Fast and Exact Simulation of Large Gaussian Lattice Systems in ...

Table 4: Criteria of the Pólya Type for Isotropic Covariance Functions in 2 . A continuous functionϕ on [0, ∞) belongs to the class Φ 2 if ϕ(0) is positive, lim t→∞ ϕ(t) = 0, and if the convexity or thenonnegativity condition in the table holds.Convexity Condition Requires ReferenceNonnegativity Conditionϕ(t 2 ) convex α ≤2 1 Gneiting (1999a)ϕ ′ (t) + 2tϕ ′′ (t) ≥ 0−ϕ ′ (t 1/2 ) convex α ≤ 1 Gneiting (1999b)ϕ ′′ (t) − tϕ ′′′ (t) ≥ 0( )t −2 2ϕ ′ (t 1/2 ) − 2t 1/2 ϕ ′′ (t 1/2 ) + tϕ ′′′ (t 1/2 ) convex α ≤ 2 Gneiting (2001)48 (ϕ ′′ (t) − ϕ ′ (t)) − 24t 2 ϕ ′′′ (t) + 7tϕ (iv) (t) − t 4 ϕ (v) (t) ≥ 0respectively. We experimented with various other forms of ψ, but the results were generally oflimited use. For instance, analogues of Theorem 1 and Theorem 2 apply when ψ(t) = b(r c − t c )where 0 < c ≤ 1 2 , and ψ(t) = b(r − t)c where c ≥ 2, respectively, with b and r chosen such that ρ issmooth. However, the cut-off embedding approach calls for small values of the cut-off, r, and thelatter is mimimal when c = 1 2and c = 2, respectively.Theorem 1 Let ϕ be a continuous function on [0, 1] such that ϕ(t 2 ) is positive and convex andϕ ′ (1) is negative. Then the function ρ on [0, ∞) defined by⎧⎪⎨ϕ(t), 0 ≤ t ≤ 1,ρ(t) = b(r⎪⎩1/2 − t 1/2 ), 1 ≤ t ≤ r,(8)0, t ≥ r,where(r = 1 − 1 )ϕ(1) 22 ϕ ′ , b = −2ϕ ′ (1), (9)(1)belongs to the class Φ 2 .Note that r and b are chosen such that ρ is continuous and differentiable at t = 1. We skip theproof of Theorem 1, which consists in a straightforward reduction to the first criterion in Table 4.For covariances of powered exponential type, ϕ(t) = exp(−(θt) α ), and for covariances of Cauchytype, ϕ(t) = (1 + (θt) α ) −β/α , the conditions of the theorem are satisfied if and only if α ≤ 1 2 . If ϕis powered exponential with α =2 1 and θ = 1, we recover eq. (3). If ϕ belongs to the Matérn class,that is, ifϕ(t) = 21−νΓ(ν) (θt)ν K ν (θt),where K ν is a modified Bessel function of index ν > 0, we getϕ ′ (t) = − 21−νΓ(ν) θν+1 t ν K ν−1 (θt).8