# Questions tagged [markov-chains]

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For the laws of two pure-jump Markov processes $\mu_1$ and $\mu_2$ on $\mathbb R^n$, which generators are $H_1f(x)=\int h(x,dy) (f(y)-f(x))$ and $H_2f(x)=\int e^{-g(x,y)} h(x,dy) (f(y)-f(x))$ (...
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### Eigenspace of Gaussian Markov operator

Consider the (one-dimensional) Gaussian distribution $Q := N(\nu,\tau^2)$ and the (Gaussian) Markov operator \begin{equation*} \begin{array}{rccc} R : & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) &...
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### Distribution of non-overlapping words in randomly generated text

The question can be described in the following way: Suppose I have a finite language $\mathcal{L}$ over alphabet $\Sigma$. I have a string that is composed of a concatenated series of $n$ instances ...
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### Reduce the asymptotic variance for a class of Metropolis-Hasting estimates

I'm running the Metropolis-Hastings algorithm with state space $E$, target distribution $\mu=p\lambda$ and proposal kernel $Q$ to estimate $\mu(hf)$ for a fixed function $f:E\to[0,\infty)^3$ and a ...
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### Show a Poincaré inequality for a Markov kernel and minimize the Poincaré constant

Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$ (see definitions ...
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### Minimizing the rate of geometric ergodicity of a Metropolis-Hastings kernel depending on a parameter

Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$. I want to ...
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### Is there any equivalent Foster-Lyapunov Theorem where expected Lyapunov drift is negative though not uniformly upper bounded by a negative number

I have a process $\{X_t\}_t$, where $x_t$ in $[0,1]$, and a Lyapunov function $V(x)=x$ such that the drift $$E[X_{t+1}|x_t] - x_t < -k(1-x_t)$$ for $x_t \in (1-\epsilon,1]$, where $k>0$. Thus ...
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### Does Chu and Hough's solution to the mixing time of the 15-puzzle carry over to the Rubik's cube?

In his 1988 book Group Representations in Probability and Statistics , Diaconis considers mixing times of the 15-puzzle. He states: Here is a simplified version: Consider the blank as a $16$th ...
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### Maximize an $L^p$-functional subject to a set of constraints

Let $(E,\mathcal E,\lambda)$ and $(E',\mathcal E',\lambda')$ be measure spaces $f\in L^2(\lambda)$ $I$ be a finite nonempty set $\varphi_i:E'\to E$ be bijective $(\mathcal E',\mathcal E)$-measurable ...
Consider a $p$ dimensional random variable with a discrete support. Consider a Markov transition kernel on the state space that is defined in terms of element-wise transition distributions. One can ...