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      Questions tagged [markov-chains]

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      Distribution of markov chain with a stopping time

      I have a Markov chain $X_0, X_1, ..., X_\tau$, where $X_0$ is sampled from the stationary distribution and $\tau$ is a stopping time. Is it true that for any fixed $k$, $X_k$, given that $k \leq \tau$,...
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      Why do we assume that $\mathcal{A}$ is an algebra in this 2003 paper of Bobkov and Tetali?

      In the following paper (extended version here), at the beginning of section 3, the authors give two axioms about $\mathcal{A}$. Axiom 1 is about $\mathcal{A}$ being an algebra. I do not see where this ...
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      Convexity of conditional relative entropy for Markov distributions

      Consider two Markov processes $p$ and $q$. The conditional relative entropy between them is \begin{align} D(p\parallel q)& =\sum_a p(a)\sum_b p(b\mid a)\log\frac{p(b\mid a)}{q(b\mid a)}\\ & =\...
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      147 views

      Markov chain and random iteration of functions

      Could anyone give me some references for the convergence rate( assuming having a stationary distribution) of such Markov chain arising from the random iteration of a FINITE number of Lipschitz ...
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      187 views

      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 ...
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      1answer
      159 views

      Existence and uniqueness of a stationary measure

      This same question was also posted on MSE https://math.stackexchange.com/questions/3327007/existence-and-uniqueness-of-a-stationary-measure. Recently I have posted the following question on MO ...
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      88 views

      Characterizing the relationship between element-wise Markov transitions and the full-conditionals of the stationary distribution

      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 ...
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      107 views

      Can we show that $\mathbb R^{\mathbb N}\ni x\mapsto\bigotimes_{n\in\mathbb N}\mathcal N_{x,\:\sigma^2}$ is a Markov kernel?

      Let $\sigma>0$ and $\mathcal N_{x,\:\sigma^2}$ denote the normal distribution with mean $x\in\mathbb R$ and variance $\sigma^2$. From the Ionescu-Tulcea theorem, we know that $$\kappa(x,\;\cdot\;):=...
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      4answers
      207 views

      Probability of traversing all other states and finally landing on one state

      This is a cross-post from math.stackexchange.com. There has been no response there. Given a Markov chain of finite states with constant transition probabilities, what is the method to compute the ...
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      243 views

      Stationary distribution of a Markov process defined on the space of permutations

      Let $S$ be the set of $n!$ permutations of the first $n$ integers. Let $p\in(0,1)$. Consider the Markov Process defined on the elements of $S$. Let $x\in S$. Choose $1\le i <i+1 < n$ uniformly ...
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      62 views

      A closed form of mean-field equations

      Assume that a system at time t, for example number of costumers in a line at time $t$ which is denoted by $q(t)$, follows a Markov chain with these dynamics (probabilities) $$P(q(t+\Delta t)-q(t)=1)=\...
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      69 views

      Distribution of a linear pure-birth process' integral

      I stumbled across the following random variable, defined as the integral of a linear pure-birth process i.e. a Yule process: $$ Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k , Y_0=1\bigg] $$ ...
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      17 views

      Simulating an inhomogeneous DTMC with transition matrix dependent on Xn

      I am simulating an agent navigating through space, where the agent's navigation strategy changes over time as a Markov chain with transition probabilities dependent on its position in space. ...
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      2answers
      223 views

      Slowest initial state for convergence of finite birth-and-death Markov chains

      Consider the continuous-time birth-and-death Markov chain on $\{1,\cdots,n\}$ with all rates equal to $1$. Is it true that the convergence to equilibrium, in total variation distance, is slowest when ...
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      1answer
      84 views

      Reference request: When is the variance in the central limit theorem for Markov chains positive?

      I'm looking for a reference which gives sufficient conditions for the variance to be positive in the central limit theorem for Markov chains (cf https://en.wikipedia.org/wiki/...

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