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A REMARK ON THE EQUIVALENCE OF GAUSSIAN PROCESSES

In this note we extend a classical equivalence result for Gaussian stationary processes to the more general setting of Gaussian processes with stationary increments. This will allow us to apply it in the setting of aggregated independent fractional Brownian motions.


Introduction and main result
It is well known that every mean-square continuous, centered, stationary Gaussian process X = (X t ) t≥0 admits a spectral representation. Indeed, by Bochner's theorem there exists a symmetric, finite Borel measure µ on the line such that EX s X t = R e i(t−s)λ µ(dλ).
The measure µ is called the spectral measure. If it admits a Lebesgue density, this is called the spectral density of the process. A classical result in the theory of continuous-time stationary Gaussian processes gives sufficient conditions for the equivalence of the laws of two centered processes with different spectral densities, see for instance [7], or [8], Theorem 17 on p. 104. The result says that if the two densities f , g involved satisfy for some R > 0 then, under a regularity condition on the tail behaviour of the densities, the laws of the associated processes on (R Unfortunately, the proof of this result, as given for instance on pp. 105-107 of [8], does not allow extension to the setting of processes with stationary increments. Processes of the latter type admit a spectral representation as well. If X = (X t ) t≥0 is a mean-square continuous, centered Gaussian process with stationary increments that starts from 0, i.e. X 0 = 0 (we call such processes Gaussian si-processes from now on), there exists a unique symmetric Borel measure µ on the line such that (1 + λ 2 ) −1 µ(dλ) < ∞, and for all s, t ≥ 0 (cf., e.g., [3]). Slightly abusing terminology we also call µ the spectral measure of the process X in this case and if it admits a Lebesgue density we call it the spectral density again. The main example is the fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1), which has spectral density (see for instance [9]). It turns out that if we just do as if equivalence result cited above is valid for si-processes, we obtain equivalence statements that are actually correct and can be proved rigorously. Consider for instance the so-called mixed fBm as introduced in [2], which is the sum W +X of a standard Brownian motion W and an independent fBm X with some Hurst index H ∈ (0, 1). The process W has spectral density f identically equal to 1/(2π) and hence the mixed fBm has spectral density g(λ) = 1/(2π) + c H |λ| 1−2H . We see that condition (1.1) becomes in this case which is fulfilled if and only if H > 3/4. This would suggest that the mixed fBm is equivalent to ordinary Brownian motion if H > 3/4. And indeed, this is exactly what [2] proved, cf. also [1]. The main purpose of this note is to show that this example is not a coincidence, and that the classical equivalence result for stationary processes indeed extends to si-processes. We call two processes equivalent on [0, T ] if the laws they induce on ( Recall that an entire function ϕ on the complex plane is said to be of exponential type τ if lim sup It is said to be of finite exponential type if it is of exponential type τ for some τ < ∞. We denote by L e T the linear span of the collection of functions Theorem 1. Let X and Y be centered, mean-square continuous Gaussian processes with stationary increments and spectral densities f and g, respectively. Suppose there exist positive constants c 1 , c 2 and an entire function ϕ of finite exponential type such that for all real λ large enough. For T > 0, suppose there exists a constant C > 0 such that Then if condition (1.1) holds for some R > 0, the processes X and Y are equivalent on [0, T ].
As explained on p. 104 of [8], condition (1.3) is for instance fulfilled if for some p ∈ (−∞, 1) it holds that c 1 |λ| p ≤ f (λ) ≤ c 2 |λ| p for |λ| large. In the stationary process result of [8] it is assumed that g satisfies condition (1.3) as well (with the same ϕ). The condition on g in Theorem 1 gives somewhat more flexibility in special cases, since it is for instance satisfied as soon as g ≥ Cf for some constant C > 0.
In the next section we present the proof of Theorem 1. Then in Section 3 the result is used to extend an equivalence result for aggregated fBm's of [10].

Proof
The proof of the theorem exploits the fact that for Gaussian si-processes, we have a reproducing kernel Hilbert space (RKHS) structure in the frequency domain. For T > 0 and a spectral measure µ, let L T (µ) be the closure in L 2 (µ) of the set of functions L e T , which is defined as the linear span of the collection {λ → (exp(iλt) − 1)/(iλ) : t ∈ [0, T ]}. Then L T (µ) is a RKHS of entire functions (see for instance [4], or [6]). We denote its reproducing kernel by S T . This function has the property that S T (ω, ·) ∈ L T (µ) for every ω ∈ R and for every ψ ∈ L T (µ) and ω ∈ R, where ϕ, ψ L 2 (µ) = ϕψ dµ. Below we will use the fact that every ψ ∈ L T (µ) has a version that can be extended to an entire function on the complex plane, that is of finite exponential type (at most T ). Conversely, the restriction to the real line of an entire function ψ of exponential type at most T that satisfies R |ψ(λ)| 2 µ(dλ) < ∞, belongs to L T (µ) (cf. [4], [6]). We shall apply the following theorem obtained in [10]. It gives sufficient conditions for equivalence of Gaussian si-processes involving spectral densities and reproducing kernels.
Theorem 2. Let X and Y be centered, mean-square continuous Gaussian processes with stationary increments and spectral densities f and g, respectively. Fix T > 0 and suppose there exists a constant C > 0 such that ψ L 2 (f ) ≤ C ψ L 2 (g) for all ψ ∈ L e T . Let S T be the reproducing kernel of L T (f ). Then if for all real ω and all real λ large enough, where C is a positive constant independent of ω and λ. In particular, for |λ| large enough.
Proof. Put f 0 = |ϕ| 2 . Then since ϕ is entire, f 0 is bounded near 0 and hence, by the first inequality in (1.3), f 0 is the spectral density of a Gaussian si-process. Let ψ k be an arbitrary orthonormal basis of L T (f 0 ). For every k the function ψ k ϕ is an entire function of finite exponential type (not depending on k), say S. Moreover, we have Hence, by the Paley-Wiener theorem, ψ k ϕ =f k for certain f k ∈ L 2 [−S, S], whereĥ denotes the Fourier transform of the function h. By the Parseval relation for the Fourier transform, the fact that the ψ k are an orthonormal basis of L T (f 0 ) implies that the f k are orthonormal in L 2 [−S, S]. By Bessel's inequality, it follows that hence |ψ k (λ)| 2 ≤ S/(πf 0 (λ)). Now fix ω ∈ R and consider S T (ω, ·). This function is entire, of exponential type at most T and belongs to L T (f ) and hence, by the first inequality in (1.3), belongs to L T (f 0 ) as well (cf. [4], Chapter 6). Expanding it in the basis ψ k of the first paragraph gives S T (ω, λ) = S T (ω, ·), ψ k L 2 (f0) ψ k (λ).
By Cauchy-Schwarz, we obtain By the first paragraph, the second factor on the right is bounded by a constant times 1/f 0 (λ), which, by the second inequality of (1.3), is bounded by a constant times 1/f (λ) for |λ| large enough. The first factor equals S T (ω, ·) 2 L 2 (f0) . To bound this last quantity, observe that since f 0 is bounded near 0, we have for every a > 0 and ψ ∈ L e T , for some c > 0. On the other hand, the Gaussian si-processes with spectral measures f and corresponding to the two densities are equivalent on L e T (e.g. Theorem 4.1 of [10]). It follows that the constant c ′ not depending on ψ. Condition (1.3) implies that for a large enough we have Together we find that for some constant c > 0, it holds that ψ L 2 (f0) ≤ c ψ L 2 (f ) for all ψ ∈ L e T . By passing to the limit we see that the bound holds in fact for all ψ ∈ L T (f ). Applying this with ψ = S T (ω, ·) and using the reproducing property yields , completing the proof of the lemma.

Application
One of the main motivations for the present paper is the equivalence result for aggregated fBm's given in [10]. Consider a linear combination X = a k X k of independent fBm's X 1 , . . . , X n with increasing Hurst indices H 1 < · · · < H n , for some nonzero constants a 1 , . . . , a n . It is proved in [10] that X is equivalent to a 1 X 1 on every interval [0, T ] if H 2 − H 1 > 1/4. Morally speaking, such an equivalence result should be true under conditions that only restrict the tails of the spectral densities of the processes involved. The proof of the result presented in [10] however relies on the explicit form of the frequency domain reproducing kernel of the fBm (cf. [5]). Using Theorem 1 we can now immediately obtain the following generalization, which shows that indeed, only conditions on the tails of the spectra are needed.
Observe that we recover the cited result of [10] if we apply the theorem with (using the same notations as above) where f H is the spectral density of the fBm given by (1.2).