Strongly consistent nonparametric forecasting and regression for stationary ergodic sequences

Let {(Xi, Yi)} be a stationary ergodic time series with (X, Y ) values in the product space R ⊗ 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[Y0|X0 = x] under the presumption that m(x) is uniformly Lipschitz continuous. Autoregression, 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.