Changeset - 0f7c68754475
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Andras Gilyen - 8 years ago 2017-09-07 19:06:22
gilyen@clayoquot.swat.cwi.nl
nicer proof
1 file changed with 4 insertions and 4 deletions:
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@@ -444,11 +444,11 @@ The process on the finite chain has the following modification at the boundary:
 
        \P^{(n)}_b(A) &= \P^{(n)}(A \;|\; \start{b}) %\\
 
        %R_{b,A} &= \mathbb{E}( \#resamples \;|\; A \; , \; \start{b})
 
    \end{align*}
 
    Furthermore, for the Markov Chain on the finite chain, define
 
    Furthermore, for $v\in[n]$ we define
 
    \begin{align*}
 
        \P^{[n]}_{\partial=1}(A) &= \P^{[n]}(A \;|\; \text{boundary is initialized to }1)
 
        \P^{[n]}_{b_v=1}(A) &= \P^{[n]}(A \;|\; v\text{ is initialized to }1),
 
    \end{align*}
 
    where the boundary of $[n]$ is site $1$ and site $n$, and the boundary of $[a,b]$ are $a$ and $b$.
 
    and we define similarly $\P^{[n]}_{b_v=b_w=1}(A)$ for $v,w\in[n]$.
 
\end{definition}
 
%Note that we have $\P^{(n)}(\start{b}) = (1-p)^{|b|}p^{n-|b|}$ by definition of our Markov Chain.
 
\begin{definition}[Vertex visiting event] \label{def:visitingResamplings}
 
@@ -628,7 +628,7 @@ The following lemma considers two vertices $v,w$ that are never ``crossed'' so t
 
        &= \sum_{b\in\{0,1\}^n}
 
            \P^{[v,w]}_{b|_{[v,w]}}(\mathrm{NZ}^{(v,w)}\cap A)
 
            \P^{[v,w]}(\start{b|_{[v+1,w-1]}})
 
            \\ &\qquad\qquad\quad
 
            \\ &\qquad\qquad\quad\cdot
 
            \P^{[w,v]}_{b|_{[w,v]}}(\mathrm{NZ}^{(v,w)}\cap B)
 
            \P^{[w,v]}(\start{b|_{[w,v]}}) \\
 
        &= \left( \sum_{\substack{b_1\in\{0,1\}^{[v,w]}\\ b_v=b_w=1}}
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