The markov-chains tag has no usage guidance.

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### The spring Markov chain on $\mathbb{N}$

I'm trying to understand and learn more about "almost surely bounded" Markov chains on countable state spaces. I'm looking for references where I can learn how to work with more complicated examples ...

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39 views

### The effect of a single Markov transition on fidelity

Let $p$ and $q$ be two probability vectors of length $n$.
The fidelity (or Bhattacharyya coefficient) of $p$ and $q$, is
$$
F(p,q) \ := \ \sum_{i=1}^n \sqrt{p_i \cdot q_i}.
$$
Let $A$ be a ...

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44 views

### Ergodicity property for continuous-time Harris positive Markov process

I have posted this question on there, but got no answer.
The following theorem is Theorem 13.3.3 of Meyn and Tweedie's Markov Chains and Stochastic Stability on page 328:
Theorem 13.3.3. If $\Phi$...

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**0**answers

67 views

### Customers and Anti-Customer Queueing Problem: What is the Customer delete probability

Hello may I ask for your help?
First the setting:
I have got a problem with some queueing theory. The whole problem would be a grid of nodes, all nodes have an operation intensity $\mu_{i,j}$. ...

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votes

**1**answer

200 views

### Stationary distribution for time-inhomogeneous Markov process

I have a two state, discrete time, time-inhomogeneous Markov process with transition matrix defined by
$$T_i=\begin{pmatrix}
1-p_i\alpha & p_i\alpha \\
p_i\beta& 1-p_i\beta
\end{pmatrix}$$
...

**1**

vote

**1**answer

127 views

### Row-stochasticity of the Jacobian matrix of a stationary distribution

Let $P_{\mathbf{p}}$ be a $n \times n$ row-stochastic matrix whose entries are a function of a probability vector $\mathbf{p} \in \mathbf{R}_{> 0}^n$, $\sum_i p_i = 1$ and define the following ...

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75 views

### Markov chains on a polyhedron

A modification of a question from Gerard Letac (1976): A m-sided q-adjacent-faced polyhedron has one of its faces "up." Each round, the polyhedron rolls so that any of the adjacent faces is now up. ...

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48 views

### Oscillating Markovprocess Transition Probabilities

Suppose we have an irreducible positive-recurrent Markov process $\{X(t), t\geq0\}$ with generator $G$. Let $P(t)$ be its transition probability matrix and $\pi$ its stationary distribution. Then we ...

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69 views

### Link Between Birkhoff Ergodic Theorem and Strong LLN for Harris Recurrent Markov chain

Is it possible to derive strong law of large numbers for a Harris recurrent stationary Markov chain form Birkhoff Ergodic Theorem? As I know that there is a link between SLLN for iid sample and ...

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54 views

### Strong Markov Property of the joint process $(B_t,L_t)_{t\ge 0}$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion and $L=(L_t)_{t\ge 0}$ be its local time in zero. Given two strictly increasing functions $\phi_1$, $\phi_2: \mathbb R_+\to\mathbb R$ such that $\phi_1(0)=\...

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52 views

### Mappings between adaptive networks and Markov processes

Are there any known mappings between adaptive networks models (i.e. graph model representations of networks where the internal vertex dynamics and connectivity topology can change subject to specific ...

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votes

**7**answers

4k views

### Deep Learning / Deep neural nets for mathematician

I am interested in finding out the math ideas behind the technologies that are under the umbrella of "Deep Learning" or "Deep neural nets".
Most of the papers/books that are often quoted in papers/...

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**1**answer

156 views

### Variation of Markov Chain Convergence Theorem

Assume the chain $\{X_n\}_{n\in\mathbb{N}}$ on the statespace $(S,\mathcal{F})$ (we may assume it is countable) is aperiodic, irreducible and positive recurrent. We denote with $\pi$ its (unique) ...

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81 views

### How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...

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169 views

### Spectral radius of a column stochastic matrix perturbed by a rank-1 matrix

$P\in \mathbb{R}^{n\times n}$ is an irreducible column stochastic matrix. $P$ is also diagonally dominant. $w \in \mathbb{R}^{n} $ is a strictly positive vector satisfying $w^T \mathbf{1} = 1$ where $\...

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91 views

### Theorems on stochastic Lyapunov function

Let $X_n$ be a sequence of random variables such that
$$P(X_{n+1} \in A|X_m,x_m,m\leq n)= \int_A p(dw|X_n,x_n)$$
It is called a controlled Markov process.
Now, suppose there exist $\epsilon_0$, $...

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238 views

### Is the stationary distribution of this Markov chain uniform?

First, a little bit of background: Since 2012, Canada has decided to phase out the penny for its coinage system. Product prices may still use arbitrary cents, especially since prices do not typically ...

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49 views

### Effects of merging states on the limiting distribution of a Markov Chain

Consider a discrete time, homogeneous, finite state Markov chain given by a stochastic $n\times n$ matrix $M$.
We also have a cost vector $w$ of size $n$ with non-negative integer costs. The cost of ...

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votes

**2**answers

306 views

### Deterministic finite-state automaton driven by a Markov chain

I've stumbled on some problem, and I have the feeling that this is closed to something well-studied in dynamical systems. The problem is the following. Consider a finite-state automaton with state ...

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**2**answers

610 views

### Can ergodic theory help to prove ergodicity of general Markov chain?

I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking ...

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**0**answers

128 views

### Worst-Case Solution to (Stochastic) Matrix Inequality

EDIT: Some specific conjectures added.
This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...

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votes

**1**answer

226 views

### minimal polynomial for a graph

I wonder if there is any result relating the degree $d$ of the minimal polynomial of a directed finite graph to any of its topological features - such as its diameter, or any other similar 'natural' ...

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**1**answer

171 views

### Mixing time of a continuous time Markov chain with arbitrary rate matrix

I would like to calculate the mixing time of a continuous time starting from the rate matrix and not necessarily assuming that the time in between jumps have rate 1 - all I have is the (finite ...

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**1**answer

172 views

### On the inverse problem of Dobrushin

Dobrushin, in this paper, looked into the following problem. Suppose We are given a Markov kernel (conditional distribution) $P_{Y|X}$. Information theorist usually call $W$ a channel. It is known ...

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37 views

### Is there an effective algorithm for finding “minimal discovery times” for large graphs?

Consider a large, probably sparse graph with Markovian random walkers on it.
Define the discovery time as the expected time to first reach a vertex by
random walk from a uniform start. Are there ...

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votes

**1**answer

230 views

### Does a irreducible set of states necessarily need to be closed in a Markov chain?

I have come across two different definitions for a 'irreducible set of states' of a Markov chain.
Definition 1: A subset of states $A$ of a Markov chain is irreducible if it is possible to access (...

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88 views

### Conditional probabilities in epidemic model

I was contemplating an epidemic model where infection and recovery rates are determined by links. Here node $i$ is infected first and recovers at a rate $\mu_i$. For all other nodes, the recovery is ...

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votes

**1**answer

260 views

### Stationary distribution of Markov chain

Suppose I have a discrete time Markov chain $\boldsymbol{X}$ with state space $\mathbb{R}^+$. The chain is $\psi$-irreducible, aperiodic, atomless and has an invariant measure $\pi$.
If $\pi$ is ...

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**1**answer

181 views

### Can ergodic theorem be used here [closed]

Suppose I have an ergodic Markov Chain $\{X_n\}$ where $X_n$ are bounded. Now, Can I say anything on the limit
$$ \lim_{n\to\infty} \frac{1}{n}\ln E\left[e^{\sum_{i=0}^{n} X_i}\right]$$
I don't ...

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**1**answer

203 views

### Double Markovity

Suppose we have a double Markov relation for three random variables $X$, $Y$ and $W$ as follows
$$X\to W\to Y,$$ and $$X\to Y\to W.$$
How to prove that there exist functions $f$ and $g$ such that
$$...

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**1**answer

92 views

### Constructing a transition matrix of a time-homogeneous, finite Markov chain with full support stationary distribution

is there a way to construct a transition matrix of a time-homogeneous, finite Markov chain such that the stationary distribution always has full support (this is equivalent to all states of the chain ...

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**1**answer

59 views

### DTMC random walk model [closed]

For a discrete Markov chain random walk with p < 0.5 with state space S= {0,1,2..}
What is the stationary distribution?
I could use any help.
Thank you

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188 views

### Maximal inequalities for square of partial sums

Let $S_n = \sum_{i \leq n} X_i$ be the partial sums of a nice sequence of random variables $X_i$. In my application, $X_i$ is a functional of a finite-state, irreducible, aperiodic Markov chain, so ...

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78 views

### Simultaneous multiple perturbations in Markov chain Monte Carlo

I'm coding a McMC algorithm for geophysical applications.
Using the Metropolis-Hastings scheme to accept/reject the proposed models is smth that i thought i completely understood, but i don't. To be ...

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votes

**1**answer

182 views

### Approximating Markov chains by Brownian motion

I would like a result along the following lines to be true, but haven't been able to locate it in the literature; pointers would be welcome.
Let $X_t$ be a finite-state, irreducible, aperiodic Markov ...

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**1**answer

117 views

### N random walkers that hit node v in a graph

Consider a finite, undirected graph G, with uniform edge weights. Assume that there are n number of random walkers that will start at different nodes (lets say n=3, hence the random walkers will start ...

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94 views

### Nonstationary Markov chain maximal inequality

Let $X_i$ be a (finite-state, irreducible, aperiodic) Markov chain, not necessarily stationary. (That is, it doesn't start from the invariant distribution; I'm happy to have it be time-homogeneous if ...

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642 views

### random walk with reflecting barriers [closed]

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are (same ...

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**1**answer

193 views

### Gibbs sampler with linear constraints

My problem concerns the estimation of truncated multivariate normal distributions under constraints.
Let $X_1$ and $X_2$ two random variables following normal distributions $\mathcal{N_1}(m_1,\...

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votes

**3**answers

1k views

### How to explain “Feller process” to an undergraduate student?

I had to explain in informal terms what a Feller process was, to undergraduate students who understand Markov property, Poisson processes and such. It was easy to define Levy process as generalisation ...

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votes

**1**answer

235 views

### First Collision Time for k Random Walkers on a Torus

I consider $k$ random walkers on $\mathbb{Z}^{d}/n \mathbb{Z}^{d}$, the $d$-dimensional torus of side length $n$. More precisely, I will define a Markov chain $Z_{t} = (X_{t}[1], \ldots, X_{t}[k])$ ...

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51 views

### Markov decision processes: action set revealed at point of decision

I have a problem which looks like a finite horizon Markov decision process (MDP), except the action space at each time is revealed at the decision making point. There is no way to know before hand the ...

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**2**answers

572 views

### References for a physicist migrating to stochastic processes

I've studied "Markov Chains" - Norris and "Measure, Integral and Probability" - Capinski, Kopp. Now, I'm looking for a couple of books (or other references) that help me bridging these two topics. ...

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102 views

### Mixing time for dimers on the square-octagon graph

Consider the "fortress graph" of order $n$ (see Figure 9 of http://faculty.uml.edu/jpropp/tiling/www/mdblum/arctic.html). It's been known empirically for twenty years that if one turns the set of ...

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**3**answers

5k views

### Time-inhomogeneous Markov Chains

I'm trying to find out what is known about time-inhomogeneous ergodic Markov Chains where the transition matrix can vary over time. All textbooks and lecture notes I could find initially introduce ...

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**1**answer

128 views

### Monte Carlo estimator with autocorrelated samples

Given an integration problem $I=\int{f(x)dx}$, we can construct an ordinary Monte Carlo estimator as
$E[I]=\sum\limits_i\frac{f(x_i)}{p(x_i)}$
where the samples $x_i$ are usually i.i.d. and drawn ...

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**1**answer

193 views

### Local Markov implies global Markov

Let $G=(V,E)$ be a finite simple graph, and let $\{X_i\}_{i \in V}$ be a collection of random variables associated with the vertices of $G$. The joint distributions of these r.v.s is a Markov Random ...

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194 views

### Is any invariant, ergodic measure with full support on an irreducible Markov shift a Markov measure?

I have this question I have been struggling with for a while. It seems rather intuitive, however, I was not able to proof it yet:
Let $\Omega = \{1,2,\cdots,N\}$ a finite alphabet, $\Sigma \subset \...

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**3**answers

374 views

### Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation
\begin{equation}
dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0,
\end{equation}
where $b,\...

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**1**answer

212 views

### convergence rate of occupation measure of ergodic Markov Chain

Given an ergodic Markov chain $(X_n)_{n\geq 1}$ in $R^d$with $\pi$ as the invariant distribution of the transition kernel, under good conditions we have that the empirical occupation measure converges ...