From 0f7c6875447566ff9029c8126dd669d20f071cf4 2017-09-07 19:06:22
From: Andras Gilyen <gilyen@clayoquot.swat.cwi.nl>
Date: 2017-09-07 19:06:22
Subject: [PATCH] nicer proof

---

diff --git a/main.tex b/main.tex
index fef9c957362806b4b9b4a5c91984cb280f79cf79..37f85a83b4baf91f0cb4138243df06aebb4ac677 100644
--- a/main.tex
+++ b/main.tex
@@ -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}}