Uniform convergence rates for a class of martingales with application in non-linear co-integrating regression

Qiying Wang and Nigel Chan

Abstract

For a class of martingales, this paper provides a framework on the uniform consistency with broad applicability. The main condition imposed is only related to the conditional variance of the martingale, which holds true for stationary mixing time series, stationary iterated random function, Harris recurrent Markov chain and $I(1)$ processes with innovations being a linear process. Using the established results, this paper investigates the uniform convergence of the Nadaraya-Watson estimator in a non-linear cointegrating regression model. Our results not only provide sharp convergence rate, but also the optimal range for the uniform convergence to be held. This paper also considers the uniform upper and lower bound estimates for a functional of Harris recurrent Markov chain, which are of independent interests.

Keywords: Harris recurrent Markov chain, Martingale, non-linearity, non-parametric regression, non-stationarity, uniform convergence.

AMS Subject Classification: Primary 62G20; secondary 62G08, 60F99.