued random variable { (@x i , y ; ) )L l is symmetric and p-stable.

Recall that a real-valued random variable < is said to be symmetric p- stable if the ch. f. of < is of the form e ~ p ( - c ) t ) ~ ) where c = c ( t ) is a non- negative number depending on <. A Rk-valued random variable ( E l , . . . , Ck) is said to be symmetric p-stable if for each finite sequence of real numbers (t,)f=, the real-valued random variables zkl t,<, is symmetric p-stable.

From definition it follows that if is a symmetric p-stable random mapping then Q,x is a Y-valued symmetric p-stable random variable for each x E X. Observe that a symmetric 2-stable random mapping is nothing but a Gaussian symmetric random mapping. For properties of stable random variables taking values in a Banach space, see [9] for details.

Let be a symmetric p-stable random mapping and [a] be the closed subspace of LO(R) spanned by the random variables { ( a x , y*), x~ X, y* E Y*). If < E [@I then J is symmetric p-stable so the ch. f. of < is of the

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RANDOM STABLE MAPPINGS 39

form exp(-cltjP) where c= c(t) is a non-negative number depending on <. The length of < denoted by I(<II, is defined by

From a theorem due to Bretagnolle et al. [3] (see also the Addendum of [5 ] ) we have

1. The correspondence ( - / ( < ( , is a F-norm (a norm for p 2 1 ) on [a]. 2. All the L,(R)-topologies (0 < r < p) coincide with the topology induced by

ll<ll*-norm on P I . 3. The F-space [a] may be isometrically embedded into some Lp (S ,A , p),

provided that the real process {(ax, y*), x E X , y* E Y * ) is separable.

DEFINITION 2.10 Let (7,) is a standard p-stable sequence (i.e., a sequence of real p-stable i.i.d. random variables with the unit length), (f,) is a sequence of deterministic mappings from X into Y. The series of the form

is called a symmetric p-stable random series of mappings whenever this series is convergent a.s. in Y for each x E X.

The following theorem is an extension of Theorem 2.7 to the case of symmetric p-stable random mappings.

THEOREM 2.11 Let be a stochastically continuous symmetric p-stable random mapping from X into Y . I f [a] is isometric to eP then can be represented as a symmetric p-stable random series of continuous mappings.

Moreover if @ is a random operator then can be represented as a symmetric p-stable random series of linear continuous operators.

Proof Let I be an isometry from [a] onto lp and J = I - ' . Put

m = J(4 ax, y * ) ) = B(x, y*) E lp.

At first we shall show that the sequence (7,) is a standard p-stable sequence. Indeed, the joint ch. f. of the random variables ( y l , . . . , y,) is equal

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as desired. Next we have the expansion

where B(x, y*) = {b,,(x, y*)),",, and the series (2) converges in the norm 1 . 1 1 , so converges in probability. Now we show that there exist deterministic mappings fn : X+ Y such that for each x E X, y* E Y * we have

Fix x E X. Since the mapping y* H (ax , y*) is linear it follows that the mapping y* H B(x, y") is also linear. The ch. f. of a x is

is T,( Y *, Y)-continuous on Y *, where T, (Y *, Y) is the topology of uniform convergence on compact sets of Y 19, Prop. 1.7.21. Consequently, the function y* H bn(x, y*) is linear and r,(Y*, Y)-continuous on Y*. This means that there exists a unique element of Y namedf, x such that bn(x, y*) = (f, x, y*) as claimed (because the dual space of E * under the topology 7, (Y*, Y) is E, see [lo]). Consequently the equality (2) becomes

The rest of the proof is carried out similarly as in the proof of Theorem 2.7. Next, we shall show that f,, is continuous for each n E N. Fix n E N and

suppose that (xk) is a sequence in X such that lim xk = s, which implies that

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RANDOM STABLE MAPPINGS 4 1

@xk 4 Qix in LT(Cl) by the stochastic continuity of Qi. To show that limk+cu fn xk = f , x we need the following lemma.

LEMMA 2.12 Suppose that a sequences En of Y-valued symmetric p-stable random variables converges in pprobability to some Y-valued symmetric p- stable random variable E. Then <, i in L F ( R ) for 0 < r < p .

Indeed, let Gp c LT(C2) be the set all Y-valued symmetric p-stable random variables. Then by [9, Theorem 6.6.41 Gp is a closed subspace of LoY(R). We also have Gp c L ,!'(Q) for 0 < I < p [9]. Hence the Lemma follows from the closed graph theorem.

Now we have

n= 1

Since p < 2 by Corrolary 7.3.6 in [9] we get

n= l where C > 0 depends on r and p. From Lemma 2.12 EJI@,xk-@XI/'+ 0 as k + m. Hence limk,, fn xk = f a x as desired.

Finally, suppose that is a random operator. From the relation

I ( ( Q ~ X . J ? * ~ ) = B ( x , ~ * ) E ep it follows that for each y* E Y * the mapping x H ( f , x , y") is linear. Because Y is separable the linearity off, follows. 1

THEOREM 2.13 Let @ be a symmetric p-stable random mapping from X into Y . Then @ is equivalent to a p-stable random series of mappings i f and only i f [Qi] can be isometrically embedded into tp.

Proof Suppose that @ is equivalent to a random mapping XP represented by the p-stable random series of mappings ( f , ) . Put B ( x , y * ) =

((fn X . ,v*))zl E tP. We have

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Since is equivalent to Q from this it follows that

From this it is easy to see that correspondence

is extended into an isometry from [a] into lp. Conversely assume that I is an isometry from [@I into lP. Put

By the same argument as in the proof ot Theorem 2.11 there exist deterministic mappings f, : X - i Y such that for each x E X y* E Y * we have

Now we prove that the series C,"=, yn f , x converges a.s. in Y for each x E X. The ch. f. of the partial y, f,x is equal to exp {- Cr=l ( f nx , y*) l P ) and

= ~ e x p { - 1 ( a x , y*)ll:) = Eexp{i(@x, y * ) ) .

Thus the series C,"=, ynfnx converges a s . in Y by Ito-Nisio Theorem. Put Qx = C,Xj=, yn fnx . It is easy to check that ?I, is equivalent to a . rn

The following theorem gives a criterion for the sample continuity of random series of continuous mappings.

THEOREM 2.14 Let X be a compact metric space and @ is a random mapping from X into Y admitting the series representation

which is convergent a s . for each x E X. In this expansion, (a,) is a sequence of independent real-valued random variables, ( f ,) is a sequence of continuous deterministic mappings from X into Y . Then is sample-continuous if and only if the series (3) converges uniformly a s .

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RANDOM STABLE MAPPINGS 43

Proof Let V = C(X, Y ) be the set of all continuous mappings from X into Y. V is a separable Banach space under the supremum norm

Suppose that the series Cr="=,J, converges a.s. to V-valued random variable T. It is easy to check that the mapping 9 given by @(x, w) = T(w) x is a continuous modification of @. Conversely, suppose that Q, has a continuous modification 9. For each pair ( x , y * ) ~ X x Y * the mapping x @ y : V+R given by

is clearly an element of V*. Let F = {(x @ y*), x E X , y* E Y * ). It is easy to check that r is a separating subset of V*. Let T : R -+ V be a mapping given by T(w) : x ++ Px(w). For each ( x @y*) E F we have the mapping

is measurable. Since V is separable and r is a separating subset of V* by Theorem 1.1 in [20] T is a V-valued random variable. Moreover for each ( x 8 y*) E r we have

In view of Ito-Nisio theorem [20, Theorem 2.41 the series C,"=, a, fn converges a.s. to T i n the norm of V.

3. SPECTRAL REPRESENTATION OF RANDOM STABLE MAPPINGS

Random integrals is a good mechanism to generate random mappings. Let us start by the following example.

Example 3.1 Let (W,) (0 st 5 I ) be the Brownian motion on [0, I]. Define a random mapping Q, from L [O, 11 into C [0, 11 by

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We shall show that is a random operator. The linearity of @ is easy. Using the martingale inequality we get

which shows that is stochastically continuous. The random integral with respect to the Brownian motion (The Wiener

integral) is a significant tool in studying many statistical and probabilistic problems. Motivated by these considerations, many authors advocated the need to develop similar random integrals with respect to more general random measures.

DEFINITION 3.2 Let (S, A, p ) be a finite measurable space. A mapping M: A + Lo(S2) is said to be symmetric p-stable random measure with the control measure p if

1. For each A E A, M(A) is a real-valued p-stable random variable with the ch. f. exp{-p(A)ltlP}.

2. For every sequence (Ai) of disjoint sets in A the random variables M(A1), M(A2), . . . are independent and

Following [4] a symmetric 2-stables random measure with the control measure p is called a white noise with variance p.

The random integral of Banach space valued functions with respect to random symmetric p-stable measure M is constructed as follows (see [13]). Let E be a separable Banach space. For a simple function f : S+ E, f - EL, siXA,, where (Ai) are disjoint sets we define

A measurable deterministic function f : S + E is said to be M-integrable if there exists simple functions (f,) such that f, converges to f in p-measure

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RANDOM STABLE MAPPINGS 45

and the sequence {Js fn dM} converges in probability. Iff is M-integrable then we put

It was shown that this value does not depend on the choice of the approximating sequence ( f , ) . Among crucial properties of this random integral is the following theorem [13], which will be used later.

THEOREM 3.3 A function f : S + E is M-integrable if and only if the function

is the ch. f. of a E-valued symmetric p-stable random variable. In this case F ( y * ) is the ch. f. of the random variables J' f d M .

Following Linde [9] we say that a linear operator T : E* + L,(S, A, p ) is an A,-operator if the function F ( y " ) = exp { - ( 1 T y * ( I p ) is the ch. f. of E- valued symmetric p-stable random variable. It is shown that ([9]).

THEOREM 3.4 If T is A,-operaotor (0 < p <2) then there exists a unique function f E L:(S, A, p ) such that for each y* E E *

T y " ( s ) = ( f ( s ) , y *) for @-almost s.

Now let { g ( s , x)), , x be a family of M-integrable Y-valued functions indexed by the parameter set X. Then the rule that associates to each x E X a stochastic integral

@I- = k g (s , x )dM(s )

clearly gives a symmetric p-stable random mapping from X into Y. Conversely we have

THEOREM 3.5 Let @ be a symmetric p-stable random mapping from X into Ysuch that the real process {(ax, y"), x E X , y* E Y * ) is separable. Then there exist a synzmetric p-stable random measure M on some measurable space

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( S , A, p ) and a family {g(s, x)),,x of M-integrable Y-valued functions indexed by the parameter set X such that the random mapping 9 given by

is equivalent to @.

Proof By Theorem 2.8 the space [@I can be embedded isometrically into some L p ( S , A, p) by an isometry I. Fix x E X and consider the mapping T,: Y * -, L p ( S , A, p) given by T,(y*) = I ( (@ x , y* )). Clearly, T , is linear and we have

Hence T , is an A,-operator. By Theorem 3.4 there exists a unique function g( . , x ) E L:(S, A, p) such that for each y* E Y*

T,y* ( s ) = ( g ( s , x ) , y*) for p-almost s.

Let M is the symmetric p-stable random measure on S with the control measure p. Since

in view of Theorem 3.3 g(., x ) is M-integrable. Define a random mapping P by

We shall show that Y is equivalent to @. Indeed, let x l , . . . ,xk E X and k y ; , . . . , y; E Y * . The ch. f. f ( t l , . . . , t k ) of { ( Y x i , ~ :))i=l is equal to

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RANDOM STABLE MAPPINGS

= Eexp i x t i ( @ x i , y , ' ) . { . I 1 1 This shows that the random variables {(*xi, y ~ ' ) ) f = ~ and { ( @ x i , y f ) ) k 1 have the same law.

The following theorem gives a sufficient condition for a symmetric p- stable random mapping to be represented by a random integral.

THEOREM 3.6 Let @ be a symmetric p-stable random mapping from X into Y (0 < p < 2). Ij" [a] is isometric to some L p ( S , A, p) then there exist a symmetric p-stable random measure M on (S , A, p) and a family {g(s, x ) ),,,r of M-integrable Y-wlzred functions indexed by the parameter set X such that for each x E X

In particular, a symmetric Gaussian random mapping can be written as a random integral w.r.t, a white noise.

Proof Let I be the isometry between [a)] and Lp(S7 A, p) and J = I - ' . Define a mapping M : A i L,(R) by

M ( A ) = J ( X A ) .

We show that M is a symmetric p-stable random measure with the control measure p. Indeed, suppose that (Ai) is a sequence of disjoint sets in A . The ch. f. of the random variable ( M ( A I ) , . . . , M(Ak)) is

Eexp i C t j ~ ( l i ) = { i : l }

Eexp i x t,J(AaZ) = Eexp iJ x ~ A A , { , I 1 )I

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48 D. H. THANG

This shows that the condition 1) in definition 3.1 is satisfied and the random

variables M(AI), M(A2), . . . are independent. Because C,"=, X A , = A,)

it follows that M(u,"_,A,) = C,"=, M(A,) a.s. As shown in the Theorem 3.5 there exists a family {g(s, x)), , , of M-integrable Y-valued functions indexed by the parameter set X such that

I ( ( ax , y*)) = (g(s, x), y*) for p-almost s

Now it is easy to check that for each f E L,(S, A, p) we have I( f ) =

Js f (s) dM(4. Consequently

Because Y is separable from this it follows that

Acknowledgements

Most of work was done during the author's stay at the Institute of Dynamical Systems, University of Bremen, supported by DAAD. The author would like to express his sincere thanks to Professor L. Arnold for his support and valuable comments concerning the manuscript, to Dr. Nguyen Ding Cong for his kind help and to the staff of the Institute of Dynamical Systems for the warm hospitality.

Finally, the author would like to thank the referee for correcting several inaccuracies from the first manuscript.

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