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 x R. This study offers what is believed to be the fir st strongly consistent (with respect to pointwise, least-squares, and unifo rm distance) algorithm for inferring m(x) = E[Y-0\X-0 = x] under the presum ption that m(x) is uniformly Lipschitz continuous. Auto-regression, or fore casting, is an important special case, and as such our work extends the lit erature of nonparametric. nonlinear forecasting by circumventing customary mixing assumptions. The work is motivated by a time series model in stochas tic finance and by perspectives of its contribution to the issues of univer sal time series estimation. (C) 1999 Academic Press.