world:robbins_monro
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world:robbins_monro [2021/03/14 10:37] rdouc [Statement] |
world:robbins_monro [2022/03/16 07:40] 127.0.0.1 external edit |
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| <WRAP center round box 80%> | <WRAP center round box 80%> | ||
| * $f$ is a bounded and continuous function | * $f$ is a bounded and continuous function | ||
| - | * $f(x)(x-x_*)>0$ for all $x\in \Xset$ | + | * $f(x)(x-x_*)>0$ for all $x\neq x_* \in \Xset$ |
| </WRAP> | </WRAP> | ||
| We assume there exist a Markov kernel $K$ on $\Xset \times \Xsigma$ and a sequence $\seq{\gamma_k}{k\in\nset}$ of real numbers such that | We assume there exist a Markov kernel $K$ on $\Xset \times \Xsigma$ and a sequence $\seq{\gamma_k}{k\in\nset}$ of real numbers such that | ||
| Line 17: | Line 17: | ||
| </WRAP> | </WRAP> | ||
| - | Define iteratively the sequence <color red>$X_{n+1}=X_{n}-\gamma_{n+1} U_{n+1}$</color> where | + | Let $X_0$ be a random variable such that $\PE[X^2_0]<\infty$. Define iteratively the sequence <color red>$$X_{n+1}=X_{n}-\gamma_{n+1} U_{n+1}$$</color> where |
| $U_{n+1}|_{\mcf_n}\sim K (X_n,\cdot)$ and $\mcf_n=\sigma(X_0,\ldots,X_n)$. | $U_{n+1}|_{\mcf_n}\sim K (X_n,\cdot)$ and $\mcf_n=\sigma(X_0,\ldots,X_n)$. | ||
| - | The aim of this short note is to prove that $X_n \psconv 0$. We follow a version of the proof proposed by François Roueff. | + | The aim of this short note is to prove that $X_n \psconv x_*$. We follow a version of the proof proposed by François Roueff. |
| ====== Proof ====== | ====== Proof ====== | ||
| Line 38: | Line 38: | ||
| ==== Convergence of $(X_n-x_*)^2$ ==== | ==== Convergence of $(X_n-x_*)^2$ ==== | ||
| Considering \eqref{eq:def:Sn}, to obtain the convergence of $(X_n-x_*)^2$, we only need to prove that $S_n$ and $\sum_{k=1}^n \gamma_k^2 U_k^2$ converge $\PP$-a.s. as $n$ goes to infinity. | Considering \eqref{eq:def:Sn}, to obtain the convergence of $(X_n-x_*)^2$, we only need to prove that $S_n$ and $\sum_{k=1}^n \gamma_k^2 U_k^2$ converge $\PP$-a.s. as $n$ goes to infinity. | ||
| - | - The convergence of $S_n$ follows from | + | - The convergence of $(S_n)$ follows from (see [[world:martingale#submartingale|some results on limits of martingales]]) |
| * $(S_n)$ is a submartingale since $\CPE{S_n-S_{n-1}}{\mcf_{n-1}}=2\gamma_n f (X_{n-1}) (X_{n-1}-x_*) \geq 0$ | * $(S_n)$ is a submartingale since $\CPE{S_n-S_{n-1}}{\mcf_{n-1}}=2\gamma_n f (X_{n-1}) (X_{n-1}-x_*) \geq 0$ | ||
| * $\sup_n \PE[S_n^+] \leq \PE[(X_0-x_*)^2]+\sum_{k=1}^n \gamma_k^2 \underbrace{\PE[U_k^2]}_{\leq M}<\infty$. | * $\sup_n \PE[S_n^+] \leq \PE[(X_0-x_*)^2]+\sum_{k=1}^n \gamma_k^2 \underbrace{\PE[U_k^2]}_{\leq M}<\infty$. | ||
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| - | ==== Proof of $\mathbb{P}(A=0)=1$ ==== | + | ==== Proof of $\mathbb{P}(A=0)=1$. ==== |
| First, write the Doob decomposition for the submartingale $(S_n)$ that is: $S_n=M_n+W_n$ where | First, write the Doob decomposition for the submartingale $(S_n)$ that is: $S_n=M_n+W_n$ where | ||
| Line 67: | Line 67: | ||
| and this implies that for all $\omega \in \lrcb{\delta/2 < |A| <\delta}$, | and this implies that for all $\omega \in \lrcb{\delta/2 < |A| <\delta}$, | ||
| \begin{equation*} | \begin{equation*} | ||
| - | W_\infty \geq \sum_{n=N(\omega)}^\infty \gamma_k f (X_{k-1}(\omega)) (X_{k-1}(\omega)-x_*) \geq \gamma \sum_{n=N(\omega)}^\infty \gamma_k=\infty | + | W_\infty \geq \sum_{n=N(\omega)+1}^\infty \gamma_k f (X_{k-1}(\omega)) (X_{k-1}(\omega)-x_*) \geq \gamma \sum_{n=N(\omega)+1}^\infty \gamma_k=\infty |
| \end{equation*} | \end{equation*} | ||
| This shows that $\lrcb{\delta/2 < |A| <\delta} \subset \lrcb{W_\infty=\infty}$ and the proof is completed. | This shows that $\lrcb{\delta/2 < |A| <\delta} \subset \lrcb{W_\infty=\infty}$ and the proof is completed. | ||
world/robbins_monro.txt · Last modified: 2023/02/10 18:17 by rdouc · Currently locked by: 114.119.154.6
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