world:non-geometric
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world:non-geometric [2024/03/04 14:40] rdouc [Proof] |
world:non-geometric [2024/03/04 14:41] rdouc [Proof] |
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| \PP_x(X_1=\ldots=X_j=x) \geq \eta^j | \PP_x(X_1=\ldots=X_j=x) \geq \eta^j | ||
| \end{equation*} | \end{equation*} | ||
| - | Using that $C$ is a small set, we can easily show that $\sup_{x \in C} P(x,\{x\})<1$ (indeed, if $C\cap A_\eta$ contains two distincts elements $x,x'$ for $\eta$ sufficiently small, then $\epsilon \nu(\{x\}^c) \leq P^m(x,\{x\}^c) \leq 1-\eta^m$ showing that $\nu(\{x\}^c) + \nu(\{x'\}^c)$ is arbitrary small which is not possible since this sum is bounded from below by $\nu(\Xset)$). This allows to choose $B=A_\eta$ with $\eta$ chosen sufficiently close to $1$ so that $B \cap A_\eta=\emptyset$. Then, there exists $w_0\in C$ and $k \in \nset$ such that $\PP_{w_0}(X_k \in B, \sigma_C >k)>0$ (which can be easily seen by contradiction). | + | Moreover, |
| + | * Using that $C$ is a small set, we can easily show that $\sup_{x \in C} P(x,\{x\})<1$ (indeed, if $C\cap A_\eta$ contains two distincts elements $x,x'$ for $\eta$ sufficiently small, then $\epsilon \nu(\{x\}^c) \leq P^m(x,\{x\}^c) \leq 1-\eta^m$ showing that $\nu(\{x\}^c) + \nu(\{x'\}^c)$ is arbitrary small which is not possible since this sum is bounded from below by $\nu(\Xset)$). | ||
| + | * This allows to choose $B=A_\eta$ with $\eta$ chosen sufficiently close to $1$ so that $B \cap A_\eta=\emptyset$. Then, there exists $w_0\in C$ and $k \in \nset$ such that $\PP_{w_0}(X_k \in B, \sigma_C >k)>0$ (which can be easily seen by contradiction). | ||
| Now, write for any $\beta>1$, | Now, write for any $\beta>1$, | ||
world/non-geometric.txt · Last modified: 2024/03/27 17:27 by rdouc
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