Spline approximation of a random process with singularity

Abstract

Let a continuous random process X defined on [0, 1] be (m + β)-smooth, 0 ≤ m, 0 < β ≤ 1, in quadratic mean for all t > 0 and have an isolated singularity point at t = 0. In addition, let X be locally like a m-fold integrated β-fractional Brownian motion for all nonsingular points. We consider approximation of X by piecewise Hermite interpolation splines with n free knots (i.e., a sampling design, a mesh). The approximation performance is measured by mean errors (e.g., integrated or maximal quadratic mean errors). We construct a sequence of sampling designs with asymptotic approximation rate n −(m+β) for the whole interval.

On Piterbarg Max-Discretisation Theorem for Standardised Maximum of Stationary Gaussian Processes

“…If further the Berman condition lim T →∞ r(T ) ln T = 0 (2) holds, then Pickands theorem (see Pickands (1969) and Piterbarg (1972) for a rigorous proof of Pickands theorem) establishes the key limit result for the continuous time maximum M T = max{X t , ∀t ∈ [0, T ]}, namely…”

Mathematical Statistics and Limit Theorems

“…Instead, one could use quasi-regular sampling designs generated by a possibly unbounded design density p(t), t ∈ (0, 1], at the singularity point t 0 = 0 (cf., Abramowicz and Seleznjev 2011). For example, if p(·) is a probability density on (0, 1] such that…”

“…The approximation performance on the entire interval is measured by the integrated mean square error (IMSE) 1 0 E{(X (t) − X n (t)) 2 }dt. We construct a sequence of sampling designs (i.e., sets of observation points) taking into account the varying smoothness of X such that on a class of processes, the IMSE decreases faster when compared to conventional regular sampling designs (see, e.g., Seleznjev 2000) or to quasi-regular designs, Abramowicz and Seleznjev (2011), used for approximation of locally stationary random processes and random processes with an isolated singularity point, respectively.…”

Approximation of a Random Process with Variable Smoothness

“…The approximation performance on the entire interval is measured by integrated mean square error (IMSE) 1 0 E{(X(t) − X n (t)) 2 }dt. We construct a sequence of sampling designs (i.e., sets of observation points) taking into account the varying smoothness of X such that on a class of processes, the IMSE decreases faster when compared to conventional regular sampling designs (see, e.g., [26]) or to quasi-regular designs, [2], used for approximation of locally stationary random processes and random processes with an isolated singularity point, respectively.…”