Expansion of fractional Ornstein-Uhlenbeck processes

4 Optimal expansion of a class of stationary Gaussian processes

4.1 Expansion of fractional Ornstein-Uhlenbeck processes

The fractional Ornstein-Uhlenbeck process X ρ = X ρ t t ∈IR of index ρ ∈ 0, 2 is a continuous station- ary centered Gaussian process having the covariance function IEX ρ s X ρ t = e −α|s−t| ρ , α 0. 4.1 We derive explicit optimal expansions of X ρ for ρ ≤ 1. Let γ ρ : IR → IR, γ ρ t = e −α|t| ρ and for a given T 0, set β ρ := 1 2T Z T −T γ ρ td t, β j ρ := 1 T Z T −T γ ρ t cosπ j tT d t, j ≥ 1. 4.2 Theorem 3. Let ρ ∈ 0, 1]. Then β j ρ 0 for every j ≥ 0 and the sequence f = p β ρ, f 2 j = p β j ρ cosπ j tT , f 2 j −1 t = p β j ρ sinπ j tT , j ≥ 1 4.3 is admissible for X ρ in C [0, T ]. Furthermore, l n X ρ ≈ n −ρ2 log n 1 2 as n → ∞ and the sequence 4.3 is rate optimal. Proof. Since γ ρ is of bounded variation and continuous on [-T,T], it follows from the Dirichlet criterion that its classical Fourier series converges pointwise to γ ρ on [-T,T], that is using symmetry of γ ρ , γ ρ t = β ρ + ∞ X j=1 β j ρ cosπ j tT , t ∈ [−T, T ]. Thus one obtains the representation IEX s X t = γ ρ s−t = β ρ+ ∞ X j=1 β j ρ[cosπ jsT cosπ j tT +sinπ jsT sinπ j tT ], s, t ∈ [0, T ]. 4.4 This is true for every ρ ∈ 0, 2. If ρ = 1, then integration by parts yields β 1 = 1 − e −αT αT , β j 1 = 2 αT 1 − e −αT −1 j α 2 T 2 + π 2 j2 , j ≥ 1. 4.5 In particular, we obtain β j 1 0 for every j ≥ 0. If ρ ∈ 0, 1, then γ ρ |[0,∞ is the Laplace transform of a suitable one-sided strictly ρ-stable distribution with Lebesgue-density q ρ . Consequently, for 1215 j ≥ 1, β j ρ = 2 T Z T e −αt ρ cos π j tT d t 4.6 = Z ∞ 2 T Z T e −t x cos π j tT d tq ρ xd x = Z ∞ 2x T 1 − e −x T −1 j x 2 T 2 + π 2 j 2 q ρ xd x. Again, β j ρ 0 for every j ≥ 0. It follows from 4.4 and Corollary 1a that the sequence f j j ≥0 defined in 4.3 is admissible for X ρ in C [0, T ]. Next we investigate the asymptotic behaviour of β j ρ as j → ∞ for ρ ∈ 0, 1. The spectral measure of X ρ still for ρ ∈ 0, 2 is a symmetric ρ-stable distribution with continuous density p ρ so that γ ρ t = Z IR e i t x p ρ xd x = 2 Z ∞ cost xp ρ xd x, t ∈ IR and the spectral density satisfies the high-frequency condition p ρ x ∼ cρx −1+ρ as x → ∞ 4.7 where c ρ = αΓ1 + ρ sinπρ2 π . Since by the Fourier inversion formula p ρ x = 1 π Z ∞ γ ρ t cost xd t, x ∈ IR, we obtain for j ≥ 1, β j ρ = 2 T Z T γ ρ t cosπ j tT d t = 2 T ‚Z ∞ γ ρ t cosπ j tT d t − Z ∞ T γ ρ t cosπ j tT d t Œ = 2 π T p ρ π jT − 2 T Z ∞ T γ ρ t cosπ j tT d t Integrating twice by parts yields Z ∞ T γ ρ t cosπ j tT d t = O j −2 1216 for any ρ ∈ 0, 2 so that for ρ ∈ 0, 1 β j ρ ∼ 2 π T p ρ π jT 4.8 ∼ 2 πT ρ c ρ π j 1+ ρ = 2 αT ρ Γ1 + ρ sinπρ2 π j 1+ ρ as j → ∞. We deduce from 4.5 and 4.8 that the admissible sequence 4.3 satisfies the conditions A2 and A3 from Proposition 4 with parameters ϑ = 1 + ρ2, γ = 0, a = 1 and b = 1 − ρ2. Furthermore, by Theorem 3 in Rosenblatt [16], the asymptotic behaviour of the eigenvalues of the covariance operator of X ρ on L 2 [0, T ], d t see 3.4 for ρ ∈ 0, 2 is as follows: λ j ∼ 2T 1+ ρ πcρ π j 1+ ρ as j → ∞. 4.9 Therefore, the remaining assertions follow from Proposition 4. ƒ Here are some comments on the above theorem. First, note that the admissible sequence 4.3 is not an orthonormal basis for H = H X ρ but only a Parseval frame at least in case ρ = 1. In fact, it is well known that for ρ = 1, ||h|| 2 H = 1 2 h0 2 + hT 2 + 1 2 α Z T h ′ t 2 + α 2 ht 2 d t so that e.g. || f 2 j −1 || 2 H = 1 − e −αT −1 j 2 1. A result corresponding to Theorem 3 for fractional Ornstein-Uhlenbeck sheets on [0, T ] d with co- variance structure IEX s X t = d Y i=1 e −α i |s i −t i | ρi , α i 0, ρ i ∈ 0, 1] 4.10 follows from Corollary 4. As a second comment, we wish to emphasize that, unfortunately, in the nonconvex case ρ ∈ 1, 2 it is not true that β j ρ ≥ 0 for every j ≥ 0 so that the approach of Theorem 3 no longer works. In fact, starting again from β j ρ = 2 π T p ρ π j T − 2 T Z ∞ T γ ρ t cosπ j tT d t, three integrations by parts show that β j ρ = 2 π T p ρ π jT + −1 j+1 2T αρe −αT ρ T ρ−1 π j 2 + O j −3 = −1 j+1 2T αρe −αT ρ T ρ−1 π j 2 + O j −1+ρ , j → ∞. 1217 This means that for any T 0, the 2T -periodic extension of γ ρ |[−T,T ] is not nonnegative definite for ρ ∈ 1, 2 in contrast to the case ρ ∈ 0, 1].

4.2 Expansion of stationary Gaussian processes with convex covariance function