world:non-geometric
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world:non-geometric [2024/03/04 14:41] rdouc [Proof] |
world:non-geometric [2024/03/05 10:07] rdouc |
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| \begin{align*} | \begin{align*} | ||
| \sup_{x\in C} \PE_x[\beta^{\sigma_C}]&\geq \PE_{w_0}[\beta^{\sigma_C}-1]+1=(\beta-1) \sum_{i=0}^{\infty} \beta^i \PP_{w_0}(\sigma_C > i ) +1\\ | \sup_{x\in C} \PE_x[\beta^{\sigma_C}]&\geq \PE_{w_0}[\beta^{\sigma_C}-1]+1=(\beta-1) \sum_{i=0}^{\infty} \beta^i \PP_{w_0}(\sigma_C > i ) +1\\ | ||
| - | & \geq (\beta-1) \sum_{i=0}^{\infty} \beta^{k+j} \PP_{w_0}(X_k\in B,\sigma_C > k+j ) +1 \\ | + | & \geq (\beta-1) \sum_{j=0}^{\infty} \beta^{k+j} \PP_{w_0}(X_k\in B,\sigma_C > k+j ) +1 \\ |
| - | & \geq (\beta-1) \sum_{i=0}^{\infty} \beta^{k+j} \PP_{w_0}(X_k\in B,\sigma_C > k, X_k=X_{k+1}= \ldots=X_{k+j} ) +1 \\ | + | & \geq (\beta-1) \sum_{j=0}^{\infty} \beta^{k+j} \PP_{w_0}(X_k\in B,\sigma_C > k, X_k=X_{k+1}= \ldots=X_{k+j} ) +1 \\ |
| - | & \geq (\beta-1) \sum_{i=0}^{\infty} \beta^{k+j} \PP_{w_0}(X_k\in B,\sigma_C > k) \eta^{j} +1 | + | & \geq (\beta-1) \sum_{j=0}^{\infty} \beta^{k+j} \PP_{w_0}(X_k\in B,\sigma_C > k) \eta^{j} +1 |
| \end{align*} | \end{align*} | ||
| which is divergent for $\eta$ sufficiently close to 1. | which is divergent for $\eta$ sufficiently close to 1. | ||
world/non-geometric.txt · Last modified: 2024/03/27 17:27 by rdouc
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