# Tagged Questions

Stochastic processes (with either discrete or continuous time dependence) on a discrete (finite or countably infinite) state space in which the distribution of the next state depends only on the current state. For Markov processes on continuous state spaces please use (markov-process) instead.

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### Proof of “strong law of large numbers” in Markov Chains

I have been given a theorem stating an analogue of the strong law of large numbers for Markov chains. It states that if $X=(X_n)_{n\in\mathbb{N}}$ is a Markov chain with transition matrix $p$ and $\pi$...
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### Eigenvectors of transition matrices in PageRank algorithm

In my probability course, we were discussing applications of Markov Chains to computer science -- in particular, how the PageRank algorithm goes about finding stationary distributions, and thus, ranks ...
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### Probability of a time-dependent set of states in Markov chain

Consider a Markov matrix $P$ defining $m$ states. For each time $n$, define a set of states $S_n$ that contains a predefined subset of the states $\left\{ {1,...,m} \right\}$. For time $n=k$, I would ...
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### Mixing time for metropolis chain on graph coloring

I'm reading the Markov Chains and Mixing Times by David Levin et al.. In section 5.4 page 71 a proof is given for a bound of mixing time for the Metropolis Chain on graph coloring. In the proof, such ...
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### Existence of limiting distribution of product of Markov chains

I have two Markov chains described by the stochastic matrices $P_1$ and $P_2$ for which a limiting distribution exists. Now I combine the two stochastic matrices using the cartesian product, this is ...
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### Rational Thief Problem, optimal stopping strategy

A thief goes out stealing every day and has a chance of $p_j$ of stealing a sum $j$ with $0\leq j \leq N$. But there's also a chance $p$ of getting caught, in which case he loses everything he got ...
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### Equilibrium distribution of Ehrenfest's urn

(I'll post my own answer to this, but others may be of interest, so post your own if you have one.) (PS: In reply to comments posted below: Stackexchange encourages posting an answer to one's own ...
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### What is the state space of this markov chain?

Consider a system where two persons sit at a table and share three books. At any point in time both are reading a book, and one book is left on the table. When a person finishes reading his/her ...
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### How to define a transition matrix mathematically?

I'm writing my master thesis. Given the adjacency matrix of a graph, I need to define the transition matrix formally. I'm not able to figure out how to define it in mathematical notation. Can you help ...
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### Can a chain with repeated nodes still be considered a Markov chain?

The well-known Markov Property is that $$P(X_n = i | X_{n-1} = k_1, \dots, X_{n-j} = k_n ) = P(X_n = i | X_{n-1} = k_1)$$ Suppose we lay out some stochastic model in the following transition ...
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### Markov chain steady state existence

Is it possible for a Markov chain to have no steady state solution ? What is an example of such system ?
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### Harmonic functions and null recurrence

Let $(X_n)_{n \geq 0}$ be a irreducible Markov chain defined on a countable state space $S$. It is known some ways to figure out if this chain is recurrent or not looking for superharmonic functions ...
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### Robustness of Markov Chains

A Markov Chain on a measurable space $X$ is uniquely determined by a stochastic kernel $P$ on $X$. Let $\mathsf P_x$ denote the probability on paths generated by $P$ and the initial condition $x\in X$....
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### Expected number of lines in use in call centre (markov process: queuing theory)

Suppose we have a call centre with infinitely many lines to be able to call to. Calls come in a rate of $\lambda$ and customers are served with rate $\mu$. It is easy to see that the $Q$-matris looks ...
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### How to solve a discrete SIR epidemic model?

Let $(S(t), I(t), R(t))$ be a continuous time Markov chain SIR model with discrete space, where $S(t)$ stands for the number of susceptible people at time $t$; $I(t)$ stands for the number of ...
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### Induced Markov chain - verify Markov property and another property

First, here is how we defined induced Markov chains: Suppose that $(X,E,P)$ is an irreducible Markov chain, where $X=(X_i)_{i\in\mathbb{N}_0}$, $E$ is the state space and $P=(p_{i,j})_{i,j\in E}$ is ...
Let $(X_n)_{n\in\mathbb{N}_0}$ be a Markov chain with state space $E$ and transition matrix $P=(p_{i,j})_{i,j\in E}$. A real valued function $h$ on $E$ is called superharmonic if $h(x)\geq Ph(x)$ ...