Open AccessWhat is a linear process?
Author(s) -
Peter J. Bickel,
Peter Bühlmann
Publication year - 1996
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.93.22.12128
Subject(s) - independent and identically distributed random variables , sequence (biology) , mathematical proof , closure (psychology) , mathematics , chaotic , metric (unit) , nonlinear system , process (computing) , set (abstract data type) , statistic , statistical physics , computer science , statistics , random variable , physics , artificial intelligence , geometry , genetics , operations management , quantum mechanics , economics , market economy , biology , programming language , operating system
We argue that given even an infinitely long data sequence, it is impossible (with any test statistic) to distinguish perfectly between linear and nonlinear processes (including slightly noisy chaotic processes). Our approach is to consider the set of moving-average (linear) processes and study its closure under a suitable metric. We give the precise characterization of this closure, which is unexpectedly large, containing nonergodic processes, which are Poisson sums of independent and identically distributed copies of a stationary process. Proofs of these results will appear elsewhere.