This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
world:martingale [2021/03/20 09:05] rdouc [Proof] |
world:martingale [2022/03/16 07:40] (current) |
||
---|---|---|---|
Line 2: | Line 2: | ||
===== Supermartingale convergence results ===== | ===== Supermartingale convergence results ===== | ||
+ | |||
<WRAP center round tip 80%> | <WRAP center round tip 80%> | ||
Line 19: | Line 20: | ||
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 |