Invariance principles for linear processes with application to isotonic regression

Abstract

In this paper, we prove maximal inequalities and study the functional central limit theorem for the partial sums of linear processes generated by dependent innovations. Due to the general weights, these processes can exhibit long-range dependence and the limiting distribution is a fractional Brownian motion. The proofs are based on new approximations by a linear process with martingale difference innovations. The results are then applied to study an estimator of the isotonic regression when the error process is a (possibly long-range dependent) time series.

Published in at http://dx.doi.org/10.3150/10-BEJ273 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (6)

• Strong invariance principles for dependent random variables
• Martingale approximations for sums of stationary processes
• Central limit theorem for stationary linear processes
• A general asymptotic scheme for inference under order restrictions
• Invariance principles for fractionally integrated nonlinear processes
• Invariance principle for stochastic processes with short memory

Cited by in corpus (8)

• Empirical central limit theorems for ergodic automorphisms of the torus
• Asymptotic properties for linear processes of functionals of reversible Markov Chains
• Functional Convergence of Linear Sequences in a non-Skorokhod Topology
• Asymptotic Properties of Self-Normalized Linear Processes with Long Memory
• Invariance principles for operator-scaling Gaussian random fields
• An invariance principle for fractional Brownian sheets
• Remarks on limit theorems for reversible Markov processes
• Convergence to the maximum process of a fractional Brownian motion with shot noise