Martingale approximations for sums of stationary processes

By Wei Biao Wu, Michael Woodroofe. 2004

116 citations recorded by the source corpus.

Approximations to sums of stationary and ergodic sequences by martingales are investigated. Necessary and sufficient conditions for such sums to be asymptotically normal conditionally given the past up to time 0 are obtained. It is first shown that a martingale approximation is necessary for such normality and then that the sums are asymptotically normal if and only if the approximating martingales satisfy a Lindeberg-Feller condition. Using the explicit construction of the approximating martingales, a central limit theorem is derived for the sample means of linear processes. The conditions are not sufficient for the functional version of the central limit theorem. This is shown by an example, and a slightly stronger sufficient condition is given.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0710c0450a7fc7e7bc0b922ad7b78aa140887e0d

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Citations in the corpus

A new maximal inequality and invariance principle for stationary sequences (2004, background): “(3) In the ergodic case, Theorem 1.1 improves upon the corresponding results of Maxwell and Woodroofe (2000) [see also Derriennic and Lin (2003) and Wu and Woodroofe (2002)].”
Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples (2011, methodology): “Then, if condition (38) holds for the sequence (Xi)i∈Z, then by a construction in Woodroofe and Zhao (2008) and by the uniqueness of the martingale approximation, the sequence (Xi − di)i∈Z is still a strictly stationary sequence and by our theorem limn→∞ n ‖max1≤k≤n |Sk − ∑k i=1 di|‖p = 0 .”
A Martingale Proof of Dobrushin's Theorem for Non-Homogeneous Markov Chains (2004, background): “Kifer [10], Kipnis and Varadhan [11], Pinsky [12], and Wu and Woodroofe [ 19 ]).”
On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria (2007, background): “…the following orthogonal decomposition is valid: (2.1) Using the stationarity, we see that p n is a svf (see Theorem 8.13 in Bradley (2002)), and the same is true for This has been proved by Wu and Woodroofe (2004) if X 0 is M 0 -measurable, and extended to the general case by Voln(cid:19)y (2005).”
TESTING FOR WHITE NOISE UNDER UNKNOWN DEPENDENCE AND ITS APPLICATIONS TO DIAGNOSTIC CHECKING FOR TIME SERIES MODELS (2010, unknown): “Note that the martingale approximation for the single array sequence ut has been well studied (see, among others, Hsing and Wu, 2004; Wu and Woodroofe, 2004; Wu and Shao, 2007), but the techniques there are not directly applicable.”
On linear processes with dependent innovations (2005, background): “The moment condition an 2 L with a42 cannot be weakened to an 2 L(2): The following example is constructed based on Example 3 in [51].”
Recent advances in invariance principles for stationary sequences (2006, methodology): “6], Wu and Woodroofe [70] and in Merlevède and Peligrad [42].”