Strongly consistent nonparametric forecasting and regression for stationary ergodic sequences

Abstract

Let $\{(X_i,Y_i)\}$ be a stationary ergodic time series with $(X,Y)$ values in the product space ${\mathbb{R}}^d⨂\mathbb{R}.$ 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.