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 Rd ⊗ 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. 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.