Spectral Conditions for Equivalence of Gaussian Random Fields with Stationary Increments

This paper studies the problem of equivalence of Gaussian measures induced by Gaussian random fields (GRFs) with stationary increments and proves a sufficient condition for the equivalence in terms of the behavior of the spectral measures at infinity. The main results extend those of Stein (2004), Van Zanten (2007, 2008) and are applicable to a rich family of nonstationary space-time models with possible anisotropy behavior.


Introduction
Space-time models have become increasingly popular in scientific studies such as geology, climatology, geophysics, environmental and atmospheric sciences, etc (Chilès and Delfiner (2012), Cressie (1993) and Stein (1999)). Gaussian random fields (GRFs) are ubiquitous in space-time modeling due to the prevalence of central limit theorems and the mathematical/computational amenability of the multivarite normal distributions. Most of the parametric models proposed in the literature are GRFs with specific parametric covariance structure (see Cressie and Huang (1999), Gneiting (2002), Stein (2005), and Gneiting, Genton and Guttorp (2007) for rich families of space-time covariance functions). One of the main objectives in statistics then is to find consistent estimates for the parameters, and finally use them for prediction of the underlying random field at unobserved locations. Given a parametric family of Gaussian random fields, an important question is to determine whether all the parameters are consistently estimable.
First step to answer this question demands an investigation on the equivalence or singularity of the corresponding Gaussian measures induced by this family of GRFs on their space of sample functions, since if two sets of parameters in the Gaussian models give equivalent Gaussian distributions, then it is impossible to find consistent estimators for these parameters involved regardless of the method chosen for estimation (see for example Zhang (2004) for a discussion on inconsistent estimation in Matérn covariance functions under the framework of fixed domain asymptotics). Another application of equivalence of Gaussian measures comes from covariance structure misspecification, and its effect on spatial interpolation (see Stein (1999)). Therefore, finding explicit conditions for deciding whether two Gaussian random fields induce equivalent Gaussian distributions on their spaces of sample functions has direct impact in evaluating the prediction error in interpolation of spatial data, and thus proving asymptotically optimal prediction under misspecified covariance structure. There are other applications of equivalence and perpendicularity of GRFs in spatial modeling. For example, we refer to Furrer, Genton and Nychka (2006), Kaufman, Schervish and Nychka (2008), Safikhani and Xiao (2014) and Safikhani (2015) for the application in covariance tapering.
Equivalence of Gaussian measures is a classical problem in probability theory that has been studied since the 1950's. We refer to the books Gikhman and Skorokhod (1965), Ibragimov and Rozanov (1978), Yadrenko (1983), Mandrekar and Gawarecki (2015) and references therein for systematic accounts. Necessary and sufficient conditions for the equivalence of Gaussian measures induced by stationary Gaussian processes in terms of their mean and covariance functions are given in Ibragimov and Rozanov (1978). Their extensions to stationary isotropic Gaussian random fields are proved in Skorokhod and Yadrenko (1973) and Yadrenko (1983). Among these results, the explicit criterion for equivalence of stationary Gaussian processes in terms of their spectral densities (cf. Theorem 17 on p. 104 of Ibragimov and Rozanov (1978)) is particularly convenient to apply. This criterion has been extended by Skorokhod and Yadrenko (1973) and Yadrenko (1983) to stationary isotropic Gaussian random fields.
However, investigation on the equivalence of nonstationary GRFs has been limited to some special cases. For instance, we refer to Cheridito (2001) on mixed fractional Brownian motion, Baudoin and Nualart (2003) on Volterra processes, Sottinen and Tudor (2006) on Gaussian random fields that are equivalent to fractional Brownian sheets, Stein (2004) on a family of intrinsic random functions with power law generalized covariance functions (including fractional Brownian fields), Van Zanten (2007, 2008 on Gaussian processes with stationary increments, and Xue (2011) on certain Gaussian random fields with stationary increments.
Our work is mainly motivated by Stein (2004) and by Van Zanten (2007, 2008 where explicit sufficient conditions for the equivalence of Gaussian processes with stationary increments in terms of their spectral densities similar to the criterion in Ibragimov and Rozanov (1978) for the stationary case have been established. The main purpose of this paper is to extend their results to the setting of Gaussian random fields with stationary increments which may have different regularities in each direction. Besides of theoretical interest, our results are applicable to anisotropic nonstationary space-time Gaussian models.
The rest of the paper is organized as follows. We start Section 2 by introducing some useful Hilbert spaces connected to the frequency domain, and study their structure. In Section 3, we state the main result of the paper, which is sufficient conditions for equivalence of GRFs with stationary increments using the tail behavior of their spectral densities. In the last section, we apply the main results to a rich family of anisotropic nonstationary spatio-temporal Gaussian models.

Preliminary
Let X = {X t : t ∈ R d } be a centered GRF with stationary increments. The covariance structure of X is fully described by Yaglom (1957). For simplicity, we assume that X(0) = 0 and the covariance function of X can be written as where F is a non-negative symmetric measure on R d \ {0}, called the spectral measure of X, that satisfies It follows from (2.1) that X has the following spectral representation: where W is a complex-valued Gaussian random measure with mean 0 and control measure F . If the spectral measure F is absolutely continuous with respect to the Lebesgue measure on R d , we will call its Radon-Nikodym derivative, denoted by f (λ), the spectral density of X. We will give conditions for the equivalence of GRFs with stationary increments in terms of their spectral densities, but first, we recall the definition of equivalence of GRFs.
is the σ-field generated by the cylinder subsets 2 of R D . Moreover, we call X and Y locally equivalent if they are equivalent on all bounded subsets of R d .
The spectral representation (2.1) makes an important bridge between the problem of equivalence of GRFs and the description of the space generated by the linear combinations of the kernel functions. For that purpose, in this section we define for a fixed and bounded set D ⊆ R d , an incomplete Hilbert space L e D = span e t (λ) := e i t,λ − 1 : t ∈ D with the inner product We denote the closure of L e D in L 2 (F ) by L D (F ). Also, for T > 0, denote by Π T = [−T, T ] d the cube with side 2T .
Observe that the functions in L e D are entire functions defined on C d (see Ronkin (1974) for definition and more properties), and they are of finite exponential type. Recall that an entire function ϕ on C d is called of finite exponential type if lim sup However, in general, the elements in the completed Hilbert space, L D (F ), may not have the same properties as the functions in L e D . This problem is discussed in details in Pitt (1973Pitt ( , 1975. In this paper, we assume that the spectral measure F has a density function f (λ) that satisfies the following condition: This assumption on the spectral density will imply the elements in L D (F ) to be entire functions of finite exponential type. These properties enable us to apply the Paley-Wiener type theorems to get nice description of the elements in the Hilbert space L D (F ) for D = Π T . The next two lemmas will prove these statements. The following lemma is taken from Xiao (2007) and we state it here again for completeness.
Lemma 2.1. Suppose that the spectral density f satisfies (C1). Then for fixed T > 0, there exists positive constants C and M such that for all functions φ of the form where a k ∈ R and t k ∈ Π T , we have for all z ∈ C d |φ(z)| ≤ C φ F exp{M |z|}. (2.5) Moreover, for fixed C 1 > 0, there exists a positive constant C 2 such that for all functions of the form (2.4), we have for all |z| ≤ C 1 .
One can use (2.5) to define the limiting functions in L ΠT (F ) in such a way that they also satisfy both (2.5) and (2.6). We will prove this in the next Lemma.
Lemma 2.2. Suppose that the spectral density f satisfies (C1). Then, for each T > 0, the space L ΠT (F ) consists of the restriction to R d of entire functions on C d of finite exponential type. Moreover, (2.6) holds for all functions φ ∈ L ΠT (F ).
Proof. The idea of the proof is similar to Pitt (1975), p. 304. Take a sequence φ n ∈ L e ΠT , such that φ n − φ F → 0 for some φ ∈ L ΠT (F ). Then, it is a Cauchy sequence in L 2 (F ), and using (2.5), we get This means for each fixed z ∈ C d , the sequence {φ n (z), n ≥ 1} is a Cauchy sequence in C. So, it is convergent and, moreover, the convergence is locally uniform. Denote the limit by φ(z). Now, since limit in L 2 (F ) sense implies the almost everywhere convergence for a subsequence, φ = φ a.e. with respect to F . From now on, we will take φ as our favorite version of the limits of functions in L e ΠT . Therefore, the elements in the space L ΠT (F ), are not only the L 2 (F ) limits of functions in L e ΠT , but also the pointwise limits as well. Thus, both (2.5) and (2.6) are true for all the elements in L ΠT (F ). The only thing left to prove is that these functions φ are entire functions on C d . But this is true since any element of the space L ΠT (F ) is the locally uniform limit of functions of the form (2.4) which are obviously entire functions, and thus, they are entire functions as well (This is called the Weirerstrass Theorem; see, for example, Proposition 2.8 on p. 52 of Ebeling (2007)).
This lemma shows that if the spectral density satisfies the assumption (C1), we can complete the space L e ΠT in such a way that the resulting functions are locally uniform limits of entire functions, and hence, they are entire functions of finite exponential type. Furthermore, since (2.5) is true for all the elements in the Hilbert space L ΠT (F ), we can see that the point evaluators, i.e. the functionals on L ΠT (F ) of the form φ → φ(z) for each fixed z ∈ C d are bounded operators. Now, we can apply the Riesz Representation Theorem (See Halmos (1957), Theorem 3. p. 31) to prove that the space L ΠT (F ) is a Reproducing Kernel Hilbert Space (RKHS) in the sense of Aronszajn (1950). This means that there exists a function K T (·, ·) : R d × R d → C such that (i) K T (ω, ·) ∈ L ΠT (F ) for all ω ∈ R d , and (ii) for every φ ∈ L ΠT (F ) and ω ∈ R d , we have the following kernel property is the orthogonal projection of ψ on L ΠT (F ) (See the proof in Aronszajn (1950), p. 345). We denote this projection by π LΠ T (F ) ψ.
Finding explicit forms of the reproducing kernels is not an easy job. However, in order to prove the results in Section 3, we only need to establish upper bounds for the growth rate of the diagonal elements of the reproducing kernels at origin and also at infinity. The following proposition proves an important growth rate for the diagonal elements in the reproducing kernels.
Proposition 2.1. Suppose that the spectral density f (λ) of F satisfies (C1). Then, for fixed T > 0 and C 1 > 0, there exists a positive constant C 2 such that Proof. Since for any fixed ω ∈ R d , K T (ω, .) ∈ L ΠT (F ), we can apply Lemma 2.2 to these functions. It follows from (2.6) that for all ω ∈ R d and λ ∈ C d with |λ| < C 1 . By taking λ = ω, we obtain the desired result.
We also need to define another Hilbert space based on the tensor product of the elements in L ΠT (F ). For this purpose, first we define L e ΠT ⊗ L e ΠT to be the span of functions (e t ⊗ e s )(ω, λ) : According to Theorem 1 on p. 361 of Aronszajn (1950), the new Hilbert space (2.11) This implies that for ψ ∈ L ΠT (F ) ⊗ L ΠT (F ), We finish this section by a lemma stating that the norm of the elements in spaces L ΠT (F ) depends essentially on the tail behavior of the spectral measure F . Lemma 2.3. Suppose f 0 and f 1 are two spectral densities satisfying the condition (C1), and further f 0 (λ) ≍ f 1 (λ) as |λ| → ∞. Then, L ΠT (f 0 ) = L ΠT (f 1 ), and further there exist positive constants C 3 and C 4 such that Proof. Suppose g ∈ L ΠT (f 0 ). There exists functions g n of the form (2.4) such that g − g n f0 → 0 as n → ∞. Now, using Lemma 2.2, we get (2.13) . Finally one can see that (2.13) holds if we replace g−g n by φ ∈ L ΠT (f 0 ) This leads to the second part of the lemma.

Main Results
In this section, we study the equivalence of GRFs with stationary increments, and clarify its connection to the Hilbert spaces constructed in Section 2. In particular, the role of the reproducing kernels of the RKHS L ΠT (F ) will be emphasized. We start this section by an extension of Theorem 5 on p. 84 in Ibragimov and Rozanov (1978) (Theorem 1 on p. 149 in Yadrenko (1983)) for stationary Gaussian processes (fields) to Gaussian random fields with stationary increments. Some extensions of the criteria for equivalence of stationary Gaussian processes (fields) have also been obtained by Van Zanten (2007, 2008 and Xue (2011). The following theorem is an extension of Theorems 3.3.9 and 3.3.10 in Xue (2011) and also Theorem 4.3 in Van Zanten (2007). (3.1) Proof. The proof is essentially a reconstruction of the proof of the Theorem 5, p. 84 of Ibragimov and Rozanov (1978), and is given here for the sake of completeness. The starting point is however, Theorem 4 on p. 80 of the same reference. The proof there can be adapted to our context with little change since it only involves the "entropy distance" between the Gaussian measures, and thus is true for general GRFs (See also Chatterji and Mandrekar (1978), Theorem 4.1, 4.4 pp. 180-185 ). After doing so, we get that two GRFs with stationary increments and spectral measures F 0 and F 1 are equivalent on . Now, since ∆ is a self-adjoint operator, if it is also a Hilbert-Schmidt operator, by the Spectral Theorem (See Dunford and Schwartz (1963), Corollary 5 p. 905 ), we can conclude that there exists an orthonormal basis for L D (F 0 ) consisting of the eigenvectors of ∆, denoting them by φ j , j = 1, 2, ..., with corresponding eigenvalues λ j , j = 1, 2, ... with j λ 2 j < ∞. Note that we can write . The square root of this quantity is called the Hilbert-Schmidt norm. This norm doesn't depend on the choice of the orthonormal basis (See Dunford and Schwartz (1963), Lemma 2, p. 1010). Therefore, we can rephrase Theorem 4 in the following form: two GRFs with stationary increments and spectral measures F 0 and F 1 are equivalent on D, if and only if, φ F0 ≍ φ F1 , ∀φ ∈ L e D , and j,k ∆φ j , φ k 2 F0 < ∞ for any orthonormal basis for L D (F 0 ). Now, take an arbitrary orthonormal basis for L D (F 0 ), φ 1 , φ 2 , ..., and suppose This shows that (3.1) holds for orthonormal basis of the space L D (F 0 ). Therefore, by continuity of inner product (3.1) will be true for all the elements in L D (F 0 ), especially for e t and e s when t, s ∈ D.
Conversely, suppose there exists a function ψ 0 This completes the proof.
Theorem 3.1 is stated in a general form for GRFs with stationary increment, with no restriction on their spectral measures. However, verifying the second condition in this theorem, which involves finding a function ψ ∈ L D (F 0 ) ⊗ L D (F 0 ) with the property (3.1), seems to be hard. If we put the condition (C1) on one of the spectral measures (say, F 0 ), we get the following theorem using the reproducing kernels of L ΠT (F 0 ). In fact, this theorem clarifies what must be the function ψ in Theorem 3.1.
Checking the first assumption in Theorem 3.2 may not be easy in general since we need to compare the norms of all the elements in the space L e ΠT under two different measures. For that purpose, in the following, we will find equivalent conditions which may be easier to verify in application.
It is well known (See Dunford and Schwartz (1963), p. 1009 ) that for ψ ∈ L 2 (F ⊗ F ), a Hilbert-Schmidt operator on L 2 (F ) can be defined as follows for every φ ∈ L 2 (F ). If we use specifically the ψ in Theorem 3.2, and restrict the domain to L ΠT (F ), we will have again a Hilbert-Schmidt operator on L ΠT (F ). Note that the image of the operator ψ is in fact inside the L ΠT (F ). To prove this, observe that for φ ∈ L ΠT (F ), This argument shows that ψφ ∈ L ΠT (F ) for any φ ∈ L ΠT (F ). Also, observe that since ψ(ω, λ) = ψ(λ, ω), the operator ψ is self-adjoint. This fact together with compactness of this operator (Since ψ is a Hilbert-Schmidt operator, it is already compact, see Dunford and Schwartz (1963), p. 1009) enable us to use the Spectral Theorem for compact normal operators (See Dunford and Schwartz (1963), Corollary 5, p. 905), which we will use in the proof of the next theorem. In fact, the next theorem is an extension of Theorem 4.3 in Van Zanten (2007) and shows that the first condition in Theorem 3.2 can be replaced by 1 / ∈ σ(ψ), where σ(ψ) is the spectrum of the operator ψ. Recall that σ(ψ) is the set of all λ ∈ C such that λI − ψ is not an invertible operator where I is the identity operator (cf. Dunford and Schwartz (1963), p. 902).
The following is an extension of Theorem 4.3 in Van Zanten (2007) to the setting of random fields.
Remark 3.1. Notice that based on the proof of Theorem 3.3, we can change the first condition in Theorem 3.2 to As Lemma 2.3 emphasizes that the behavior of the spectral measure at origin does not affect the structure of the space L ΠT (F ), one might expect the same formation in terms of the equivalence of Gaussian measures. The following theorem shows that changing the spectral measure on bounded subsets of R d will not affect the equivalence of the corresponding GRFs. In other words, for checking the equivalence of GRFs, only the behavior of their spectral measures at infinity is important.
Theorem 3.4. Suppose two GRFs with stationary increments have spectral measures F 0 and F 1 such that F 0 satisfies the condition (C1), and F 0 = F 1 on I c , where I is a bounded subset of R d . Then, these two GRFs are locally equivalent.
Proof. Define F 1 (dλ) = 1 I c (λ)F 0 (dλ). First, we show that F 0 and F 1 will produce locally equivalent GRFs with stationary increments. For that, fix T > 0. We will investigate the equivalence of measures on Π T . The function ψ appearing in Theorem 3.3 in this case is given by Notice that by the reproducing kernel property, The specific representation of the function ψ(ω, λ) in (3.7) helps us to show that ψ(ω, λ) ∈ L ΠT (F 0 ) ⊗ L ΠT (F 0 ). The idea of the proof is to first show that ψ(ω, λ) ∈ L 2 (F 0 ⊗ F 0 ) and then, use the projection technique to further derive that ψ(ω, λ) ∈ L ΠT (F 0 ) ⊗ L ΠT (F 0 ). Note that (3.8) The second inequality in (3.8) is based on Proposition 2.1 and the fact that I is bounded. Now, we prove that the projection of ψ(ω, λ) to the space L ΠT (F 0 )⊗ L ΠT (F 0 ) is in fact itself. This verifies that ψ(ω, λ) ∈ L ΠT (F 0 ) ⊗ L ΠT (F 0 ). To this end, observe that It remains to show that 1 / ∈ σ(ψ). For that purpose, take an arbitrary φ ∈ L ΠT (F 0 ), and observe that (We use the fact that K 0 T (ω, λ) = K 0 T (λ, ω), see Aronszajn (1950), p. 344.) Therefore, if ψφ = φ, it implies in particular that φ F0 ≤ φ1 I F0 . This means φ = 0 almost everywhere with respect to F 0 in I c . Hence, since φ is an entire function, this implies that φ = 0. Thus, 1 cannot be in the spectrum of ψ. So far, we proved GRFs with spectral measures F 0 and F 1 are locally equivalent, but since F 0 = F 1 on I, similarly, we can say that F 1 and F 1 produce locally equivalent GRFs. Putting these two together, we get the desired result.
Theorems 3.2 and 3.3 give necessary and sufficient conditions for equivalence of GRFs with stationary increments, but it might be difficult to verify the conditions in these theorems. In the literature, there are sufficient conditions for equivalence of certain GRFs in terms of their spectral densities. These conditions are easily verifiable once the two spectral densities are known. For example, we refer to Ibragimov and Rozanov (1978), Theorem 17, p. 104, Yadrenko (1973), Theorem 4, andYadrenko (1983), theorem 4, p. 156 for stationary GRFs; and to Van Zanten (2007, 2008, and Stein (2004) for some nonstationary cases. The following is the main theorem of this paper which gives an explicit sufficient condition in terms of the spectral measures for the equivalence of GRFs with stationary increments.
Theorem 3.5. Suppose that the spectral measure F 0 and F 1 have positive densities f 0 and f 1 with respect to the Lebesgue measure, and F 0 satisfies the condition (C1). If there exists a finite constant C > 0 such that φ F0 ≤ C φ F1 for all φ ∈ L e ΠT , and for some k > 0, then GRFs with stationary increments and spectral measures F 0 and F 1 are equivalent on Π T .
Proof. Applying Theorem 3.4, we can change the value of f 1 on any bounded set, without having any consequences on the equivalence. So, we assume here L ΠT (F 0 ) at infinity also plays an important role. Since finding explicit forms of reproducing kernels are difficult, we need at least to find upper bounds for the growth rate of the diagonal terms. The following condition on spectral density helps us to accomplish this task: (C2) For spectral density f , there exist an entire function φ 0 on C d of finite exponential type such that f (λ) ≍ |φ 0 (λ)| 2 as |λ| → ∞ on R d .
The following lemma shows that under (C2) we have an upper bound for the behavior of the reproducing kernels on the diagonal at infinity.
Lemma 3.1. Suppose f 0 is a spectral density such that it satisfies (C1) and (C2) for some entire function φ 0 . Then, for T > 0, there exists a finite constant for all ω, λ ∈ R d with |λ| large enough. In particular, for all λ ∈ R d with |λ| large enough.
Proof. The idea of the proof is similar to the one in Lemma 3 in Van Zanten (2008). Put f (λ) = |φ 0 (λ)| 2 . Since f and f 0 are comparable at ∞, and f is bounded around 0, it is clear that f is satisfying both conditions (2.2) and (C1). This means we can define L ΠT (f ) the same way, and this space is also a RKHS. Consider an arbitrary orthonormal basis for this space, and denote them by ψ k , k = 1, 2, ... . Now, by Lemma 2.2, they are entire functions on C d with finite exponential type which doesn't depend on k. Further, we know Therefore, we can apply the Paley-Wiener Theorem (Ronkin (1974), Theorem 3.4.2, p. 171) to get ψ k φ 0 = g k for certain functions g k ∈ L 2 (B) where B is a bounded subset of R d (Here h stands for the Fourier transform of h). By Parseval's identity, we can deduce that g k 's are orthonormal in L 2 (B). It follows from Bessel's inequality that where m(B) is the Lebesgue measure of B. Therefore, for λ ∈ R d with |λ| large enough. Now, consider the reproducing kernels of L ΠT (f 0 ), denoting them by K 0 T (ω, .). Since f (λ) ≍ f 0 (λ) as |λ| → ∞, by Lemma 2.3, K 0 T (ω, .) belong to L ΠT (f ) as well for ll ω ∈ R d . Thus, we can expand it using the basis ψ k , and get K 0 , ψ k f ψ k (λ), and then by Cauchy-Schwarz and Lemma 2.3, we get which makes the proof complete.
Theorem 3.5 in combination with Lemma 3.1 leads to an appealing result. If the relative difference between two spectral densities is square integrable at infinity, then the corresponding GRFs with stationary increments will be locally equivalent. We finish this section by proving this fact.
Theorem 3.6. Suppose that the spectral measures F 0 and F 1 have positive densities f 0 and f 1 with respect to the Lebesgue measure, with f 0 satisfying (C1) and (C2) for some entire function φ 0 on C d of finite exponential type. If there exists a finite constant k > 0 such that (3.11) then GRFs with stationary increments having spectral measures F 0 and F 1 are locally equivalent.
Proof. Thanks to Lemma 3.1 and Theorem 3.5, it is sufficient to prove that 1 / ∈ σ(ψ). In spirit of Theorem 3.4, we can assume that f 0 = f 1 on |λ| ≤ k. Now, take an arbitrary element φ ∈ L D (f 0 ), and observe that by using the multidimensional Paley-Wiener Theorem (Ronkin (1974), Theorem 3.4.2. p. 171), we derive that φφ 0 is the inverse Fourier transform of a squared integrable function g with bounded support, B in R d . This implies that for all λ ∈ R d . This means φφ 0 is bounded on R d . This fact together with (3.11) imply that φ f0−f1 f0 ∈ L 2 (f 0 ). Now, observe that . Letting k → ∞, by the Dominated Convergence Theorem, we get φ F0 = 0. This implies that φ = 0. Thus, 1 cannot be in the spectrum of ψ and this concludes proof.

Application
In this section, we apply the results in Section 3 to some anisotropic GRFs with stationary increments. In particular, we consider GRFs with stationary increments and spectral density of the form where λ = (λ 1 , ..., λ d ) ∈ R d \{0}, β j > 0 for all j = 1, ..., d, and γ > d j=1 1 βj . The latter condition guaranties the integrability condition in (2.2) for spectral measures (See Proposition 2.1 in Xue and Xiao (2011)). Fractal and smoothness properties of this family of GRFs are discussed in Xue and Xiao (2011). Now we apply Theorem 3.6 to determine the equivalence of Gaussian measures induced by these GRFs.
To this end, first notice that (C1) is obviously satisfied for spectral densities of the form (4.1). The next lemma shows that these spectral densities also satisfy (C2). Proof. First of all, it is obvious that as |λ| → ∞. Therefore, it suffices to prove the lemma for functions of the form on the right hand side. Now, similar to the construction made in the proof of Lemma 2.3 in Luan and Xiao (2012), we can find a function φ ∈ L 2 (B) for some bounded subset B ⊆ R d such that as |λ| → ∞, which φ is the Fourier transform of φ. By the Paley-Wiener Theorem (Ronkin (1974), Theorem 3.4.2. p. 171), φ is actually the restriction on R d of an entire function on C d with finite exponential type. This finishes the proof.
The next theorem proves that, under certain conditions, the mixture of spectral densities of the form (4.1) will be equivalent to the one with the lowest decay rate at infinity. Similar results of this type have been proved by Van Zanten (2007) and Cheridito (2001) for linear combinations of independent fractional Brownian motions.
Theorem 4.1. Suppose X and Y are two independent centered GRFs with stationary increments with spectral densities of the form (4.1) with parameters (β j , γ) and (β ′ j , γ ′ ), respectively. Then, if

2)
then X and X + Y are locally equivalent.
Proof. Using Lemma 4.1 and Theorem 3.6, we only need to show By using the inequality (a + b) p ≤ 2 p (a p + b p ), we can break the integral in (4.3) into d integrals. Thus, it's enough to show for each fixed j = 1, ..., d (4.4) Since |λ| > 1, this implies that |λ k | > 1/ √ d for some k ∈ {1, ..., d}. We distinguish two cases: Case I is when k = j, and Case II is when k = j. In both cases, we use the following fact that, for positive constants β and γ, and nonnegative constant b, there exists a finite positive constant c such that for all a > 0 (4.5) First, let's consider case I. By applying (4.5) d times, we have Next, we consider case II, where k = j. Similar to case I, we use (4.5) iteratively, but we take the integration in different order. Denote the integration in λ i for i = j, k by dλ \j,k , and observe where the second and the third inequality follow since 2γ ′ β ′ j > 2β j γ + 1 and > 1, respectively using the assumption (4.2). This finishes the proof.
Next, we consider a similar situation as in Theorem 4.1, but this time we put discrete spectral measure mixed with the ones of the form (4.1). For that purpose, consider discrete spectral measure of the form where γ n ∈ R d , α n ≥ 0, for n ≥ 1, and ∞ n=1 |γ n | 2 1+|γ n | 2 α n < ∞. If {γ n , n = 1, 2, ...} is a bounded subset of R d , then in view of Theorem 3.4, this spectral measure will not affect the equivalence of Gaussian measures. Therefore, we consider here only the case where |γ n | → ∞ as n → ∞.
Theorem 4.2. Let X and Y be two independent centered GRFs with stationary increments with spectral measures F X and F Y . Suppose F X has density with respect to Lebesgue measure on R d , denoted by f , which satisfies both conditions (C1) and (C2), and F Y is a discrete measure of the form (4.6). Then, if n>N α n f (γ n ) < ∞, (4.7) for some N ≥ 1, then X and X + Y are locally equivalent.
Proof. First of all, using Theorem 3.4, we can assume α n = 0 for n = 1, ..., N , without having any consequences on the equivalence of Gaussian measures. Second, observe that for all φ ∈ L e ΠT , φ FX ≤ φ FX +FY , which by Remark 3.1, is equivalent to condition (i) in Theorem 3.2.
All we need to prove is then to show that the function ψ in Theorem 3.2 is in L ΠT (f ) ⊗ L ΠT (f ). For that, note that the function ψ can be written as ψ(ω, γ) = n>N α n K T (ω, γ n )K T (λ, γ n ). (4.8) Observe that functions of the form K T (., γ n )K T (., γ n ) belong to the space L e ΠT ⊗ L e ΠT since K T (., γ n )'s are the reproducing kernel elements of L e ΠT . Therefore, if we can show that ψ 2 f ⊗f < +∞, it implies that the function ψ defined in (4.8) is the L 2 (f ⊗ f ) limit of the partial sums and hence, ψ ∈ L ΠT (f ) ⊗ L ΠT (f ) since L ΠT (f ) ⊗ L ΠT (f ) is the closure in L 2 (f ⊗ f ) of the space L e ΠT ⊗ L e ΠT . To this end, observe that by the assumption (4.7). The proof is complete.
The following theorem shows that a similar phenomenon occurs for equivalence of these Gaussian random fields. Proof. The sufficiency is obvious, so we only need to prove the necessity. Suppose for some k ∈ {1, ..., d}, H 0 k < H 1 k . By Lemma 3.2 in Xue and Xiao (2011), there exist c 1 , c 2 > 0, such that for all t ∈ R d for i = 0, 1. If we simply choose t ∈ R d with t k = l, and t j = 0 for j = k, we get This violates the necessary condition for equivalence of Gaussian measures in Theorem 3.1.