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I'm looking at a Martingale (actually a Martingale difference sequence), $$ M_n = \sum \delta M_n, $$ and I'd like to prove something about convergence. If Martingale is Hilbert space valued (infinite dimensional), are there immediate analogs of the convergence theorems? I'm particularly interested in generalizing quadratic variation results to the Hilbert space case.

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  • $\begingroup$ Intuitively, quadratic variation seems more likely to be a useful object if you are working in a Hilbert space. $\endgroup$ Jun 15, 2015 at 2:13
  • $\begingroup$ Yes, I am working in a Hilbert space. I've edited to add some more details. $\endgroup$ Jun 15, 2015 at 2:21
  • $\begingroup$ We proved a few basic results along these lines in this paper, and one place to look for more material on Hilbert space-valued martingales is Métivier's book Semimartingales. If this seems to be helpful I can add it as an answer. $\endgroup$ Jun 15, 2015 at 2:35
  • $\begingroup$ I'm interested in what I think is a simpler problem. My process is discrete in time, and those works appear to handle the continuous case. $\endgroup$ Jun 15, 2015 at 2:48
  • $\begingroup$ I guess I don't know specifically, but often one can prove such things (assuming your Hilbert space is separable) by letting $P_k$ be a sequence of finite-rank orthogonal projections with $P_k \to I$ strongly, proving something for the finite-dimensional process $P_k M_n$ with uniformity in $k$, and letting $k \to \infty$. $\endgroup$ Jun 15, 2015 at 6:01

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