Forward estimation for ergodic time series

Details Forward estimation for ergodic time series Forward estimation for ergodic time series

2007-11-24

arxiv.org/abs/0711.3856v1

Ann. Inst. H. Poincare Probab. Statist. 41 (2005), no. 5, 859--870

Mathematics

Probability

Scientific paper

The forward estimation problem for stationary and ergodic time series $\{X_n\}_{n=0}^{\infty}$ taking values from a finite alphabet ${\cal X}$ is to estimate the probability that $X_{n+1}=x$ based on the observations $X_i$, $0\le i\le n$ without prior knowledge of the distribution of the process $\{X_n\}$. We present a simple procedure $g_n$ which is evaluated on the data segment $(X_0,...,X_n)$ and for which, ${\rm error}(n) = |g_{n}(x)-P(X_{n+1}=x |X_0,...,X_n)|\to 0$ almost surely for a subclass of all stationary and ergodic time series, while for the full class the Cesaro average of the error tends to zero almost surely and moreover, the error tends to zero in probability.

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