AENC/resampling_chain Changeset - 9b0f04598ca9 · Centrum Wiskunde & Informatica (CWI)

Changeset - 9b0f04598ca9

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Andras Gilyen - 8 years ago 2017-09-07 16:40:09
gilyen@clayoquot.swat.cwi.nl

nicer proof

1 file changed with 11 insertions and 11 deletions:

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 			R^{(n)}
 			&= \E^{(n)}(\Res{1}) \tag{by translation invariance}\\
 			&= \sum_{k=1}^{\infty}\P^{(n)}(\Res{1}\geq 1) \\
 			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}\P^{(n)}(\Res{1}\geq 1\& P\in\mathcal{P}) \tag{partition}\\
 			&= \frac{1}{n}\sum_{v\in[n]}\sum_{t=1}^{\infty}\sum_{P\text{ patch}}t\cdot\P^{(n)}(v \text{ is resampled }t\text{ times and }v\in P | P\in\mathcal{P}) \; \P^{(n)}(P\in\mathcal{P})\\
 			&= \frac{1}{n}\sum_{P\text{ patch}}\E^{(n)}(\# \text{ resamples in }P|P\in \mathcal{P}) \; \P^{(n)}(P\in\mathcal{P})\\
 			&= \sum_{s=1}^{n-1}\E^{(n)}(\# \text{ resamples in }[s] \;|\; [s]\in \mathcal{P}) \; \P([s]\in\mathcal{P}) +\bigO{p^{n}}
 			\tag{by translation symmetry}\\
 			&= ???? \\
 			&= \sum_{s=1}^{n-1}\E^{[0,s+1]}(\# \text{ resamples in }[s]|[s]\in \mathcal{P})\P^{[s+1,n]}(\NZ{s+1}\cap\NZ{n})/(1+p)^2+\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}}\\
 			&= \sum_{s=1}^{n-1}\E^{[0,s+1]}(\# \text{ resamples in }[s]|[s]\in \mathcal{P})\left(\P^{[s+1,n]}(\NZ{s+1})\right)^2/(1+p)^2+\bigO{p^{n}} \tag{by Lemma~\ref{lemma:independenetSidesNew}}\\
 			&= \sum_{s=1}^{n-1}\E^{[0,s+1]}(\# \text{ resamples in }[s]|[s]\in \mathcal{P})\left(\P^{[s+1,N]}(\NZ{s+1})\right)^2/(1+p)^2+\bigO{p^{n}} \tag{by Corollary~\ref{cor:probIndepNew}}\\
 			&= \sum_{s=1}^{n-1}\E^{[-N,N]}(\# \text{ resamples in }[s]|[s]\in \mathcal{P})+\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}, Corollary~\ref{cor:probIndepNew}}\\
 			&= \sum_{s=1}^{N}\E^{[-N,N]}(\# \text{ resamples in }[s]|[s]\in \mathcal{P})+\bigO{p^{n}}.
 			%&= \sum_{k=1}^{\infty}\sum_{\underset{\ell\geq r-1}{\ell,r\in[n]}}\P^{(n)}(\Res{1}\geq 1\,\&\, [\ell+1,r-1]\in\mathcal{P}) \tag{partition}\\
 			%&= \sum_{k=1}^{\infty}\sum_{\underset{\ell\geq r}{\ell,r\in[n]}}\P^{(n)}(\Res{1}\geq 1\,\&\, [\ell+1,r-1]\in\mathcal{P})  +\bigO{p^{n}} \\
 			%&= \sum_{k=1}^{\infty}\sum_{\underset{\ell\geq r}{\ell,r\in[n]}}\P^{[l,r]}_{b_{\ell}=b_{r}=1}(\Res{1}\geq 1\,\&\, [\ell+1,r-1]\in\mathcal{P}) \P^{[r,\ell]}(\NZ{\ell,r}) +\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}}\\
 			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}\P^{(n)}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) \tag{partition}\\
 			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}^{|P|<n}\P^{(n)}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) +\bigO{p^{n}}\\
 			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}^{|P|<n}\P^{[P\cup \partial P]}_{b_{\partial P}=1}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) \P^{[\overline{P}]}(\NZ{\partial P}) +\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}}\\
 			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}^{|P|<n}\P^{[P\cup \partial P]}_{b_{\partial P}=1}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) \left(\left(\P^{[|\overline{P}|]}(\NZ{1})\right)^2+\bigO{p^{|\overline{P}|}}\right) +\bigO{p^{n}} \tag{by Lemma~\ref{lemma:independenetSidesNew}}\\
 			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}^{|P|<n}\P^{[P\cup \partial P]}_{b_{\partial P}=1}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) \left(\left(\P^{[N]}(\NZ{1})\right)^2+\bigO{p^{|\overline{P}|}}\right) +\bigO{p^{n}} \tag{by Corollary~\ref{cor:probIndepNew}}\\
 			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}^{|P|<n}\P^{[-N,N]}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) +\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}}\\
 			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}\P^{[-N,N]}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) +\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}}\\
 			&= \E^{[-N,N]}(\Res{1})+\bigO{p^{n}}.
 		\end{align*}
 		Repeating the same calculation with $m$, and comparing the two expressions completes the proof.

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