Ams 245 Spatial Statistics Homework 2

1. Prove the results about the smoothness of the members of the Matèrn family. Proof. From the class we know that for small values of τ , we have K ν (τ) ≈ Γ(ν)2 ν−1 τ −ν. Also, for the derivatives we have that d dτ (τ ν K ν (τ)) = −τ ν K ν−1 (τ). If the Matèrn family has the correlation function ρ(τ) = 1 Γ(ν)2 ν−1 τ ν K ν (τ), then ρ (τ) = −1 Γ(ν)2 ν−1 τ ν K ν−1 (τ) Note that K ν (τ) = K −ν (τ), and so for 0 < ν < 1, ρ (τ) = −1 Γ(ν)2 ν−1 τ ν K 1−ν (τ), and hence for 0 < ν < 1, lim τ →0 ρ (τ) ≈ lim τ →0 −1 Γ(ν)2 ν−1 τ ν Γ(1 − ν)2 1−ν−1 τ −(1−ν) = −Γ(1 − ν) Γ(ν)2 2ν−1 lim τ →0 τ 2ν−1. If 0 < ν < 1/2, lim τ →0 ρ (τ) = −∞. When ν = 1/2, ρ(τ) is an exponential correlation function , and lim τ →0 ρ (τ) = −Γ(1 − 0.5) Γ(0.5)2 2(0.5)−1 lim τ →0 τ 2(0.5)−1 = −1. When ν = 1, lim τ →0 ρ (τ) = 0. Since Γ(0) = +∞, when ν → 1, lim τ →0 ρ (τ) = −∞. Here we need to justify τ 2ν−1 converges to 0 slower than Γ(1 − ν) approaching to ∞ when 1/2 < ν < 1. So when 1/2 ≤ ν < 1, ρ (0) ∈ (−∞, 0) (WHY?). The second derivative is ρ (τ) = −1 Γ(ν)2 ν−1 τ ν−1 K ν−1 (τ) + τ (−τ ν−1 K ν−2 (τ)) = −1 Γ(ν)2 ν−1 τ ν−1 K ν−1 (τ) − τ ν K ν−2 (τ). If ν = 1, Using the fact that K 0 (τ) ≈ ln(τ) for small τ , lim τ →0 ρ (τ) = − lim τ →0 [ln τ ] = −∞. (2d−1) (0) = 0 and ρ (2d) (0) ∈ (−∞, 0) (WHY?).