We show that every multiparameter Gaussian process with integrable variance
function admits a Wiener integral representation of Fredholm type with respect
to the Brownian sheet. The Fredholm kernel in the representation can be
constructed as the unique symmetric square root of the covariance. We analyze
the equivalence of multiparameter Gaussian processes by using the Fredholm
representation and show how to construct series expansions for multiparameter
Gaussian processes by using the Fredholm kernel.Comment: Published at http://dx.doi.org/10.15559/15-VMSTA39CNF in the Modern
  Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA)
  by VTeX (http://www.vtex.lt/

Sottinen, Tommi

Viitasaari, Lauri

Modern Stochastics Theory and Applications

English

arXiv

We  show  that  every  multiparameter  Gaussian  process  with  integrable  variance function admits a Wiener integral representation of Fredholm type with respect to the Brownian sheet. The Fredholm kernel in the representation can be constructed as the unique symmetric square root of the covariance. We analyze the equivalence of multiparameter Gaussian processes by using the Fredholm representation and show how to construct series expansions for multiparameter Gaussian processes by using the Fredholm kernel.© 2015 The Author(s). Published by VTeX. Open access article under the CC BY license.fi=vertaisarvioitu|en=peerReviewed

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Modern Stochastics: Theory and Applications 2 (2015) 287–295DOI: 10.15559/15-VMSTA39CNFFredholm representation of multiparameter Gaussianprocesses with applications to equivalence in law andseries expansions✩Tommi Sottinena,∗, Lauri Viitasaarib,1aDepartment of Mathematics and Statistics, University of Vaasa, P.O. Box 700,FIN-65101 Vaasa, FinlandbDepartment of Mathematics and System Analysis, Aalto University School ofScience, P.O. Box 11100, FIN-00076 Aalto, Finlandtommi.sottinen@iki.fi (T. Sottinen), lauri.viitasaari@aalto.fi (L. Viitasaari)Received: 25 June 2015, Revised: 21 September 2015, Accepted: 21 September 2015,Published online: 2 October 2015Abstract We show that every multiparameter Gaussian process with integrable variancefunction admits a Wiener integral representation of Fredholm type with respect to the Browniansheet. The Fredholm kernel in the representation can be constructed as the unique symmetricsquare root of the covariance. We analyze the equivalence of multiparameter Gaussian pro-cesses by using the Fredholm representation and show how to construct series expansions formultiparameter Gaussian processes by using the Fredholm kernel.Keywords Equivalence in law, Gaussian sheets, multiparameter Gaussian processes,representation of Gaussian processes, series expansions2010 MSC 60G15, 60G60✩The authors thank the referees for their useful comments.∗Corresponding author.1Lauri Viitasaari was partially funded by Emil Aaltonen Foundation.© 2015 The Author(s). Published by VTeX. Open access article under the CC BY license.www.i-journals.org/vmsta288 T. Sottinen, L. Viitasaari1 IntroductionIn this article, we consider multiparameter processes, that is, our time is multidimen-sional. Throughout the paper, the dimension of time n ≥ 1 is arbitrary but fixed.We use the following notation throughout this article: t, s, u ∈ Rn are n-dimen-sional multiparameters of time: t = (t1, . . . , tn), s = (s1, . . . , sn), u = (u1, . . . , un);0 is an n-dimensional vector of 0s, and 1 is an n-dimensional vector of 1s. We denotes ≤ t if sk ≤ tk for all k ≤ n. For s ≤ t, the set [s, t] ⊂ Rn is the n-dimensionalrectangle {u ∈ Rn; s ≤ u ≤ t}.Let X = (Xt)t∈[0,1] be a real-valued centered Gaussian multiparameter processor field defined on some complete probability space (Ω,F ,P). We assume that theGaussian field X is separable, that is, its linear space, or the 1st chaos,H1 = cl(span{Xt ; t ∈ [0, 1]})is separable. Here cl means closure in L2(Ω,F ,P).Our main result, Theorem 1, shows when the Gaussian field X can be representedin terms of the Brownian sheet. Recall that the Brownian sheet W = (Wt)t∈[0,1] is thecentered Gaussian field with the covarianceE[WtWs] =n∏k=1min(tk, sk).The Brownian sheet can also be considered as the Gaussian white noise on [0, 1]with the Lebesgue control measure. This means that dW is a random measure on([0, 1],B([0, 1]), Leb([0, 1])) characterized by the following properties:1.∫AdWt ∼ N (0, Leb(A)),2.∫AdWt and∫BdWs are independent if A ∩ B = ∅.Then, if f, g : [0, 1] → R are simple functions, then we have the Wiener–Itô isometryE[∫[0,1]f (t) dWt∫[0,1]g(s) dWs]=∫[0,1]f (t)g(t) dt. (1)Consequently, the integral∫[0,1] f (t) dWt can be extended for all f ∈ L2([0, 1]) byusing the isometry (1), and the isometry (1) will also hold for this extended integral.In this article, we show the Fredholm representation for Gaussian fields satisfyingthe trace condition (3) in Section 2, Theorem 1. In Section 3, we apply the Fredholmrepresentation to give a representation for Gaussian fields that are equivalent in law,and in Section 4, we show how to generate series expansions for Gaussian fieldsby using the Fredholm representation. The Fredholm representation of Theorem 1can also be used to provide a transfer principle that builds stochastic analysis andMalliavin calculus for Gaussian fields from the corresponding well-known theory forthe Brownian sheet. We do not do that in this article, although it would be quitestraightforward given the results for the one-dimensional case provided in [9].Fredholm representation 2892 Fredholm representationRecall that X is a separable centered Gaussian field with covariance function R andW is a Brownian sheet. Suppose that X is defined on a complete probability space(Ω,F ,P) that is rich enough to carry Brownian sheets.The following theorem states that the field X can be realized as a Wiener inte-gral with respect to a Brownian sheet. Let us note that it is not always possible toconstruct the Brownian sheet W directly from the field X. Indeed, consider the triv-ial field X ≡ 0 to see this. As a consequence, the Karhunen representation theorem(see, e.g., [2, Thm. 41]) cannot be applied here. Consequently, the Brownian sheet inrepresentation (2) is not guaranteed to exist on the same probability space with X.In any case, representation (2) holds in law. This means that for a given Browniansheet W , the field given by (2) is a Gaussian field with the same law as X.Theorem 1 (Fredholm representation). Let (Ω,F ,P) be a probability space suchthat σ {ξk; k ∈ N} ⊂ F , where ξk , k ∈ N, are i.i.d. standard normal random variables.Let X be a separable centered Gaussian field defined on (Ω,F ,P). Let R be thecovariance of X.Then there exist a kernel K ∈ L2([0, 1]) and a Brownian sheet W , possibly,defined on a larger probability space, such that the representationXt =∫[0,1]K(t, s) dWs (2)holds if and only if R satisfies the trace condition∫[0,1]R(t, t) dt < ∞. (3)Proof. From condition (3) it follows that the covariance operatorRf (t) =∫[0,1]f (s)R(t, s) dsis Hilbert–Schmidt. Indeed, the Hilbert–Schmidt norm of the operator R satisfies, bythe Cauchy–Schwarz inequality,‖R‖HS =√∫[0,1]∫[0,1]R(t, s)2 dt ds≤√∫[0,1]∫[0,1]R(t, t)R(s, s) dt ds=∫[0,1]R(t, t) dt.Since Hilbert–Schmidt operators are compact operators, it follows from, for example,[7, p. 233] that the operator R admits the eigenfunction representationRf (t) =∞∑k=1λk∫[0,1]f (s)φk(s) ds φk(t). (4)290 T. Sottinen, L. ViitasaariHere (φk)∞k=1, the eigenfunctions of R, form an orthonormal system on L2([0, 1]). Inparticular, this means thatR(t, s) =∞∑k=1λk φk(t)φk(s). (5)From this it follows that the square root of the covariance operator R admits a kernelK if and only if∞∑k=1λk < ∞. (6)Note that condition (6) is equivalent to condition (3). Consequently, we can defineK(t, s) =∞∑k=1√λk φk(t)φk(s) (7)since the series in the right-hand side of (7) converges in L2([0, 1]), and the eigenval-ues (λk)∞k=1 of a positive-definite operator R are nonnegative.Now,R(t, s) =∞∑k=1λk φk(t)φk(s)=∞∑k=1∞∑=1√λk√λφk(t)φ(s)∫[0,1]φk(u)φ(u) du=∫[0,1]( ∞∑k=1√λkφk(t)φk(u)∞∑=1√λφ(s)φ(u))du=∫[0,1]K(t, u)K(s, u) du,where the interchange of summation and integration is justified by the fact that series(7) converges in L2([0, 1]). From this calculation and from the Wiener–Itô isometry(1) of the integrals with respect to the Brownian sheet it follows that the centeredGaussian processes on the left-hand side and the right-hand side of Eq. (2) have thesame covariance function. Consequently, since they are Gaussian fields, they have thesame law. This means that representation (2) holds in law.Finally, we need to construct a Brownian sheet W associated with the field X suchthat representation (2) holds in L2(Ω,F ,P). Let (φ̃k)∞k=1 be any orthonormal basison L2([0, 1]). Setφk(t) =∫[0,1]φ̃k(s)K(t, s) ds.Then (φk)∞k=1 is an orthonormal basis (possibly finite or even empty!) on the repro-ducing kernel Hilbert space (RKHS) of the Gaussian field X (see further for a defi-nition). Let Θ be an isometry from the RKHS to L2(Ω, σ(X),P). Set ξk = Θ(φk).Fredholm representation 291Then ξk are i.i.d. standard normal random variables, and by the reproducing propertywe have thatXt =∞∑k=1φk(t) ξkin L2(Ω,F ,P). Now, it may be that there are only finitely many ξk developed thisway. If this is the case, then we augment the finite sequence (ξk)nk=1 with independentstandard normal random variables. Then setWt =∞∑k=1∫[0,t]φ̃k(s) ds ξk.For this Brownian sheet, representation (2) holds in L2(Ω,F ,P). Indeed,∫[0,1]K(t, s) dWs =∫[0,1]K(t, s) d∞∑k=1∫[0,t]φ̃k(s) ds ξk=∞∑k=1∫[0,1]K(t, s)φ̃k(s) ds ξk=∞∑k=1φk(t) ξk= Xt.Here the change of summation, differentiation, and integration is justified by the factthat the everything is square integrable.Remark 1. 1. The eigenfunction expansion (5) for the kernel (t, s) → K(t, s) issymmetric in t and s. Consequently, it is always possible to have a symmetrickernel in representation (2), that is, in principle it is always possible to transferfrom a given representationXt =∫[0,1]K(t, s) dWstoXt =∫[0,1]K̃(t, s) dW̃swhere W̃ is some other Brownian sheet, and the kernel K̃ is symmetric. Unfor-tunately, for a given kernel K and Brownian sheet W , the authors do not knowhow to do this analytically.2. In general, it is not possible to choose a Volterra kernel K in (2). By a Volterrakernel we mean a kernel that satisfies K(t, s) = 0 if sk > tk for some k. Tosee why a Volterra representation is not always possible, consider the followingsimple counterexample: Xt ≡ ξ , where ξ is a standard normal random variable.This field cannot have a Volterra representation since Volterra fields vanish in292 T. Sottinen, L. Viitasaarithe origin. A Fredholm representation for this field is simply Xt =∫[0,1] dWs(with suitable Brownian sheet W depending on ξ).For a more complicated counterexample (with X0 = 0) see [9, Example 3.2].Consequently, in general, it is not possible to generate a Gaussian field X onthe rectangle [0, t] from the noise W on the same rectangle [0, t]. Instead, thewhole information on the cube [0, 1] may be needed.3. If the family {K(t, · ) ; t ∈ [0, 1]} is total in L2([0, 1]), then a Brownian sheetin representation (2) exists on the same probability space (Ω,F ,P). Moreover,in this case, it can be constructed from the Gaussian field X. Indeed, in thiscase, we can apply the Karhunen representation theorem [2, Thm. 41].The reproducing kernel Hilbert space (RKHS) of the Gaussian field X is theHilbert space H that is isometric to the linear space H1, and the defining isometry isR(t, ·) → Xt. In other words, the RKHS is the Hilbert space of functions over [0, 1]extended and closed linearly by the relation〈R(t, ·), R(s, ·)〉H = R(t, s).The RKHS is of paramount importance in the analysis of Gaussian processes. In thisrespect, the Fredholm representation (2) is also very important. Indeed, if the kernel Kof Theorem 1 is known, then the RKHS is also known as the following reformulationof Lifshits [6, Prop. 4.1] states.Proposition 1. Let X admit representation (2). ThenH ={f ; f (t) =∫[0,1]f̃ (s)K(t, s) ds, f̃ ∈ L2([0, 1])}.Moreover, the inner product in H is given by〈f, g〉H = inff̃ ,g̃∫[0,1]f̃ (t)g̃(t) dt,where the infimum is taken over all such f̃ and g̃ thatf (t) =∫[0,1]f̃ (s)K(t, s) ds,g(t) =∫[0,1]g̃(t)K(t, s) ds.3 Application to equivalence in lawTwo random objects ξ and ζ are equivalent in law if, their distributions satisfy P[ξ ∈B] > 0 if and only if P[ζ ∈ B] > 0 for all measurable sets B. On the contrary, therandom objects ξ and ζ are singular in law if there exists a measurable set B suchthat P[ξ ∈ B] = 1 but P[ζ ∈ B] = 0. For centered Gaussian random objects there isthe well-known dichotomy that two centered Gaussian objects are either equivalentor singular in law; see [4, Thm. 6.1].Fredholm representation 293There is a complete characterization of the equivalence by any two Gaussian pro-cesses due to Kallianpur and Oodaira; see [5, Thms. 9.2.1 and 9.2.2]. It is possible toextend this to Gaussian fields and formulate it in terms of the operator K. The resultwould remain quite abstract, though. Therefore, we due not pursue in that direction.Instead, the following Proposition 2 gives a partial solution to the problem what doGaussian fields equivalent to a given Gaussian field X look like. Proposition 2 usesonly the Hitsuda representation theorem, which is, unlike the Kallianpur–Oodairatheorem, quite concrete.Let X̃ = (X̃t)t∈[0,1] be a centered Gaussian field with covariance function R̃, andlet X = (Xt)t∈[0,1] be a centered Gaussian field with covariance function R.Proposition 2 (Representation of equivalent Gaussian fields). Suppose that X hasrepresentation (2) with kernel K and Brownian sheet W . IfX̃t =∫[0,1]K(t, s) dWs −∫[0,1]∫[s,1]K(t, s)L(s, u) dWu ds (8)for some L ∈ L2([0, 1]), then X̃ is equivalent in law to X.Proof. By [8, Prop. 4.2] we have the following multiparameter version of the Hitsudarepresentation theorem: A centered Gaussian field W̃ = (W̃t)t∈[0,1] is equivalent inlaw to a Brownian sheet if and only if it admits the representationW̃t = Wt −∫[0,t]∫[0,s]L(s, u) dWu ds (9)for some Volterra kernel L ∈ L2([0, 1]).Let then X have the Fredholm representationXt =∫[0,1]K(t, s) dWs. (10)Then X̃ is equivalent to X if it admits the representationX̃t =∫[0,1]K(t, s) dW̃s, (11)where W̃ is related to W by (9). But Eq. (8) implies precisely this.Remark 2. On the kernel level, Eq. (8) states thatK̃(t, s) = K(t, s) −∫[s,1]K(t, u)L(u, s) dufor some Volterra kernel L ∈ L2([0, 1]).294 T. Sottinen, L. Viitasaari4 Application to series expansionsThe Mercer square root (7) can be used to build the Karhunen–Loève expansion forthe Gaussian process X. But the Mercer form (7) is seldom known. However, if wecan somehow find any kernel K such that representation (2) holds, then we can con-struct a series expansion for X by using the Fredholm representation of Theorem 1 asfollows.Proposition 3 (Series expansion). Let X be a separable Gaussian process with repre-sentation (2), and let (φk)∞k=1 be any orthonormal basis on L2([0, 1]). Then X admitsthe series expansionXt =∞∑k=1∫[0,1]φk(s)K(t, s) ds · ξk, (12)where the (ξk)∞k=1 is a sequence of independent standard normal random variables.The series (12) converges in L2(Ω,F ,P) and also almost surely uniformly if andonly if X is continuous.The proof below uses reproducing kernel Hilbert space technique. For more de-tails on this, we refer to [3], where the series expansion is constructed for fractionalBrownian motion by using the transfer principle.Proof. The Fredholm representation (2) implies immediately that the reproducingkernel Hilbert space of X is the image KL2([0, 1]) and K is actually an isometryfrom L2([0, 1]) to the reproducing kernel Hilbert space of X. Indeed, this is whatProposition 1 states.The L2-expansion (12) follows from this due to [1, Thm. 3.7] and the equiva-lence of almost sure convergence of (12), and the continuity of X follows from [1,Thm. 3.8].References[1] Adler, R.J.: An Introduction to Continuity, Extrema, and Related Topics for General Gaus-sian Processes. Lect. Notes Monogr. Ser., vol. 12. Institute of Mathematical Statistics Hay-ward, CA (1990), 160 p. MR1088478 (92g:60053)[2] Berlinet, A., Thomas-Agnan, C.: Reproducing Kernel Hilbert Spaces in Probabilityand Statistics. Kluwer Academic Publishers, Boston, MA (2004), 355 p. MR2239907.doi:10.1007/978-1-4419-9096-9[3] Gilsing, H., Sottinen, T.: Power series expansions for fractional Brownian motions. TheoryStoch. Process. 9(3–4), 38–49 (2003). MR2306058[4] Hida, T., Hitsuda, M.: Gaussian Processes. Transl. Math. Monogr., vol. 120. AmericanMathematical Society, Providence, RI (1993), 183 p. Translated from the 1976 Japaneseoriginal by the authors. MR1216518 (95j:60057)[5] Kallianpur, G.: Stochastic Filtering Theory. Stoch. Model. Appl. Probab., vol. 13, p. 316.Springer (1980). MR583435 (82f:60089)[6] Lifshits, M.: Lectures on Gaussian Processes. Springer Briefs Math. Springer (2012),121 p. MR3024389. doi:10.1007/978-3-642-24939-6Fredholm representation 295[7] Riesz, F., Sz.-Nagy, B.: Functional Analysis. Frederick Ungar Publishing Co., New York(1955), 468 p. Translated by Leo F. Boron. MR0071727 (17,175i)[8] Sottinen, T., Tudor, C.A.: On the equivalence of multiparameter Gaussian processes.J. Theor. Probab. 19(2), 461–485 (2006). MR2283386 (2008e:60100). doi:10.1007/s10959-006-0022-5[9] Sottinen, T., Viitasaari, L.: Stochastic analysis of Gaussian processes via Fredholm repre-sentation. Preprint, arXiv:1410.2230 (2014)

Fredholm representation of multi-parameter Gaussian processes with applications to equivalence in law and series expansions

Abstract We show that every multiparameter Gaussian process with integrable variance function admits a Wiener integral representation of Fredholm type with respect to the Brownian sheet. The Fredholm kernel in the representation can be constructed as the unique symmetric square root of the covariance. We analyze the equivalence of multiparameter Gaussian processes by using the Fredholm representation and show how to construct series expansions for multiparameter Gaussian processes by using the Fredholm kernel

Fredholm representation of multiparameter Gaussian processes with applications to equivalence in law and series expansions

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