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world:forward-downward-martingale
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world:forward-downward-martingale [2026/02/03 15:58] rdouc [Proof] |
world:forward-downward-martingale [2026/02/07 15:03] (current) rdouc |
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| ====== Upcrossing Inequality and Martingale Convergence ====== | ====== Upcrossing Inequality and Martingale Convergence ====== | ||
| - | **Intuition.** | + | ===== Doob's Upcrossing Inequality ===== |
| - | A process cannot oscillate infinitely many times between two fixed levels $a<b$ without paying a cost. | + | |
| - | Each complete upcrossing from $a$ to $b$ forces an increase of at least $b-a$. | + | |
| - | The upcrossing inequality makes this idea precise and leads to almost sure convergence results for martingales and supermartingales. | + | |
| - | + | ||
| - | ===== Theorem 1 (Doob's Upcrossing Inequality) ===== | + | |
| Let $a<b$ and let $(X_k)_{k\ge0}$ be a real-valued process. | Let $a<b$ and let $(X_k)_{k\ge0}$ be a real-valued process. | ||
| Line 34: | Line 29: | ||
| * The process $Y_k$ accumulates only the increments of $X$ occurring during these upcrossing phases. | * The process $Y_k$ accumulates only the increments of $X$ occurring during these upcrossing phases. | ||
| - | Let $U_n[a,b]$ denote the number of completed upcrossings from $a$ to $b$ between times $0$ and $n$. | + | <WRAP center round tip 80%> |
| + | 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)^-. |
| $$ | $$ | ||
| + | |||
| + | </WRAP> | ||
| ===== Proof ===== | ===== Proof ===== | ||
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| Summing the contributions of all completed upcrossings and accounting for the final correction yields the stated inequality. ∎ | Summing the contributions of all completed upcrossings and accounting for the final correction yields the stated inequality. ∎ | ||
| - | ===== Theorem 2 (Doob's Forward Convergence Theorem) ===== | + | ===== Theorem 1 (Doob's Forward Convergence Theorem) ===== |
| + | <WRAP center round todo 80%> | ||
| Let $(X_n)_{n\ge0}$ be a supermartingale such that | Let $(X_n)_{n\ge0}$ be a supermartingale such that | ||
| $$ | $$ | ||
| Line 60: | Line 59: | ||
| $$ | $$ | ||
| exists almost surely. | exists almost surely. | ||
| + | |||
| + | </WRAP> | ||
| ===== Proof ===== | ===== Proof ===== | ||
| Line 65: | 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 73: | Line 74: | ||
| The $L^1$ boundedness assumption implies that $\sup_n \mathbb E[(X_n-a)^-]<\infty$. | The $L^1$ boundedness assumption implies that $\sup_n \mathbb E[(X_n-a)^-]<\infty$. | ||
| - | Therefore, | + | By the monotone convergence theorem, |
| $$ | $$ | ||
| - | \sup_n \mathbb E[U_n[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 86: | Line 87: | ||
| The conditional expectations stabilize and converge to the conditional expectation with respect to the remaining common information. | The conditional expectations stabilize and converge to the conditional expectation with respect to the remaining common information. | ||
| - | ===== Theorem 3 (Lévy's Downward Convergence Theorem) ===== | + | ===== Theorem 2 (Lévy's Downward Convergence Theorem) ===== |
| - | Let $(\mathcal G_{-n})_{n\ge1}$ be a decreasing sequence of $\sigma$-fields such that | + | <WRAP center round todo 80%> |
| + | Let $(\mathcal F_{-n})_{n\ge1}$ be a decreasing sequence of $\sigma$-fields such that | ||
| $$ | $$ | ||
| - | \mathcal G_{-\infty} | + | \mathcal F_{-\infty} |
| = | = | ||
| - | \bigcap_{k\ge1}\mathcal G_{-k} | + | \bigcap_{k\ge1}\mathcal F_{-k} |
| - | \subseteq \cdots \subseteq \mathcal G_{-(n+1)} \subseteq \mathcal G_{-n} \subseteq \cdots \subseteq \mathcal G_{-1}. | + | \subseteq \cdots \subseteq \mathcal F_{-(n+1)} \subseteq \mathcal F_{-n} \subseteq \cdots \subseteq \mathcal F_{-1}. |
| $$ | $$ | ||
| For any random variable $Z$ satisfying $\mathbb E[|Z|]<\infty$, we have | For any random variable $Z$ satisfying $\mathbb E[|Z|]<\infty$, we have | ||
| $$ | $$ | ||
| - | \lim_{n\to\infty}\mathbb E[Z\mid\mathcal G_{-n}] | + | \lim_{n\to\infty}\mathbb E[Z\mid\mathcal F_{-n}] |
| = | = | ||
| - | \mathbb E[Z\mid\mathcal G_{-\infty}] | + | \mathbb E[Z\mid\mathcal F_{-\infty}] |
| \quad\text{almost surely}. | \quad\text{almost surely}. | ||
| $$ | $$ | ||
| + | |||
| + | </WRAP> | ||
| ===== Proof ===== | ===== Proof ===== | ||
| Line 108: | Line 112: | ||
| Define the process | Define the process | ||
| $$ | $$ | ||
| - | X_k := \mathbb E[Z\mid\mathcal G_k], \qquad -n\le k\le -1. | + | X_k := \mathbb E[Z\mid\mathcal F_k], \qquad -n\le k\le -1. |
| $$ | $$ | ||
| 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 exactly the same reasons, | + | 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: |
| - | The number of upcrossing $U_n[a,b]$ between $-n$ and $-1$ can be bounded as follows: | + | |
| $$ | $$ | ||
| - | Y_n \geq (b-a) U_n[a,b] - (X_n-a)^- | + | Y_{-1} \geq (b-a) U_{-n:-1}[a,b] - (X_{-1}-a)^- |
| $$ | $$ | ||
| - | Applying the same upcrossing construction to $(X_k)$ on the interval $[-n,-1]$, we obtain for any $a<b$: | + | As before, the integrability of $Z$ ensures that the expected number of upcrossings is finite, which implies that $(X_k)$ converges almost surely as $k\to-\infty$. |
| - | $$ | + | |
| - | Y_n \ge (b-a)\,U_n[a,b] - (X_n-a)^-. | + | |
| - | $$ | + | |
| - | As before, the integrability of $Z$ ensures that the expected number of upcrossings is finite, which implies that $(X_k)$ converges almost surely as $k\to-\infty$. | + | The limit must coincide with $\mathbb E[Z\mid\mathcal F_{-\infty}]$, which completes the proof. ∎ |
| + | |||
| + | [[?do=backlink|Linked pages]] | ||
| - | The limit must coincide with $\mathbb E[Z\mid\mathcal G_{-\infty}]$, which completes the proof. ∎ | ||
world/forward-downward-martingale.1770130733.txt.gz · Last modified: 2026/02/03 15:58 by rdouc
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