A LIMIT THEOREM FOR RANDOM FIELDS WITH A SINGULARITY IN THE SPECTRUM

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1 Teor Imov r. ta Matem. Statist. Theor. Probability an Math. Statist. Vip. 81, 1 No. 81, 1, Pages S )816- Article electronically publishe on January, 11 UDC A LIMIT THEOREM FOR RANDOM FIELDS WITH A SINGULARITY IN THE SPECTRUM A. YA. OLENKO AND B. M. KLYKAVKA Abstract. Homogeneous isotropic ranom fiels with singularities in spectra at nonzero frequencies are stuie. This class of fiels generalizes the case of ranom fiels with long range epenence where the spectrum has a singularity at the origin. We obtain a limit theorem for integral weight functionals of the fiel. We also iscuss the ifference between this class an the long range epenence. 1. Introuction Let ξx), x, be a real, measurable mean square continuous, wie sense homogeneous, an isotropic ranom fiel see [1, ]) with the zero mean an covariance function B n r) =B n x ) =E ξ)ξx), x, where r = x = n ) 1/. i=1 x i It is known see, for example, [1, ]) that there exists a boune nonecreasing function Φλ), λ calle the spectral function of the fiel ξx)) such that 1) B n r) = n n ) Γ J n rλ) Φλ), rλ) n where J ν z) is the Bessel function of the first kin an of orer ν> 1. A function ϕλ), λ, such that ϕλ) L 1 ), ϕλ), B n x ) = e iλ,x) ϕλ) λ, x, is calle the spectral ensity of the ranom fiel. The spectral ensity of a homogeneous an isotropic ranom fiel epens on λ only. In what follows we use the same notation ϕ ) forbothcasesϕλ) anϕ λ ). It will be clear from the context whether ϕ ) isa function of several parameters or of a single argument. For a function of a single argument, we have λ n 1 ϕλ) L 1 [, + )), ) Φλ) = πn/ Γn/) λ z n 1 ϕz) z, λ. 1 Mathematics Subject Classification. Primary 6G6; Seconary 6F17. Key wors an phrases. Ranom fiels, limit theorem, integral weight functionals, spectral functions, long range epenence. Supporte by the Sweish Institute grant SI-144/ c 11 American Mathematical Society

2 148 A. YA. OLENKO AND B. M. KLYKAVKA Various properties of ranom fiels with long range epenence in other wors, fiels with the long memory) are stuie in [1, 3, 4, 5]. The spectra of such ranom fiels have a singularity at the point λ =. For example, the spectral ensity amits the following representation: 3) ϕλ) = h λ ) λ n α, where h ) is a function efine on R + := [, + ), continuous in a neighborhoo of the origin, an such that h ) anh ) is boune in R +. Several limit theorems are known for integral functionals over multiimensional spheres or balls for such ranom fiels. Homogeneous an isotropic ranom fiels are consiere in the paper [6] for the case where the spectrum has a singularity at an arbitrary point a. The asymptotic behavior of the ifference Φa + λ) Φa λ) is stuie in [6] as λ +. Consier the function Φ a λ), λ, efine by Φ a λ) := { Φa + λ) Φa λ), λ<a, Φa + λ), λ a. Then Φ a ) is a spectral function an the asymptotic behavior of Φλ) at the point a is reuce to the asymptotic behavior of Φ a λ) at zero. Thus one can apply the results of [3, 4] where a uality between the asymptotic behavior of a spectral function at zero an that of the variance of integrals of the ranom fiel over a sphere or a ball whose raius increases to infinity is prove. Introucing the notation J ba r) :=π) n n/ rλ) rλ) n Φ a λ), it is prove in [6] that there exists a real raial function f n,r,a ) such that [ ] J ba r) =Var f n,r,a x )ξx) x =π) n n/ rλ a)) rλ a)) n Φλ). A particular case of a fiel with long range epenence is given by the case where a =,Φ a λ) =Φλ), an f n,r, x ) = { r n, x <r,, x r; see [6]. In this case, f n,r, x )ξx) x = 1 R r n ξx) x, n v n r) where v n r) ={x : x r} is the ball of raius r in. It was also prove in [6] that the weight function f n,r,a x ) amits the following representation: f n,r,a x ) = 1 x n/ 1 λ n/ J n/rλ a)) rλ a)) n/ J n/ 1 x λ) λ, x r. We are intereste in stuying the asymptotic behavior as r + of the following integral functional: f n,rt,a x ) ξx) x.

3 A LIMIT THEOREM FOR RANDOM FIELDS WITH A SINGULARITY IN THE SPECTRUM 149 We consier the ranom fiels whose spectra Φλ) have singularities at a point λ = a, that is, on the sphere S n a) = { x : x = a }. In contrast to the case of ranom fiels with long range epenence, the correlation functions of the fiels satisfying the above property oscillate with ecreasing amplitues. This is emonstrate by the following example. Example. Let n =3anα = 1. Consier a ranom fiel with long range epenence whose spectral ensity is given by { 1/λ 3 α, λ 1, ϕλ) =, λ > 1. Accoringto1),),an sinz) J 1/ z) =, π z we have 1 B 3 r) = π)3/ J 1/ rλ) λ = 4π 1 sinrλ) λ r λ r λ 3/ see [7]). The function B 3 r) epicte in Figure 1a is ecreasing an oes not oscillate. Suppose another ranom fiel has a singularity at the point λ = 1 an its spectral ensity is given by 1 ϕλ) = λ 1, λ 1,, λ > 1. Then the correlation function of this ranom fiel is such that 1 B 3 r) = π)3/ J 1/ rλ)λ 3/ r λ λ = 4π 1 λ sinrλ) 1 r λ 1 λ. The function B 3 r) epicte in Figure 1b has an oscillating behavior an its amplitue is ecreasing. Figure 1a Figure 1b

4 15 A. YA. OLENKO AND B. M. KLYKAVKA. Main result We nee an analog of representation 3) for the spectral ensity corresponing to the spectrum Φλ) with a singularity at a point a. Conition A. ξ :Ω R is a mean square continuous homogeneous isotropic Gaussian ranom fiel whose spectral ensity is 4) ϕλ) = h λ a) λ a 1 α, where ϕλ) L 1 ), α>, an h ) is a function efine in the interval [ a, + ), continuous in a neighborhoo of an such that h) an h ) is boune in [ a, + ). Similarly to the case of the function ϕ ), we use the same notation h ) for both cases hλ) anh λ ). Remark 1. In what follows, we consier the case where the function h ) is continuous in a neighborhoo of an where the inex α in representation 4) is the same for both cases λ >aan λ <a. Nevertheless all the results can be generalize to the case where the function h ) has istinct left an right limits at zero as well as to the case where the number α is ifferent for λ >aan λ <a. Remark. Comparing with the case of stochastic processes case of n = 1), the singularities of spectra for ranom fiels are essentially ifferent. The spectra of the stochastic processes may only have a singularity at a unique point. In the case of ranom fiels, the singularity is present at a unique point only if a =. Otherwise that is, if a ) the spectral ensity has singularities at all points of the n-imensional ball S n a). Remark 3. In contrast to the case of ranom fiels with long range epenence, the inex in the representation 4) of the spectral ensity is equal to 1 α, as compare to the inex n α in 3). This is explaine by that the spectral ensity in representation 3) is integrable if the following integral having a singularity at zero) + + λ n 1 h λ) ϕλ) λ = λ λ1 α is finite. The latter integral appears after the spherical change of variables. The spectral ensity in 4) is integrable for a> if the following integral having a singularity at the point a) + λ n 1 ϕλ) λ = + λ n 1 h λ a) λ a 1 α λ is finite. In both cases, the integral of the spectral ensity is finite only if α>. The ifference between the inices is explaine by that the spectral ensity for the case of long range epenence i.e., for the case of a = ) has a singularity at a unique point λ =, while the spectral ensity has singularities at all points of the n-imensional ball S n a) ifa>. Theorem 1. Assume that conition A hols an that <α<1. Then the finiteimensional istributions of the process tr α/ X r t) = π) n/ f h)a n 1)/ n,rt,a x )ξx) x

5 A LIMIT THEOREM FOR RANDOM FIELDS WITH A SINGULARITY IN THE SPECTRUM 151 weakly converge as r to the finite-imensional istributions of the process ) u t Xt) = Zu), where Z ) is the white noise in, B n ). J n/ R u n α/ n 3. Proof of the theorem We use the spectral ecomposition of the ranom fiel 5) ξx) = e i λ,x ϕ λ ) W λ) see [1]), where W ) isthewienermeasurein, B n ). Consier the following integral of the ranom fiel: 6) f n,r,a x )ξx) x. Substituting the spectral ecomposition 5) for ξx) in the integral 6), we get f n,r,a x) e i λ,x ϕ λ ) W λ) x R 7) n R n = f n,r,a x)e i λ,x x ϕ λ ) W λ). To justify the transformation in 7), we note that e i λ,x ϕ λ ) W λ) is the Fourier transform of the stochastic measure μλ) = ϕ λ ) W λ), while f n,r,a x)e i λ,x x is the usual Fourier transform of the function f n,r,a x). Thus equality 7) takes the form f n,r,a x) μx) = fn,r,a λ) μλ), which is a stochastic analogue of the Parseval equality. Since f n,r,a ) L )see[6]) an ϕ ) L ), equality 7) follows; see [8]. The function f n,r,a x ) is efine in [6] as a solution of the equation 8) f n,r,a x )e i λ,x x =π) n/ J n/r λ a)) R r λ a)). n n/ Substituting 8) in 7), we obtain f n,r,a x)e i λ,x x ϕ λ ) W λ) R 9) n =π) n/ J n/r λ a)) ϕ λ ) W λ). R r λ a)) n n/ The ratio J ν x)/x ν is an even function. If the spectrum is absolutely continuous, the finite-imensional istributions of the stochastic integral o not change uner the change of the integran on a set of zero Lebesgue measure.

6 15 A. YA. OLENKO AND B. M. KLYKAVKA Thus the integral in 9) is the sum of two integrals, π) n/ J n/r λ a)) ϕ λ )W λ) R r λ a)) n n/ =π) n/ J n/ r λ a)) ϕ λ ) W λ) λ >a r λ a)) n/ +π) n/ J n/ ra λ )) ϕ λ ) W λ) λ <a ra λ )) n/ =: I 1 r)+i r). In the integral I 1 r), we change the variables { u = λ 1 a 1) λ ), u = λ a, λ = u 1+ u ) a, λ = u + a. Note that this transformation is bijective an maps the set \ v n a) ={λ : λ >a} into the set := \{,...,)}. Since λ i = u i + a u i u = u u i i + a n i=1 u i in 1), we have λ i =1+a u u i u i u 3, λ i = a u iu j u j u 3, i j. Hence the Jacobian matrix of transformation 1) is given by ) n λi J n u) = u j i,j=1 ) = 1 u 3 u 3 + a u u 1 au 1 u... au 1 u n au u 1 u 3 + a ) u u au n u u 3 + a ). u u n Lemma 1. The Jacobian matrix of transformation 1) is equal to etj n u)) = 1+ a ) n 1, u. u Proof. Note that transformation 1) is raial. Thus for all pairs of vectors λ 1 an λ whose lengths are the same, i.e., λ 1 = λ, the scale images are the same in a neighborhoo of these vectors. This means that the Jacobian eterminant is the same for all vectors of the same length. This allows us to evaluate it along a single irection chosen appropriately. The calculations simplify if we consier the irection u 1,,...,). We have u 3... J n u) = 1 u 3 + a u... u u 3 + a u

7 A LIMIT THEOREM FOR RANDOM FIELDS WITH A SINGULARITY IN THE SPECTRUM 153 Thus etj n u)) = 1 u 3n u 3 u 3 + a u ) n 1 = 1+ a ) n 1. u We apply the formula for the change of variables in the stochastic integral, [9, Proposition 4.], [1, Theorem 4.4], an Lemma 1. Note that [9, Proposition 4.] an [1, Theorem 4.4] are simpler for our case, since we eal with a single function: I 1 r) =π) n/ J n/ r λ a)) h λ a) W λ) r λ a)) n/ λ a) 1 α =π) n/ =π) n/ =π) n/ λ >a J n/ r u ) h u ) r u ) n/ u J n/ r u ) h u ) r u ) n/ J n/r u ) r u ) n/ 1 α W u 1+ a )) u etjn u 1 α u)) W u) h u ) u 1 α 1+ a ) n 1 W u), u where W ) is the Wiener measure in, B n ). Now we change the variables ru i = ũ i, i =1,...,n, in the latter integral an use the property that a Gaussian white noise is semistable of orer n/: 11) W cx) =c n/ W x). Then the integral I 1 ) epens on the argument rt, t [, 1], namely I 1 rt ) 1) n/ r1 α n)/ =π) t J n/ ) ũ ) n/ h ũ /r) ũ 1 α 1+ ar ) n 1 W ũ). ũ Further asymptotic behavior of the integral I ) 1 rt is stuie similarly to the proof of Theorem.1.1 in [1]. Let J ) n/ 13) Y t) := W ũ), t [, 1]. R ũ n n α/ Relations 1) an 13) imply that tr α/ R r t) :=E π) n/ h)a I n 1)/ 1 rt ) ) Y t) 14) where = ) Rn Jn/ ũ n α Q r ũ ) := h ũ /r) h) Q r ũ ) ũ, 1+ ũ ar ) n 1 1. Let ψr) an ψr)/r asr. We represent the integral in 14) as the sum of two integrals, R r t) =R r,1 t) +R r, t). The omains of integration are B 1 := {ũ : ũ ψr)} for R r,1 t) anb := {ũ : ũ >ψr)} for R r, t).

8 154 A. YA. OLENKO AND B. M. KLYKAVKA Accoring to conition A, for every ε>thereexistsr such that Q r ũ ) <εfor r>r an ũ B 1.Since Jn/ s) 1 s n n Γ n as s, +1) Jn/ s) ) 1 s n = O s n+1 as s + see [7]), the integral Rn J n/ ) ũ n α ũ = u = t ũ = = πn/ t 1 α/n Γn/) Rn J n/ u ) u n α t1 α/n u Jn/ s) s n+1 α s is uniformly boune in t. Thus we can make R r,1 t) as small as we want by ecreasing ε. The Bessel functions are boune, that is, Jn/ u) 1 see [7]), thus there exists r such that the secon integral can be estimate as follows: R r, t) sup u [ a,+ ) hu) J ) n/ h) B ũ n α 1 + ũ ) n 1 ũ ) ) ũ + s = O B ũ 1 α+n = O ψr) s α for r>r. The latter expression approaches zero as r +. Hence lim r R r t) = an p lim E r + j=1 tj r α/ a j π) n/ h)a I n 1)/ 1 rt j ) Y t j )) = for all a j R an t j [, 1], j =1,...,p. This implies the convergence as r + of the finite-imensional istributions of the integral tr α/ π) n/ h)a I n 1)/ 1 rt ) to those of the process Y t), t [, 1]. The integral I r) is consiere similarly. Since the function J n/ x)/x n/ is even, the integral amits the following representation: I r) =π) n/ J n/ ra λ )) ϕ λ ) W λ). ra λ )) n/ Now we change the variables λ <a 15) { u = λ a λ 1), λ = u a u 1), u = a λ, λ = a u. This transformation is a bijection of the ball with center remove, { λ :< λ <a }, to itself.

9 A LIMIT THEOREM FOR RANDOM FIELDS WITH A SINGULARITY IN THE SPECTRUM 155 Relation 15) implies that λ i = a u i n i=1 u i u i, λ i = a u u i u i u 3 1, λ i = a u iu j u j u 3, i j. Thus the Jacobian matrix of transformation 15) is equal to ) n λi J n u) = u j i,j=1 a u u1) u 3 au 1 u... au 1 u n = 1 u 3 au u 1 a u u ) u au n u a u u n) u 3 Lemma. The Jacobian eterminant of transformation 15) is equal to et Jn u) ) ) n 1 a = u 1, u. Proof. Similarly to the proof of Lemma 1, it is sufficient to consier the Jacobian matrix only for the irection u 1,,...,): u 3... J n u) = 1 a u u 3... u a u u 3 Thus. et Jn u) ) = u 3 a u u 3n u 3) ) n 1 n 1 a = u 1. Applying the formula for the change of variables in the stochastic integral together with Lemma, we get ) I r) =π) n/ J n/ r u ) n 1 h u ) a r u ) n/ u 1 α u 1 W u), u <a where W ) is Wiener measure in, B n ), which is inepenent of W ). Making the change of variables ru i = ũ i, i =1,...,n, in the latter integral an using the semistability of the Gaussian white noise 11), we prove that I rt ) 16) 1 α n n/ r =π) t Now let 17) V t) := ũ <ra J n/ J n/ ) h ũ ) n/ ) ũ r ũ 1 α ũ n α/ W ũ), t [, 1]. ) ) n 1 ar ũ 1 W ũ).

10 156 A. YA. OLENKO AND B. M. KLYKAVKA Relations 16) an 17) imply that 18) where tr Rrt) 1 α/ :=E π) n/ h) a I n 1)/ rt ) V t) ) = Rn J n/ ũ n α Q 1 r ũ ) := h ũ /r) h) x) + = Q 1 r ũ ) ũ, 1 ũ ) n 1 ar + { x if x, if x<. 1, ) As in the case of R r t), we represent the integral on the right han sie of 18) as the sum of two integrals, R 1 rt) =R 1 r,1t)+r 1 r,t), whose omains of integration are B 1 for R 1 r,1t) anb for R 1 r,t). Similarly to what we i for R r,1 t) weprovethatr 1 r,1t) can be mae as small as we want uniformly in t [, 1]. For the integral R 1 r, t), we use the estimate J n/ u) 1 an note that This yiels Q 1 r ũ ) sup u [ a,+ ) hu). h) J ) n/ B Rr,t) 1 sup u [ a,+ ) hu) h) ) ũ + = O B ũ n α = O ψr) ũ n α ũ s s n+1 α The latter expression approaches zero as r +. Similarly to the proof for I 1, the above reasoning implies the convergence as r + of finite-imensional istributions of the integral tr α/ π) n/ h)a I n 1)/ rt ) to those of the process V t), t [, 1]. Thus the finite-imensional istributions of the process Xr t) = tr α/ π) n/ I 1 rt ) + I rt )) h)a n 1)/ converge to those of the process Y t)+v t), t [, 1]. Since the Wiener measures W ) anw ) are inepenent, there exists another Wiener measure Z ) in, B n ) such that Z ) = W )+W ) ).

11 A LIMIT THEOREM FOR RANDOM FIELDS WITH A SINGULARITY IN THE SPECTRUM 157 if we write formally). Thus Y t)+vt) = The theorem is prove. = ) J n/ R ũ n α/ n Jn/ ) ũ n α/ W ũ)+ ) J n/ R ũ n α/ n Zũ) = Xt). W ũ) 4. Concluing remarks We prove a limit theorem for ranom fiels whose spectra have singularities at nonzero frequencies. This theorem generalizes the results for ranom fiels with long range epenence see [1, 5]) whose spectra have singularities at the origin. Despite that the limit processes in our case are similar to those in the case of long range epenence, passing to the limit as a in Theorem 1 oes not prove the result. This can be explaine by the ifference in the nature of singularities in the spectra for these two cases; see Remarks an 3. One can prove other results generalizing an explaining the main theorem. In particular, the limit theorems for the functionals f n,rt,ax)h m ξx)) x, m > 1, can be stuie, where H m ) isthemth Chebyshev Hermite polynomial. These results will be publishe elsewhere. Among interesting open problems are proving limit theorems for ranom fiels whose spectra have more than one singularity point; consiering limit theorems for ranom fiels with OR spectra see [6]); obtaining properties of the fiels consiere in this paper an comparing them with known results for the case of long range epenence. Bibliography 1. N. N. Leonenko an A. V. Ivanov, Statistical Analysis of Ranom Fiels, KluwerAcaemic Publishers, Dorrecht Boston Lonon, MR g:635). M. I. Yarenko, Spectral Theory of Ranom Fiels, Optimization Software Inc., New York, 1983 istribute by Springer-Verlag). MR f:63) 3. N. N. Leonenko an A. Ya. Olenko, Tauberian theorems for correlation functions an limit theorems for spherical averages of ranom fiels, Ranom Oper. Stoch. Eqs ), no. 1, MR a:668) 4. A. Ya. Olenko, Tauberian an Abelian theorems for ranom fiels with strong epenence, Ukrain.Mat.Zh ), no. 3, ; English transl. in Ukrainian Math. J ), no. 3, MR k:6143) 5. N. N. Leonenko, Limit Theorems for Ranom Fiels with Singular Spectrum, KluwerAcaemic Publishers, MR16879 k:61) 6. A. Ya. Olenko, Tauberian theorems for ranom fiels with OR asymptotics. II,Teor.Imovirnost. Matem. Statist. 74 6), 81 97; English transl. in Theory Probab. Math. Statist. 74 7), MR i:685) 7. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambrige University Press, MR1746 6:64a) 8. C. Houre, Linear Fourier an stochastic analysis, Probab. Theory an Relate Fiels ), MR e:696)

12 158 A. YA. OLENKO AND B. M. KLYKAVKA 9. R. L. Dobrushin, Gaussian an their suborinate self-similar ranom generalize fiels, The Annals of Probability ), no. 1, 1 8. MR e:669) 1. P. Major, Multiple Wiener Itô Integrals, Lecture Notes in Math., vol. 849, Springer, New York, MR i:699) Department of Mathematics an Statistics, La Trobe University, Victoria 386, Australia aress: a.olenko@latrobe.eu.au Department of Probability Theory an Mathematical Statistics, Faculty for Mechanics an Mathematics, National Taras Shevchenko University, Acaemician Glushkov Avenue, Kiev 317, Ukraine aress: bklykavka@yahoo.com Receive 31/AUG/9 Translate by O. KLESOV