By Serge Cohen; et al
Fractional Levy Fields.- the speculation of Scale features for Spectrally detrimental Levy approaches
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Additional info for Lévy matters II : recent progress in theory and applications: fractional Lévy fields, and scale functions
0 Let us illustrate this construction with a simple example: d = 1 and μ(du) = 1 (δ 2 −1 (du)+δ1 (du)), where δ’s are Dirac masses. In this case M (ds) is a compound random Poisson measure and can be written as an infinite sum of random Dirac masses M (ds) = δSn (ds)εn , n∈Z where Sn+1 − Sn are identically independent random variables with an exponential law, and εn are identically distributed independent Bernoulli random variables such that P(εn = 1) = P(εn = −1) = 1/2. The εn ’s are independent of the Sn ’s.
Cohen The change of variables λ = e −i p−p n ·ξ k K −i n ·ξ k=0 ak e ξ leads to n 2 ||ξ||d+2H Rd dξ = n −2H e K k=0 −i(p−p )·λ ak eik·λ ||λ||d+2H Rd 2 dλ. (99) d Define the operator D = ∂ . Let us suppose that ∀j, ∂x i j=1 pj = pj , integrating by parts leads to e−i(p−p )·λ Rd d = id j=1 1 (pj − pj ) Rd K ||λ||d+2H ⎡ ⎢ e−i(p−p )·λ D ⎣ 2 dλ K ik·λ k=0 ak e ||λ||d+2H 2 ⎤ ⎥ ⎦ dλ. (100) K a = 0, The conditions K ik·λ k=0 ak e a = 0 ensure the convergence of the integral. =0 =0 Since there exists a constant C1 such that, as n → +∞, ⎛ ⎞ n−K 1 ⎝1 ⎠ → C1 , n (m − m )2 m,m =0,m=m nd Gn → C2 .
This shows that at large Moreover the limit field is a rhfsf with parameter H. scales the behavior of rhfLf can be very far from the Gaussian model even if the rhfLfs are fields that have moments of order 2. The rhfLf with control measure dρ |ρ|1+α 1(|ρ| < 1) can be viewed roughly speaking as in between a rhfsf at large scales and a fractional Brownian field at low scales. Let us now state precisely the asymptotic self-similarity. 14. Let us assume that H α 2 ˜ 0 < H < 1. The real harmonizable fractional L´evy field, with control measure μρ (dρ), dρ 1(|ρ| < 1), |ρ|1+α ˜ is asymptotically self-similar at infinity with parameter H lim R→+∞ XH (Ru) RH˜ (d) u∈Rd = (YH˜ (u))u∈Rd , (79) where the limit is in distribution for all finite dimensional margins of the fields, and the limit is a real harmonizable fractional stable field that has a representation: YH˜ (u) = e−iu·ξ − 1 Rd ξ d ˜ α +H Mα (dξ), (80) where Mα (dξ) is complex isotropic α-stable random measure defined in (28).
Lévy matters II : recent progress in theory and applications: fractional Lévy fields, and scale functions by Serge Cohen; et al