APPROXIMATING A HARMONIZABLE ISOTROPIC RANDOM FIELD

The class of harmonizable fields is a natural extension of the class of stationary fields. This paper considers a stochastic series approximation of a 
harmonizable isotropic random field. This approximation is useful for 
numerical simulation of such a field.


Introduction
In recent years the study of harmonizable processes has played a central role in the development of the theory of nonstationary processes.Crucial to this development is the pioneering work of Chang and Rao [1] on bimeasures and Morse-Transue integration.Their paper set the stage for the recent advances in the theory.A recent account of the development of harmonizable processes and some of their applications may be found in Swift [5].That paper also contains a detailed bibliography of the existing work on harmonizable processes.
In this paper, a stochastic series approximation of a harmonizable isotropic random field is considered.

Background
Let (a,Z,P) be a probability space.For p >_ 1, define L(P) to be the set of all centered complex valued f E LP(f, E, P), that is E(f) 0, where E(f) fnf(co)dP(w)is the expectation.We will consider second order random fields, more specifically, mappings X:Nn--Lg(P).
1Presented at the Fifth International Conference on Combinatorics, Statistics, Pattern Recognition and Related Areas, Mysore India, December 28-30, 1998.250 RANDALL J. SWIFT A random field X(t)is (weakly)stationary if its covariance function t) is translation invariant and continuous.Thus r(s,t)= (s-t) and is also positive definite.The power of the stationary assumption is seen in the following classical theorem of Bochner.Theorem 2.1: If Y (.) is a continuous stationary covariance then there exists a unique bounded Borel measure F:n---C such that (t) / ei"dF().
This representation bring8 to bear the powerful Fourier analytic methods in the study of stationary processes and fields.
A motivation for the concept of harmonizability is to enlarge the applications of stationary processes and fields while retaining the Fourier analytic methods.This notion is seen in the following definition.Definition 2.1: A random field X:nL(P) is weakly harmonizable if its covariance r(., is expressible as r(s,t) //ei'-ix"'dF(,,,V) (1) where F'IRnxlRn--+C is a positive semi-definite bimeasure of bounded Frchet variation.
a random field, X(.), is strongly aoi;* ir the bimeasure F(., .)in(1) extends to a complex measure and hence is of bounded Vitali variation.Random fields often admit an additional property: Definition 2.2: a random field X:[R+L2(P) with mean m(.) and covariance r(.,. is isotropic if for each orthogonal matrix t/acting on [R n, one has m(gt) re(t) and r(gs, gt) r(s, t).
The representation of the covariance of a harmonizable isotropic random field was first obtained by R.J. Swift [4] and more recently by M.M. Rao [3], using a dilation procedure.
Theorem 2.2: The covariance r:nxn---*C of a weakly harmonizable isotropic random field has representation 0 0 where Ju(') is the Bessel function (of the first kind) of order u-(n-2)/2 and F(., .) is of complex bounded Frchet variation.
Note that when the spectral measure F(.,.A useful characterization in spherical-polar form for the covariances of harmonizable isotropic random fields was given by Swift [4] as: Theorem 2.3: A random field x:n--L(P) is weakly harmonizable isotropic iff the covariance function r(.,.) is expressible for n >_ 2 as: where uand (i) 8-(T1, ll), -(7"2, V) are the spherical polar coordinates of s, in Nn, here 7"1 I I S II, 7"2 [1 /11 and u rl v rare unit vectors.
Using (2) and a form of Karhunen's Theorem, a spectral representation of a har- monizable isotropic random field is given by: Theorem 2.4: A random field X'Nn--*Lg(P) is weakly harmonizable isotropic iff it admits a spectral representation of the form E( Zm(B1)Zm,B(2)) 6mm,61l,F(B1, B2)   with F(. .) a function of bounded Frdchet variation and 7" I I t [I u-7, and (5 mr.a the Kronecker delta, the stochastic integral being in the Dunford-Schwartz sense.
With these details now in place, we will consider a series approximation of a harmonizable isotropic random field.

A Series Approximation
Recently, Yadrenko and Rakhimov [6], considered a method of simulating a stationary isotropic random field using a stochastic series.With the above background in place, we will consider a similar notion for a harmonizable isotropic random field.
Let , be a nonnegative random variable with density function fl()) and let .' be another nonnegative random variable with density function f2(1').
Let {rml} and {m2 } be two mutually independent sequences of independent random variables such that for each m 254 RANDALL J. SWIFT But, the growth inequality (cf., Lebedev  Thus, the error in approximating the random field (3) by its partial sum depends upon a boundedness condition upon its moments and is given by: Theorem 3.2: Suppose that the two-dimensional random field X(. with represent- ation satisfies / / .,.'fl(/)f2(/')d/d,t_ 0 0 then the partial sum XN(t), given by (8), converges in the L2-sense to X(t) as In practice, it is often desirable to guarantee not only the closeness of the partial sum, but also the closeness of partial derivatives up to a given order.The following theorem is stated here for completeness.The proof follows using similar, (though tedious), computations as above and the inequality (of., Lebedev [2]) (Jn(z))2 1/2(J2m-1 (z) -}-Or2rn -I-(z))" concentrates on the diagonal ,-,V the representation of the covariance becomes J(r) o Approximating a Harmonizable Isotropic Random Field 251 which is the representation of a stationary isotropic covariance obtained by Bochner.

1 :
The error of the L2-approximation of the random field (3) by its partial sum (8) is bounded by (2 I I t I I 2#l,1)/N.