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world:martingale [2021/03/20 09:04] rdouc [Proof] |
world:martingale [2022/03/16 07:40] (current) |
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| ===== Supermartingale convergence results ===== | ===== Supermartingale convergence results ===== | ||
| + | |||
| <WRAP center round tip 80%> | <WRAP center round tip 80%> | ||
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| Let $a<b$ and define $C_1=\indiacc{X_0<a}$ and for $n\geq 2$, | Let $a<b$ and define $C_1=\indiacc{X_0<a}$ and for $n\geq 2$, | ||
| $$ | $$ | ||
| - | C_n=\indiacc{C_{n-1}=1,X_{n-1} \leq b}+\indiacc{C_{n-1}=0,X_{n-1} \leq a} | + | C_n=\indiacc{C_{n-1}=1,X_{n-1} \leq b}+\indiacc{C_{n-1}=0,X_{n-1} < a} |
| $$ | $$ | ||
| In words, the first time $C_n $ flags $1$ is when $X_{n-1}<a$. Then it flags $1$ until $X_{n-1}$ goes above $b$. Then it flags $0$ until $X_n$ goes below $a$. So consecutive sequences of $C_n=1$ are linked with upcrossings of $[a,b] $ for $(X_n) $. Now, define | In words, the first time $C_n $ flags $1$ is when $X_{n-1}<a$. Then it flags $1$ until $X_{n-1}$ goes above $b$. Then it flags $0$ until $X_n$ goes below $a$. So consecutive sequences of $C_n=1$ are linked with upcrossings of $[a,b] $ for $(X_n) $. Now, define | ||
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| which shows that $X_\infty=\lim_{n \to \infty} X_n$ exits almost surely. | which shows that $X_\infty=\lim_{n \to \infty} X_n$ exits almost surely. | ||
| - | Morevoer, $\PE[X_0] \geq \PE[X_n]=\PE[X^+_n]-\PE[X^-_n]$ so that | + | Moreover, $\PE[X_0] \geq \PE[X_n]=\PE[X^+_n]-\PE[X^-_n]$ so that |
| $$ | $$ | ||
| - | \sup_n \PE[|X_n|]=\sup_n \lr{\PE[X^+_n]+\PE[X^-_n]} \geq \PE[X_0] +2 \sup_n \PE[X^-_n] \leq \PE[X_0] +2 M<\infty | + | \sup_n \PE[|X_n|]=\sup_n \lr{\PE[X^+_n]+\PE[X^-_n]} \leq \PE[X_0] +2 \sup_n \PE[X^-_n] \leq \PE[X_0] +2 M<\infty |
| $$ | $$ | ||
| - | which implies by Fatou's lemma that $\PE[|X_\infty|]=\PE[\liminf_{n}|X_\infty|] \leq \liminf_{n} \PE[|X_\infty|] \leq \sup_n \PE[|X_n|]<\infty$. The proof is completed. | + | which implies by Fatou's lemma that $\PE[|X_\infty|]=\PE[\liminf_{n}|X_n|] \leq \liminf_{n} \PE[|X_n|] \leq \sup_n \PE[|X_n|]<\infty$. The proof is completed. |
| ===== Corollary: Submartingale convergence results ===== | ===== Corollary: Submartingale convergence results ===== | ||
world/martingale.1616227474.txt.gz ยท Last modified: 2022/03/16 01:37 (external edit)
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