AENC/resampling_chain Changeset - b43c29412d5d · Centrum Wiskunde & Informatica (CWI)

Changeset - b43c29412d5d

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Tom Bannink - 8 years ago 2017-05-29 08:51:35
tombannink@gmail.com

Add note on probability independence claim

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         \mathbb{P}[\mathrm{NZ}_{j_1} , \mathrm{NZ}_{j_2} \;|\;\text{start in }b_1]
         \; \cdot \;
         \mathbb{P}[\mathrm{NZ}_{j_1} , \mathrm{NZ}_{j_2} \;|\;\text{start in }b_2]
     \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}}
     Let us interpret $[n]$ as the vertices of a length-$n$ cycle, and interpret operations on vertices mod $n$ s.t. $n+1\equiv 1$ and $1-1\equiv n$.
     %\begin{definition}[Resample sequences]
     %	A sequence of indices $(r_\ell)=(r_1,r_2,\ldots,r_k)\in[n]^k$ is called resample sequence if our procedure performs $k$ consequtive resampling, where the first resampling of the procedure resamples around the mid point $r_1$ the second around $r_2$ and so on. Let $RS(k)$ the denote the set of length $k$ resample sequences, and let $RS=\cup_{k\in\mathbb{N}}RS(k)$.

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