Strongly consistent nonparametric forecasting and regression for stationary ergodic sequences

TL;DR

This paper introduces the first strongly consistent nonparametric algorithm for estimating the regression function in stationary ergodic sequences, applicable to forecasting and inference without mixing assumptions.

Contribution

It presents a novel strongly consistent nonparametric method for regression and forecasting in stationary ergodic sequences, extending existing approaches beyond mixing conditions.

Findings

Achieves strong consistency in pointwise, least-squares, and uniform distances.

Applicable to nonparametric, nonlinear forecasting in ergodic time series.

Extends forecasting literature by removing mixing assumptions.

Abstract

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.