On Quadratic Variation of Processes with Gaussian Increments

Ambit Processes; with Applications to Turbulence and Tumour Growth

“…(In the present context, key references to that literature are Cairoli and Walsh (1975), Khoshnevisan (2002), Klein and Giné (1975), Walsh (1986a,b) and Wong and Zakai (1974 For the particular specification of X considered in Section 2.4, we have argued as if these questions had been positively resolved. In fact, under mild assumptions on g, h, I and J the manipulations in that Section can be verified by direct calculations (details to be given elsewhere).…”

Sample functions of Pólya processes

“…The proof is quite same as in [6] which contributes to find a sequence {ε n [ 0} satisfying being based on the bound of Hanson and Wright (1971). Hence, only some different points from the proof in [6] will be noted here.…”

“…Hence, only some different points from the proof in [6] will be noted here. In the proof, it necessitates to evaluate E [(X(tl n) …”

Singularity functions for fractional processes: application to the fractional Brownian sheet

“…In practice we have to compute m −2 Γ. Actually the computation of Γ seems delicate but (12) with Z in place of ξ. And we have an upper bound for N d N .…”

“…In [8] the mesh of the increments that define the quadratic 1 variation is 1/2 n and an almost sure convergence result is obtained when n → ∞. Actually it is known (see [12]) that the almost convergence is true when the mesh is o(1/ log(n)). Nevertheless quadratic variations are not suitable when one is interested in a central limit theorem for fractional Brownian motion.…”