Let (εk)_{k∈Z} be a weak stationary, ergodic process with values in a Hilbert space H and let (Xk)_{k∈Z} be another process with values in H. Does nonanticipativity of (Xk)_{k∈Z} w.r.t. (εk)_{k∈Z}, that is X_k=g(ε_k,ε_k−1,...) holds for all k a.s. where g:H∞→H is a measurable function, under a certain moment condition of g imply that (Xk)_{k∈Z} is weak stationarity and ergodic?
According to Theorem 3.1.8 of the book "Almost Sure Convergence" by Stout (1974), (Xk)_{k∈Z} is strictly stationary and ergodic if (Xk)_{k∈Z} is nonanticipative w.r.t. (εk)_{k∈Z} and if (εk)_{k∈Z} is strictly stationary and ergodic. Unfortunately, I don`t have access to fee-based literature, so I couldn't look it up myself.
Thank you in advance.
Dear Sebastian, The best to ask general questions on RG, so you will get more answers. Here, I suggest some work that may help you: https://arxiv.org/pdf/1801.08256.pdf
http://www.maths.lth.se/matstat/staff/georg/Publications/lecture2006.pdf
Sebastian Kühnert The book "Almost Sure Convergence" by William F. Stout, E. Lukacs, Z. W. Birnbaum may be downloaded for free here:
https://b-ok.cc/book/2867475/81e378
If by some reason this does not work, I can e-mail it to you directly.
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