## Bernoulli

• Bernoulli
• Volume 20, Number 1 (2014), 207-230.

### Uniform convergence rates for a class of martingales with application in non-linear cointegrating regression

#### 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 chains 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.

#### Article information

Source
Bernoulli, Volume 20, Number 1 (2014), 207-230.

Dates
First available in Project Euclid: 22 January 2014

https://projecteuclid.org/euclid.bj/1390407286

Digital Object Identifier
doi:10.3150/12-BEJ482

Mathematical Reviews number (MathSciNet)
MR3160579

Zentralblatt MATH identifier
06282548

#### Citation

Wang, Qiying; Chan, Nigel. Uniform convergence rates for a class of martingales with application in non-linear cointegrating regression. Bernoulli 20 (2014), no. 1, 207--230. doi:10.3150/12-BEJ482. https://projecteuclid.org/euclid.bj/1390407286

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