The markov-chains tag has no usage guidance.

**7**

votes

**1**answer

117 views

### Equalizing Geometric means of Graph Cycles

Consider a strongly connected directed graph $G$. I have been stuck on the following question: can you assign real numbers in $[0,1]$ to each edge of $G$ so that the geometric mean of all cycles are ...

**0**

votes

**0**answers

26 views

### positive Harris recurrent, aperiodic, stationary Markov chain

How to proof that every positive Harris recurrent, aperiodic, stationary Markov chain is alpha-mixing (strong-mixing)?

**-1**

votes

**0**answers

14 views

### Under what conditions does a continuous-time Markov chain is also a Feller process? [on hold]

It is known that not all continuous-time Markov chains are Feller, but are there any discussions on the sufficient conditions? For example, is a continuous-time Markov chain with finite number of ...

**1**

vote

**0**answers

24 views

### Approximating logarithm of operator norm of a submatrix

This is a matrix approximation-theoretic question that arose in my study of Markov chains.
Let $\|B\|:=\sup_{v: \|v\|=1} \|Bv\|$ denote the operator norm of a matrix $B.$
Let $\sqrt{B}$ be the ...

**1**

vote

**1**answer

108 views

### Two types of random walkers on square lattice

Consider a two dimensional square lattice ($n$ by $n$), which is our space $S$ (each point labelled by an index $1\to n^2$), containing two types of particles, distinguished here by either an index ...

**3**

votes

**1**answer

93 views

### Markov-semigroup sobolev inequality

I have a question about the following definition:
A probability measure $\mu$, such that the Markov semi-group $e^{Lt} \in L(L^2)$ exists and is symmetric, satisfies the Sobolev inequality iff for ...

**0**

votes

**0**answers

28 views

### maximum for a nonstationary Markov Chain

Let $\{X_j\}$ be a nonstationary Markov Chain with transition matrix $P$.
What is a relation between
$P(M_{1:n} \leq x)\quad \text{and}\quad P(M_{1:T} \leq x), \quad P(M_{T:n} \leq x)$
where $M_{k:n} ...

**3**

votes

**0**answers

45 views

### How much larger than the relaxation time can the mixing time be?

The notation is mostly taken from the book "Markov chains and mixing times" by Levin, Peres, and Wilmer.
Consider an irreducible, aperiodic, time-reversible, discrete-time Markov chain on a finite ...

**1**

vote

**1**answer

61 views

### General Markov Chains on same Probability Space?

A Markov chain $(X_i)_{i\in \mathbb{N}}$ on a measurable space $(E,\Sigma)$ is (see e.g. Revuz or Meyn/Tweedie) constructed on the following probabilty space.
$$ \Omega = \{ (x_l)_{l \in \mathbb{N}} ...

**0**

votes

**0**answers

37 views

### Validating a probability density distribution forecast model for a Markov process

Let's say we have a Markov process $X_t$, and we come up with a forecast model that takes some information from outside world and says: "value $X_{t+1}$ has probability density distribution $P_t(x)$". ...

**3**

votes

**0**answers

49 views

### Variance of a functional of transition probabilities using spectral gap of Markov chain

Consider an irreducible, aperiodic, time-reversible, discrete-time Markov chain on a finite state space $S$ whose Markov kernel is $K$ and unique stationary distribution is $\pi.$ Then, reversibility ...

**0**

votes

**0**answers

43 views

### Markov chain matching local time

Let $\left(X_{t}\right)_{t\geq0}$ be a Markov process taking values in
a finite state space $E$. Its local time at $y\in E$ started at
$x\in E$ is defined as
$$
...

**3**

votes

**0**answers

92 views

### 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 ...

**0**

votes

**0**answers

38 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 ...

**0**

votes

**0**answers

38 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 ...

**2**

votes

**0**answers

66 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 $\mu_{i,j}$. ...

**2**

votes

**1**answer

139 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

124 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 ...

**0**

votes

**0**answers

66 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. ...

**1**

vote

**0**answers

40 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 ...

**1**

vote

**0**answers

49 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 ...

**0**

votes

**0**answers

48 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 ...

**2**

votes

**0**answers

46 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 ...

**30**

votes

**6**answers

2k 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 ...

**1**

vote

**1**answer

136 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) ...

**1**

vote

**0**answers

76 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) ...

**1**

vote

**0**answers

100 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 ...

**0**

votes

**0**answers

44 views

### Reference request for specific POMDP examples

Following is strictly for discrete-time discrete-space Markov chain.
Consider a partially observed Markov decision process (POMDP) $P = \{X,O,A,P,B_a\}$.
Here $X = \{x_1, \cdots, x_n\}$ refers to ...

**1**

vote

**0**answers

66 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 $\epsilon_0$, ...

**1**

vote

**0**answers

171 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 ...

**1**

vote

**0**answers

31 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 ...

**4**

votes

**1**answer

229 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 ...

**5**

votes

**1**answer

421 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 ...

**3**

votes

**0**answers

118 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 ...

**4**

votes

**0**answers

133 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' ...

**1**

vote

**1**answer

136 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 ...

**0**

votes

**1**answer

170 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 ...

**1**

vote

**0**answers

35 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 ...

**0**

votes

**1**answer

139 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 ...

**1**

vote

**0**answers

86 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 ...

**2**

votes

**1**answer

226 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 ...

**1**

vote

**1**answer

174 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 ...

**1**

vote

**1**answer

198 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
...

**0**

votes

**1**answer

81 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 ...

**0**

votes

**1**answer

52 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

**5**

votes

**0**answers

167 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 ...

**1**

vote

**0**answers

69 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 ...

**3**

votes

**1**answer

167 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 ...

**1**

vote

**1**answer

114 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 ...

**1**

vote

**0**answers

77 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 ...