The markov-chains tag has no usage guidance.

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### Convergence of sets

Let $E$ be a compact subset of $\mathbb{R}^n$. Let the density function $\phi(x,y)$ be Lipschitz continuous and such that
$$
\int\limits_E \phi(x,y)dy=1
$$
for all $x\in E$. Let us consider the ...

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### maximum variance unfolding

Consider positive weights $\pi_1, \ldots, \pi_n$ (one can suppose that they add up to $1$) and $n-1$ lengths $d_1, \ldots, d_{n-1}$.
Is there an analytical solution to the following problem:
find the ...

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643 views

### Exist closed forms of the distribution of return time in markov chains?

Hi, I am interested in the distribution of return times in simple random walks on finite graphs.
Let $G$ be a connected finite graph with, with two independent random walks. If both random walks ...

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673 views

### Counterexample Markov process

Let $X$ be a homogeneous Markov process in a continuous time with value in the set $E$. Suppose that for some $T>0,x\in E, A\subset E$ we have
$$
P_x[X_t\in A] = 0
$$
for all $t\in [0,T]$ but
$$
...

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446 views

### Reachability for Markov process

Let $X$ be a Markov process (in continuous or discrete time) and define an event
$$
R(T,A) = (\exists t\leq T: X_t \in A).
$$
I have seen in one paper that
$$
\Pr[R(\infty,A)] = \sup\limits_{\tau} ...

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### Statistics of a simple Markov chain

Imagine a two-state Markov chain which hops between the states $\pm 1$ with probability $p<1/2$, so that the autocorrelation function after $k$ steps is
$\rho_k = (2p-1)^k$
If I take an ...

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**1**answer

161 views

### scalar diffusions are reversible

It is well known that under mild assumptions a scalar diffusion $dX_t = a(X_t) dt + \sigma(X_t) dW_t$ with invariant probability distribution $\pi$ is reversible. This is indeed not true for ...

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### Bounds on $||P^{k+1} - P^k||$ for $n$ by $n$ stochastic matrix $P$ with trace $n-1$ and integer $k>>n$.

The problem:
We have a $n$-state Markov chain with arbitrary initial distribution and transition matrix $P$ that is arbitrary except that we know that $P$ has trace $n-1$. Of course $P$ is also a ...

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**1**answer

732 views

### A simple problem in markov chains

I'm trying to understand a 1954 paper of Kubo intitled "Note on the stochastic theory of resonance absorption". The specific problem can be stated mathematically as follows: let $X(t)$ be a random ...

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### Is there MDPs (Markow Decision Process) which have a non deterministic optimal policy ?

I'm working on Markov Decision Process and I have not found yet an example of MDP that has a stochastic (non deterministic) optimal policy. Is there MDPs that have a stochastic optimal policy or is it ...

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### Expectation of first positive value in random walk

Let $p$ be a parameter in $]0,1[$. Let $(X_k)_{k\geq 0}$ be an independent, identically distributed sequence of random variables, such that each $X_k$ takes values only in
$\lbrace -1, \frac{1-p}{p} ...

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**1**answer

102 views

### how to find a sequence of digits in base b such that each consecutive block of size k appears exactly once?

My question is most precisely stated in the title. As an example, if we consider base 10, and k=4, then I am asking, is it possible to have a sequence of length 10^4 + 3, such that each 4 digit number ...

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**1**answer

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### How to do integration using MCMC?

I want to evaluate $I = \int_V f(\vec{x}) d\vec{x}$. The classical Monte Carlo method is to sample uniformly from within the integration volume $V$, and then compute $I \approx V \frac{1}{N} ...

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### Potts model simulation

I was wondering what were the state-of-the-art methods to simulate low temperature configurations of Potts-like models that exhibit a discontinuous phase transition. For models with a continuous phase ...

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476 views

### Markov chain convergence problem.

Consider a markov chain matrix P of size n x n (n states).
P is known to be:
1- there are at least two absorbent states. one of them is denoted by null. (thus, we have that P_null,null = 1)
2- For ...

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### Convergence of a markov matrix

Consider a markov chain matrix P of size n x n (n states).
P is known to be:
1- Not irreducible (i.e. there exist at least a pair of states i, j such that we cannot go from i to j)
2- Not all ...

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**5**answers

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### Where to publish a paper on the Mafia game?

I wrote a research paper "A mathematical model of the Mafia game" (arXiv:1009.1031 [math.PR]). However, I do not know where to publish it. As an undergraduate studying majorly physics, I have little ...

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**1**answer

351 views

### Lower bound on the convergence rate of a specific Markov chain

I have a Markov chain $\mathbf{A} = (A_0, A_1, \ldots)$ with state space $\{0, \ldots, n\}$ which converges towards a stationary distribution $\pi$. There are a lot of well-known results on ...

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### (Stochastic) matrix for which a stochastic matrix logarithm exists?

I think this is basically the inverse question of Matrices whose exponential is stochastic.
i.e. what are sufficient conditions on the matrix representation of an evolution operator of a (finite) ...

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### representation theoretic interpretation of Jack polynomials

Monomial symmetric polynomials on $n$ variables $x_1, \ldots x_n$ form a natural basis of the space $\mathcal{S}_n$ of symmetric polynomials on $n$ variables and are defined by additive symmetrization ...

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### Diagonalizing some matrices arising from Fourier transform on $S_n$.

Consider the function $f$ on $S_n$ which equals $1/n$ on all adjacent transpositions $(i,i+1)$, where we let $n+1 = 1$, and $0$ otherwise, and its Fourier transform $\hat{f}(\rho)$ evaluated at the ...

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### Markov Chain Patterns

Hi
I would like to detect repetitive patterns and deviations from these repetitions. I have historical data and can calculate probabilities for the transitions between my many states. I have ...

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### Random walk is to diffusion as self-avoiding random walk is to …?

One can view a random walk as a discrete process whose continuous
analog is diffusion.
For example, discretizing the heat diffusion equation
(in both time and space) leads to random walks.
Is there a ...

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### Gromov's list of 7 constructions in differential topology

At the 2010 Clay Research Conference, Gromov explained that we know of only 7 different methods for constructing smooth manifolds. Working from memory, and hence not necessarily respecting the order ...

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**1**answer

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### initial condition of a diffusion approximation

I am trying to prove that a certain sequence of Markov chains $x^N_k$ converges towards a diffusion process. The invariant measure of $x^N$ is $\pi^N$ and the Markov chain $x^N$ is started in ...

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### Existence of Limiting Distribution for Moving Regions in Stat. Phys. Models

As the title (hopefully) suggests, I've been trying to prove (or disprove) the existence of a limiting distribution for a certain projection in a statistical physics model. I'll give the details of ...

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**1**answer

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### “Induced” arrivals in an M/M/1 queue?

I'm a newcomer to the realm of queueing theory, so please bear with me :)
I'd like to model web server traffic with a modified M/M/1 queue.
In the simple case we have two parameters - $\lambda$ for ...

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### When is a 1-block factor of a non-Markovian process Markov?

Let $Y$ be a discrete stationary stochastic process. Suppose that $Y$ is not $n$-step Markov for any positive integer $n$. Let $Z$ be a 1-block factor of $Y$. For what condition on $Y$ or the ...

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### Examples of Slowly Mixing Chains in Statistics

This should probably be community wiki, but I don't know how to set that myself.
I'm looking for examples or Markov chains that are used in statistics or statistical physics, and which are known to ...

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### Finitarily Markovian Finite Factors of Bernoulli Schemes

By processes, I mean discrete, stationary stochastic processes, that is $(X,\mathcal{U},\mu,T)$ where $X$ is the set of doubly infinite sequences of some alphabet $A$, $\mathcal{U}$ is the ...

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### Markov chain on Groups

Let $G$ be a permutation group on the finite set $\Omega$. Consider the Markov chain where you start with an element $\alpha \in \Omega$ chosen from some arbitrary starting probability distribution. ...

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### Is there a way to analytically compute the recurrence time of a finite Markov process?

Let $X_t$ be an ergodic (time-homogeneous) Markov process (in discrete or continuous time) on a finite state space $\{1,\dots,n\}$. Let $T(X_0)$ be the stopping time given by the infimum of times such ...

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315 views

### Where can I learn about master equation?

I am reading a paper by Dorogovstev on structure of growing complex networks with preferential linking. I need to learn master equation for this.
I need a reference for the same.

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### Efficiently sampling points from an integer lattice.

Let $\mathcal{L}$ = {x$\in$ $N^n$ : ||x||$_1$ $\leq$ m} denote the set of integer points in the positive orthant of the $\ell_1$ ball of radius $m$, where $m < n$. For each $x \in \mathcal{L}$, let ...

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### How can I generate random permutations of [n] with k cycles, where k is much larger than log n?

I've been thinking a lot lately about random permutations. It's well-known that the mean and variance of the number of cycles of a permutation chosen uniformly at random from Sn are both ...