Examples and counterexamples to almost-sure convergence of bilateral martingales (Q699979)
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scientific article; zbMATH DE number 1808182
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Examples and counterexamples to almost-sure convergence of bilateral martingales |
scientific article; zbMATH DE number 1808182 |
Statements
Examples and counterexamples to almost-sure convergence of bilateral martingales (English)
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26 September 2002
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The author deals with a stationary process \((X_p)_{p\in\mathbb{Z}}\) taking values in a finite alphabet, and an event \(B\) which is measurable with respect to the \(\sigma\)-algebra \(\sigma(X_p, p\in\mathbb{Z})\). Observing a finite number of coordinates of the process \(X_{-n}\), \(X_{-n+1},\dots, X_{m-1}, X_m\) (where \(m\) and \(n\) are positive integers), the author discusses whether the two-parameter martingale \(\mathbb{E}[1_B\mid X_{-n},X_{-n+ 1},\dots\), \(X_{m-1}, X_n]\) converges almost surely to \(1_B\) as \(n\) and \(m\) go to infinity.
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two-parameter martingales
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generating process
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stationary process
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entropy
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almost sure convergence
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