Approximation of a random process with variable smoothness

TL;DR

This paper investigates the optimal rate of piecewise constant approximation for a locally stationary process with variable smoothness, proposing a new observation design and establishing asymptotic error bounds.

Contribution

It introduces a composite dilated design for observation points and proves the optimality of the approximation rate for processes with variable smoothness.

Findings

Proposed a new observation point construction method.

Derived asymptotics for the integrated mean square error.

Proved the optimality of the approximation rate.

Abstract

We consider the rate of piecewise constant approximation to a locally stationary process $X(t),t\in [0,1]$, having a variable smoothness index $\alpha(t)$. Assuming that $\alpha(\cdot)$ attains its unique minimum at zero and satisfies the regularity condition, we propose a method for construction of observation points (composite dilated design) and find an asymptotics for the integrated mean square error, where a piecewise constant approximation $X_n$ is based on $N(n)\sim n$ observations of $X$. Further, we prove that the suggested approximation rate is optimal, and then show how to find an optimal constant.