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b/stopping.tex index 989bb9e..aa13c9a 100644 --- a/stopping.tex +++ b/stopping.tex @@ -33,7 +33,7 @@ P=\dfrac{1}{6} \left( 0&0&0&0&1&0&4&1 \\ 0&0&0&1&0&1&0&4 \end{array} -\right) +\right). \] \end{xpl} @@ -69,9 +69,13 @@ and $$t_{\rm mix}(\varepsilon)=\min\{t \mid d(t)\leq \varepsilon\}.$$ -Intuitively speaking, $t_{\rm mix}$ is a mixing time -\textit{i.e.}, is the time until the matrix $X$ of a Markov chain -is $\epsilon$-close to a stationary distribution. +%% Intuitively speaking, $t_{\rm mix}$ is a mixing time +%% \textit{i.e.}, is the time until the matrix $X$ of a Markov chain +%% is $\epsilon$-close to a stationary distribution. + +Intutively speaking, $t_{\rm mix}(\varepsilon)$ is the time/steps required +to be sure to be $\varepsilon$-close to the staionary distribution, wherever +the chain starts. @@ -115,7 +119,7 @@ $$\P_X(X_\tau=Y)=\pi(Y).$$ A stopping time $\tau$ is a {\emph strong stationary time} if $X_{\tau}$ is -independent of $\tau$. +independent of $\tau$. The following result will be useful~\cite[Proposition~6.10]{LevinPeresWilmer2006}, \begin{thrm}\label{thm-sst} @@ -231,7 +235,8 @@ This probability is independent of the value of the other bits. Moving next in the chain, at each step, the $l$-th bit is switched from $0$ to $1$ or from $1$ to $0$ each time with the same probability. Therefore, for $t\geq \tau_\ell$, the -$\ell$-th bit of $X_t$ is $0$ or $1$ with the same probability, proving the +$\ell$-th bit of $X_t$ is $0$ or $1$ with the same probability, and +independently of the value of the other bits, proving the lemma.\end{proof} \begin{thrm} \label{prop:stop} @@ -345,7 +350,7 @@ direct application of lemma~\ref{prop:lambda} and~\ref{lm:stopprime}. \end{proof} Now using Markov Inequality, one has $\P_X(\tau > t)\leq \frac{E[\tau]}{t}$. -With $t=32N^2+16N\ln (N+1)$, one obtains: $\P_X(\tau > t)\leq \frac{1}{4}$. +With $t_n=32N^2+16N\ln (N+1)$, one obtains: $\P_X(\tau > t_n)\leq \frac{1}{4}$. Therefore, using the defintion of $t_{\rm mix)}$ and Theorem~\ref{thm-sst}, it follows that $t_{\rm mix}\leq 32N^2+16N\ln (N+1)=O(N^2)$. @@ -354,11 +359,11 @@ $t_{\rm mix}\leq 32N^2+16N\ln (N+1)=O(N^2)$. Notice that the calculus of the stationary time upper bound is obtained under the following constraint: for each vertex in the $\mathsf{N}$-cube there are one ongoing arc and one outgoing arc that are removed. -The calculus does not consider (balanced) Hamiltonian cycles, which +The calculus doesn't consider (balanced) Hamiltonian cycles, which are more regular and more binding than this constraint. Moreover, the bound -is obtained using Markov Inequality which is frequently coarse. For the -classical random walkin the $\mathsf{N}$-cube, without removing any +is obtained using the coarse Markov Inequality. For the +classical (lazzy) random walk the $\mathsf{N}$-cube, without removing any Hamiltonian cylce, the mixing time is in $\Theta(N\ln N)$. We conjecture that in our context, the mixing time is also in $\Theta(N\ln N)$. -- 2.39.5

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