2004/04/01 by Wei Biao Wu, Michael Woodroofe · 3 citations
Economics, Econometrics and Finance · Mathematics · #Financial Risk and Volatility Modeling #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60F05 #msc:60F17 #msc:60G42 #msc:60J10.
paper · pdf · doi:10.1214/009117904000000351
published as Annals of Probability 2004, Vol. 32, No. 2, 1674-1690 · Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000351
openalex publication_date 2004/04/01 · arxiv created 2004/10/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.