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world:forward-downward-martingale
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world:forward-downward-martingale [2026/02/07 08:37] rdouc |
world:forward-downward-martingale [2026/02/07 15:03] (current) rdouc |
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| <WRAP center round tip 80%> | <WRAP center round tip 80%> | ||
| - | Let $U_n[a,b]$ denote the number of completed upcrossings from $a$ to $b$ between times $0$ and $n$. | + | Let $U_{0:n}[a,b]$ denote the number of completed upcrossings from $a$ to $b$ between times $0$ and $n$. |
| Then the following inequality holds: | Then the following inequality holds: | ||
| $$ | $$ | ||
| - | Y_n \ge (b-a)\,U_n[a,b] - (X_n-a)^-. | + | Y_n \ge (b-a)\,U_{0:n}[a,b] - (X_n-a)^-. |
| $$ | $$ | ||
| Line 66: | Line 66: | ||
| By the upcrossing inequality, for any $a<b$, | By the upcrossing inequality, for any $a<b$, | ||
| $$ | $$ | ||
| - | (b-a)\,\mathbb E[U_n[a,b]] | + | (b-a)\,\mathbb E[U_{0:n}[a,b]] |
| \le | \le | ||
| \mathbb E[Y_n] + \mathbb E[(X_n-a)^-]. | \mathbb E[Y_n] + \mathbb E[(X_n-a)^-]. | ||
| Line 76: | Line 76: | ||
| By the monotone convergence theorem, | By the monotone convergence theorem, | ||
| $$ | $$ | ||
| - | \mathbb E[U_\infty[a,b]]<\infty, | + | \mathbb E[U_{0:\infty}[a,b]]<\infty, |
| $$ | $$ | ||
| which implies that the total number of upcrossings is almost surely finite. | which implies that the total number of upcrossings is almost surely finite. | ||
| Line 116: | Line 116: | ||
| This is a martingale bounded in $L^1$. | This is a martingale bounded in $L^1$. | ||
| - | Similarly as before, we define $C_1=\mathsf{1}_{X_{-n} < a}$ . For all $-n < k \leq -1$, we set $C_k=\mathsf{1}_{C_{k-1}=1} \mathsf{1}_{X_{k-1} \leq b}+ \mathsf{1}_{C_{k-1}=0} \mathsf{1}_{X_{k-1} <a}$. Define also $Y_k=\sum_{\ell=-n}^k C_\ell (X_\ell - X_{\ell-1})$. Then, for the same reasons, the number of upcrossing $U_n[a,b]$ between $-n$ and $-1$ can be bounded as follows: | + | Similarly as before, we define $C_1=\mathsf{1}_{X_{-n} < a}$ . For all $-n < k \leq -1$, we set $C_k=\mathsf{1}_{C_{k-1}=1} \mathsf{1}_{X_{k-1} \leq b}+ \mathsf{1}_{C_{k-1}=0} \mathsf{1}_{X_{k-1} <a}$. Define also $Y_k=\sum_{\ell=-n}^k C_\ell (X_\ell - X_{\ell-1})$. Then, for the same reasons, the number of upcrossing $U_{-n:-1}[a,b]$ between $-n$ and $-1$ can be bounded as follows: |
| $$ | $$ | ||
| - | Y_{-1} \geq (b-a) U_n[a,b] - (X_{-1}-a)^- | + | Y_{-1} \geq (b-a) U_{-n:-1}[a,b] - (X_{-1}-a)^- |
| $$ | $$ | ||
world/forward-downward-martingale.1770449861.txt.gz · Last modified: 2026/02/07 08:37 by rdouc
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