From b43c29412d5dea02928ebef3b0fc35a50080d892 2017-05-29 08:51:35
From: Tom Bannink <tombannink@gmail.com>
Date: 2017-05-29 08:51:35
Subject: [PATCH] Add note on probability independence claim

---

diff --git a/main.tex b/main.tex
index 5c6414dbface5eab59263ce53252227058cd85a8..b826b535b105828b8fda669d5ff0bf40f744870d 100644
--- a/main.tex
+++ b/main.tex
@@ -414,6 +414,12 @@ The proof of claim \ref{claim:expectationsum} also proves the following claim
     \end{align*}
 up to any order in $p$.
 \end{claim}
+Since the left hand side is defined as
+\begin{align*}
+    \mathbb{P}[\mathrm{NZ}_{j_1} , \mathrm{NZ}_{j_2} |\;\text{start in }b]
+    = \sum_{\substack{\xi\in\paths{b}\\j_1,j_2 \text{ not 0 in } \xi}} \mathbb{P}[\xi]
+\end{align*}
+we see that all such paths $\xi$ can be split into paths $\xi_1\in\paths{b_1}$ and $\xi_2\in\paths{b_2}$ and by the same reasoning as in the proof of claim \ref{claim:expectationsum}, we obtain the right hand side.
 
 \newpage
     \subsection{Sketch of the (false) proof of the linear bound \ref{it:const}}