# Questions tagged [markov-chains]

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### 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 two distinct integers $1\le i <...

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### 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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35 views

### Distribution of a linear pure-birth process's integral

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

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14 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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205 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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77 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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75 views

### Joint drunkard walks

The drunkard walk is a game where two players have $a$ and $b$ dollars, respectively, and they play a series of fair games (both risking one dollar in each game) until one of them goes broke.
My ...

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65 views

### Show convergence of a sequence of resolvent operators

Let
$E$ be a locally compact separable metric space
$(\mathcal D(A),A)$ be the generator of a strongly continuous contraction semigroup on $C_0(E)$
$E_n$ be a metric space for $n\in\mathbb N$
$(\...

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79 views

### Lower bounds on discrete time finite Markov chains hitting probabilities

I am interested in some general theorems related to lower bounds on discrete time finite Markov chains hitting probabilities (preferably ergodic chains , but not necessarily ), with references . ...

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29 views

### Must this upper bound on mixing time depend on the minimum stationary probability?

It is known fact that for a finite-state, reversible and ergodic Markov chain with transition matrix $M$, the following control on the mixing time holds
$$\left( \frac{1}{\gamma_\star - 1}\right)\ln{...

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83 views

### Total offspring of Poisson multitype branching process

A normal branching process $Z_n$ initialized with $Z_0=1$ and offspring generated from $Pois(p),p<1,$ has a total progeny / total off spring distribution
$$X=\sum_{n=0}^\infty Z_n$$
$X\in \mathbb{...

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85 views

### Comparing mixing time of lazy and non-lazy Markov chains

Suppose we have a probability distribution $\pi : X \rightarrow [0,1]$ where $X$ is finite and let $Q : X \times X \rightarrow [0,1]$ be a Markov kernel that is reversible with respect to $\pi$. That ...

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177 views

### Obtaining generator matrix and first-passage time distribution for CTMC?

Setup:
I have a model of a biological process described by two ODEs as follows:
$$\dot{X_1} = (\beta_1-d-1)X_1 + 2X_1^2 - X_1^3 + dX_2$$
$$\dot{X_2} = (\beta_2-d-1)X_2 + 2X_2^2 - X_2^3 + dX_1$$
I ...

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202 views

### Expectation of a linear operator

We define $T: C[0,1]\to C[0,1]\ni T(f(x))= \sum\limits_{k=1}^{m} p_k (f\circ f_k)(x):=\mathbb E( f(X_{n+1}|X_n=x)$ for a system $X_{n+1}=f_{\omega_n}(X_n), n=0,1,2\dots.$ and $\omega_n$ are i.i.d ...

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### If $X^n$ is a sequence of càdlàg processes whose FDDs converge to a continous process $X$, does $X^n$ converge to $X$ in the Skorohod topology?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a complete locally compact separable metric space, $(X^n_t)_{t\ge0}$ be an $E$-valued càdlàg process on $(\Omega,\mathcal A,\...