Renewal type bootstrap for increasing degree U-process of a Markov chain

Abstract

In this paper, we investigate the uniform limit theory for a U-statistic of increasing degree, also called an infinite-degree U-statistic. Infinite-degree U-statistics (IDUS) (or infinite-order U-statistics (IOUS)) are useful tool for constructing simultaneous prediction intervals that quantify the uncertainty of several methods such as subbagging and random forests. The stochastic process based on collections of U-statistics is referred to as a U-process, and if the U-statistic is of infinite-degree, we have an infinite-degree U-process. Heilig and Nolan (2001) provided conditions for the pointwise asymptotic theory for the infinite-degree U-processes. The main purpose here is to extend their findings to the Markovian setting. The second aim is to provide the uniform limit theory for the renewal bootstrap for the infinite-degree U-process, which is of its own interest. The main ingredients are the decoupling technique combined with symmetrization techniques of Heilig and Nolan (2001) to obtain uniform weak law of large numbers and functional central limit theorem for the infinite-degree U-process. The results obtained in this paper are, to our knowledge, the first known results on the infinite-degree U-process in the Markovian setting.