Estimating Trajectories

Abstract

Let $f$ be a continuously differentiable function from [0, 1] to the complex plane. Suppose that "at time $n$" we are given the random set $\{f(k/n) + e_{n, k}: 1 \leq k \leq n\}$, where the random errors $e_{n, k}$ are i.i.d. and $(\mathrm{Re} e_{n, 1}, \mathrm{Im} e_{n, 1})$ is $N \big( (0, 0), \sigma^2 \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} \big)$ with $\sigma^2$ known. We do not know which datum belongs to which position $\theta = k/n, k = 1, 2, \cdots, n$. In general, $f$ cannot be determined. In this paper it is shown that a random set $T_n$ can be constructed such that with probability one, $T_n$ converges in the Hausdorff sense to the trajectory $f(\lbrack 0, 1 \rbrack)$.