标题： 具有可变平滑度的随机过程的逼近 显示英文标题

标题： Approximation of a random process with variable smoothness

摘要： 我们考虑对局部平稳过程$X(t),t\in [0,1]$的分段常数逼近速率，该过程具有可变的光滑指数$\alpha(t)$。假设$\alpha(\cdot)$在零处达到唯一的最小值并满足正则性条件，我们提出了一种构造观测点（复合扩张设计）的方法，并找到了积分均方误差的渐近表达式，其中基于$N(n)\sim n$次观测的分段常数逼近$X_n$用于$X$。进一步地，我们证明了所提出的逼近速率是最优的，然后说明如何找到最优常数。 显示英文摘要

摘要： 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.