Approximation of a Random Process with Variable Smoothness

We consider the rate of piecewise constant approximation to a locally stationary process $$X(t),t\in [0,1]$$ X ( t ) , t ∈ [ 0 , 1 ] , having a variable smoothness index $$\alpha (t)$$ α ( t ) . Assuming that $$\alpha (\cdot )$$ α ( · ) attains its unique minimum at zero and satisfies $$ \alpha (t)=\alpha _0+b t^\gamma + \mathrm{o}(t^\gamma ) \quad \text{ as } t \rightarrow 0, $$ α ( t ) = α 0 + b t γ + o ( t γ ) as t → 0 , we propose a method for construction of observation points (composite dilated design) such that the integrated mean square error $$ \int _0^1 {\mathbb E}\{(X(t)-X_n(t))^2\} dt \sim \frac{K}{n^{\alpha _0}(\log n)^{(\alpha _0+1)/\gamma }} \qquad \text{ as } n\rightarrow \infty , $$ ∫ 0 1 E { ( X ( t ) - X n ( t ) ) 2 } d t ∼ K n α 0 ( log n ) ( α 0 + 1 ) / γ as n → ∞ , where a piecewise constant approximation $$X_n$$ X n is based on $$N(n)\sim n$$ N ( n ) ∼ n observations of $$X$$ X . Further, we prove that the suggested approximation rate is optimal, and then show how to find an optimal constant $$K$$ K .