References: - [[martingale-convergence-theorem-proof-r-isaac-pdf]] - [[martingale-convergences-mit-pdf]] ### Martingale Convergence Theorems
The martingale convergence theorems give the conditions for which a martingale converges (almost surely or in norm) to some limit \(X:=\lim_{n \to \infty}X_{n}\).
Let \((X_{n})\) be a martingale. Then there is a representation \(X_{n}=U_{n}-V_{n}\) where \(U_{n}\) and \(V_{n}\) are non-negative martingales if and only if \(\lim_{n \to \infty}\mathbb{E}|X_{n}|< \infty\).
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The idea is that we show that a non-negative submartingale with bounded second moments is a mean-square Cauchy sequence. Then we show that such a submartingale converges almost surely. We then prove convergence for non-negative martingales and finally [[]] provides the general case.
Let \((X_{n})\) be a non-negative submartingale with \(\lim_{n \to \infty }\mathbb{E}X_{n}^2<\infty\). Then \[ \lim_{n,m\to \infty}\mathbb{E}\lvert X_{n}-X_{m} \rvert^2 = 0. \]
\begin{proof} Proof: Martingale Convergence Theorem Proof (R.Isaac).pdf
\end{proof} :::{.theorem data-title=“Non-Negative Submartingale Convergence”}
Let \((X_{n})\) be a non-negative submartingale with \(\lim_{n \to \infty}\mathbb{E}X_{n}^2< \infty\). Then \[ X_{n}\stackrel{a.s.}{\to}X. \]
:::
\begin{proof} Proof: Martingale Convergence Theorem Proof (R.Isaac).pdf \end{proof}
Let \(\{ X_{n} \}\) be a non-negative martingale. Then \[ X_{n}\stackrel{a.s.}{\to}X. \]
\begin{proof} Proof: Martingale Convergence Theorem Proof (R.Isaac).pdf \end{proof}
Let \(\{ X_{n} \}\) be a martingale satisfying \(\lim_{n \to \infty}\mathbb{E}\lvert X_{n} \rvert< \infty\). Then
The Martingale Convergence Theorem states that martingales
If \(\{ M_{n},~n\geq 1 \}\) is a martingale with respect to its own filtration \(\mathcal{F}_{n}=\sigma(M_{1}, \dots, M_{n})\) and with \(\mathbb{E}[M_{n}^2]<K<\infty,~n\geq 1\) for some \(K>0\). Then there exists in \(\mathbb{R}\) an almost sure limit \(M_{\infty}:=\lim_{n\to \infty}M_{n}\) and \[\lim_{n\to\infty}\mathbb{E}[|M_{n}-M_{\infty}|^2]=0,\]that is \(M_{n}\stackrel{L^2}{\to}L^2\).
Proof: [[martingale-convergence-theorem-proof-r-isaac-pdf]]
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