On the Komlós, Major and Tusnády strong approximation for some classes of random iterates

Christophe Cuny, Jérôme Dedecker and Florence Merlevède

Stochastic Processes and their Applications, 2018, vol. 128, issue 4, 1347-1385

Abstract: The famous results of Komlós, Major and Tusnády (see Komlós et al., 1976 [15] and Major, 1976 [17]) state that it is possible to approximate almost surely the partial sums of size n of i.i.d. centered random variables in Lp (p>2) by a Wiener process with an error term of order o(n1∕p). Very recently, Berkes et al. (2014) extended this famous result to partial sums associated with functions of an i.i.d. sequence, provided a condition on a functional dependence measure in Lp is satisfied. In this paper, we adapt the method of Berkes, Liu and Wu to partial sums of functions of random iterates. Taking advantage of the Markovian setting, we shall give new dependent conditions, expressed in terms of a natural coupling (in L∞ or in L1), under which the strong approximation result holds with rate o(n1∕p). As we shall see our conditions are well adapted to a large variety of models, including left random walks on GLd(R), contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains. We also provide some examples showing that our L1-coupling condition is in some sense optimal.

Keywords: Strong invariance principle; KMT approximation; Random iterates; Markov chains; Left random walk on GLd(R) (search for similar items in EconPapers)
Date: 2018
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414917301795
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:128:y:2018:i:4:p:1347-1385

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2017.07.011

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().