Strongly consistent nonparametric forecasting and regression for stationary ergodic sequences

Let ${(X_i,Y_i)}$ be a stationary ergodic time series with $(X,Y)$ values in the product space $\R^d\bigotimes \R .$ This study offers what is believed to be the first strongly consistent (with respect to pointwise, least-squares, and uniform distance) algorithm for inferring $m(x)=E[Y_0|X_0=x]$ under the presumption that $m(x)$ is uniformly Lipschitz continuous. Auto-regression, or forecasting, is an important special case, and as such our work extends the literature of nonparametric, nonlinear forecasting by circumventing customary mixing assumptions. The work is motivated by a time series model in stochastic finance and by perspectives of its contribution to the issues of universal time series estimation.

💡 Research Summary

The paper addresses the fundamental problem of non‑parametric regression and one‑step ahead forecasting for a stationary ergodic time series ({(X_i,Y_i)}_{i\in\mathbb Z}) taking values in (\mathbb R^d\times\mathbb R). The target of inference is the conditional mean function
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