Abstract
The main objective of this paper is to establish bootstrap uniform functional central limit theorem for Harris recurrent Markov chains over uniformly bounded classes of functions. We show that the result can be generalized also to the unbounded case. To avoid some complicated mixing conditions, we make use of the well-known regeneration properties of Markov chains. We show that in the atomic case the proof of the bootstrap uniform central limit theorem for Markov chains for functions dominated by a function in $L^{2}$ space proposed by Radulović (2004) can be significantly simplified. Finally, we prove bootstrap uniform central limit theorems for Fréchet differentiable functionals in a Markovian setting.
Highlights
The naive bootstrap for indentically distributed and independent random variables introduced by Efron (1979) has gradually evolved and new types of bootstrap schemes in both i.i.d. and dependent setting were established
We use the bootstrap asymptotic results for empirical processes indexed by classes of functions to derive bootstrap uniform central limit theorems for Frechet differentiable functionals in a Markovian case
Bootstrap uniform central limit theorems for Frechet differentiable functionals
Summary
The naive bootstrap for indentically distributed and independent random variables introduced by Efron (1979) has gradually evolved and new types of bootstrap schemes in both i.i.d. and dependent setting were established. A detailed review of various bootstrap methods such as moving block bootstrap (MBB), nonoverlapping block bootstrap (NBB) or cilcular block bootstrap (SBB) for dependent data can be found in Lahiri (2003). The main idea of block bootstrap procedures is to resample blocks of observations in order to capture the dependence structure of the original sample. Popular MBB method requires the stationarity for observations that usually results in failure of this method in non-stationary setting (see Lahiri (2003) for more details). The asymptotic behaviour of MBB method is highly dependent on the estimation of the bias and of the asymptotic variance of the statistic
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