Nonparametric change-point estimation for data from an ergodic sequence

In the framework of the series scheme we assume that an observations sequence $\{X_i^n,1\le i\le n\} $ is such that $X_i^n=U_i I(1\le i\le[\theta n])+V_i I([\theta n]+1\le i\le n)$, where $(U_i,V_i)$ is a stationary ergodic sequence the marginal distributions of which are different, and $\theta $ is a change-point in the probabilistic characteristics such that $\theta\in(0;1)$. The main result of this paper is the proof of the fact that the sequence $(\theta n)_{n\ge1} $ of nonparametric estimations constructed here is consistent $(\theta n\to\theta)$.