The markov-chains tag has no usage guidance.

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### Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...

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### Markov Chain: Number of communicating classes of a power of the irreducible transition matrix [on hold]

Suppose $P$ is an irreducible transition matrix, with period $d$. Consider the transition matrix $P_k$. In terms of $d$ and $k$, how many communicating classes does $P_k$ have, and what is the period ...

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### Is there an easy way to convert a non-deterministic optimal policy to a deterministic optimal policy for a given MDP?

For a MDP (Markov Decision Process) is there an easy way to convert a non-deterministic optimal policy into a deterministic optimal policy?
The trivial way will take $O(|\mathcal{A}|^{|\mathcal{S}|}$...

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66 views

### Looking for an exposition of a certain theorem of Talagrand

The following is a theorem by Talagrand (as stated here, http://arxiv.org/pdf/1511.08609v1.pdf),
Let $(X, \mu)$ be a probability space. Let $F : X \rightarrow \{0,1\}$
be a family of functions ...

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### Uniqueness of invariant measure for equivalent transition probabilities

Suppose $P(x,dy)$ and $Q(x,dy)$ are two Markov transition kernels on a topological space $E$ equipped with Borel $\sigma$-algebra $\mathcal B(E)$. Suppose for every $x \in E$, $P(x,\cdot)$ and $Q(x, \...

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### 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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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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### The necessary sufficient condition for recurrence of a Markovian random walk

Suppose $\sigma_{1},\sigma_{2},...$are i.i.d random variables.$S_{0}=0$. Define $S_{n}=S_{0}+\sum_{i=1}^{n}\sigma_{i}$, then ${S_{n}}$ is a Markovian random walk.
I want to figure out the necessary ...

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### Dependency of the error term on the states, in the definition of the transition rates of a continuous time Markov chain

I think this is certainly not a research or graduate level question. But I didn't get any answer from math.stackexchange.com.
I'm studying G.F.Lawler's stochastic process book. There he defines the ...

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60 views

### How can I efficiently approximate the stationary distribution of an infinite CTMC with a sparse rate matrix?

I am looking for methods to approximate the stationary distribution of an infinite CTMC with a sparse rate matrix. Each row and column of the rate matrix has a finite number of non-zero elements. ...

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### Ising model - phase transition vs rapid mixing

Consider a graph $G=(V,E)$ and Ising model on that graph, i.e. configuration space is $\Omega=${$-1,+1$}$^V$ and energy of a configuration $s \in \Omega$ is given by:
$H(s) = -\beta \sum_{u \sim v}s(u)...

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### 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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### Solving a Rubik's cube via a series of randomly selected (quarter-turn) Singmaster moves

In July of 2010, Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge demonstrated (computationally) that a $3\times3\times3$ Rubik's cube, starting in an arbitrary configuration, can ...

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### Why does the overhand shuffle converge to the uniform distribution on $S_n$?

Pemantle 1989 proves, among other things, that the Markov chain on $S_n$ induced by repeatedly and independently performing an overhand shuffle on a deck of $n$ cards is ergodic and has limiting ...

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### markov processes and ergodic theory

For an ergodic Markov Chain
$$
\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]
$$
where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...

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### Discrete time hidden markov process [closed]

I am dealing with a hidden Markov model for variable $X_{t+1}$ where
$X_{t+1}$ = $\alpha_{t}$$X_{t}$ + $(1-\alpha_{t})$$Z_{t}$
$X_{t}$ is an indicator variable indicating wether an individual is ...

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### 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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### Does every (generalized?) Markov chain admit transition probabilities?

To pose the question let us start by recalling the following notions:
Transition Probabilities. A transition probability matrix between two measurable spaces $(S,\mathcal{S})$ and $(V,\mathcal{V})$...

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### Neat definition of Harris Ergodicity

I can't find any reference where the definition of Harris Ergodicity for Continuous time Markov processes is defined.
a) What would be exactly the definition?
b) What reference could be helpful?
...

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### Choose uniformly from fixed-length paths in $[0,n]\cap\mathbb{Z}$ with fixed start and end

Let $X_k$ be a symmetric (discrete time) random walk on $\mathbb{Z}$ and let $m,n\in\mathbb{N}$. I want to chose uniformly from the paths of $X_k$, which
start at $0$
stay in $[0,n]\cap\mathbb{Z}$ ...

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### Must rows of a transition matrix be distinct?

Is it true that for all continuous time Markov processes on a countable state space $S$, we have
all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ?
This ...

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### Strong Markov vector-valued process from component strong Markov process and independence

I want to prove that if $X$ and $Y$ are (continuous time) independent strong markov $\mathbb{R}$-valued processes w.r.t. their natural filtrations $\mathcal{F}^X_t$ and $\mathcal{F}^Y_t$, that the ...

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### Most visited vertex in a random walk with place dependent drift

Consider the following Markov chain on $\mathbb{Z}$:
$$
P(x,x+1)=1-P(x,x-1)=\frac{1}{2}+e^{-|x|}\cdot \mathbf{1}_{\{x\neq 0\}}
$$
Do there exist constants $c,C>0$ such that
$$
c\cdot P^t(z,z) \...

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### 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 ...

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### Sufficient moment conditions to make $E[\sup_n |X_n|]< \infty$ for Markov process $X_n$

Is there any Markov process $X_n$ for which we can impose sufficient moment condition which will imply $E[\sup_n |X_n|]< \infty$

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### Expected number of forward jumps to reach a given quantile of a rv [closed]

I'm a noob in randomized algorithm and ran into a problem(definitely not home work. I'm doing a self study out of my interest with help of my friends. I'm pursuing research career in a machine ...

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### Does an infinite chain of a.s. eventual transitions between states necessarily implies a.s transitions along the whole chain?

Given a Markov process among a (possibly infinite) set of states $S$, with possibly infinite depth (that is, the transition probabilities from $s_i \to s_j$ at time $t$ are permitted to depend not ...

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### Random Walk 2D with dependent weights [closed]

I have spent a lot of time trying to solve this problem but have had no luck so far! Any help would be highly appreciated!
Suppose I have a 3x3 grid as shown below.
(3,1) (3,2) (3,3)
(2,1) (2,2) (...

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### 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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### Maximal inequality for Markov process

For a Markov process $\{X_n\}$ is there any inequality available for
$$ E[\sup_{0 \leq n \leq k} X_{n}]$$
in terms of moments of $E[X_n], 0 \leq n \leq k$

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### how to resolve the infinite nesting of interactive POMDP

I am reading papers about I-POMDP. I cant understand the finitely nested I-POMDPs given in these papers.
The belief update of the algorithm has a problem that agents' belief updates mutually depend ...

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### Can MDPs over functions be solved?

I understand that dynamic programs are difficult to be solved in general. However I have an MDP, for which intuitively I have a solution, I am curious to know if there is a formal approach to get a ...

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### Markov Chains and Simple Machine Learning [closed]

Suppose I have a large training set consisting of many strings of symbols.
$TS = \{Str_0, Str_1, ..., Str_n\}$
$Str_i = \{Sym_0 ... Sym_{len}\}$
These strings of symbols are each generated by the ...

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### Eigenvectors of a perturbed reducible stochastic matrix

Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix $$\tilde{Q}\,=\,(1-\alpha)Q+\...

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### Properties of a map regarding the space of invariant probability measures for controlled Markov process

Let us consider a controlled Markov process with the transition kernel $p(dy|x,\theta)$ ($\theta$ being the control parameter. Now, consider the map
$\theta \to I(\theta)$ where $I(\theta)$ is the ...

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### Markov Modulated Markov Chain

Consider a discrete time Markov chain $X_t$ on some finite state space $\mathcal{S}$ with transition matrix $P$. Now consider a process $Y_t$ also on $\mathcal{S}$, which conditioned on $X_{t}=s$ ...

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### Circular process ergodic?

Let us define a continuous-time Markov process on a circle consisting of $m-$ equally spaced points, i.e. every point has two neighbours.
Now, we define a space of functions $S:= \{-1,1\}^{\{1,...,m\}...

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### Mixing time of lazy random walk on the directed cycle $C_n$

Briefly: A hint (if this is easy), reference or derivation would be of great help.
The question
Let $C_n$ be the directed cycle with loops in each of its $n$ vertices, and consider the random walk ...

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### Bounding Hidden Markov model Bayesian filter error with inexact models

In context of a hidden Markov model, I am interested in bounding the error of a Bayesian filter when using inexact state transition and observation models.
Consider a hidden Markov model (HMM) with ...

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### Regularity of the entrance measure of SRW

Let $S(n)$ be the discrete sphere of radius $n$ (i.e., the internal boundary of the Euclidean discrete ball $B(n)$) centered in the origin, and consider a simple random walk starting at some $x\in\...

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### Regarding Ricci curvature of Markov chains

In Ricci curvature of Markov chains on metric spaces Yann Ollivier, defines a coarse Ricci curvature for a Markov chain with transition kernels $\{m_x\}$ defined on a metric space $(X,d)$ as follows: ...

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### Order statistic of Markov chain sample path and related probabilities

Consider a 1D sample path, denoted as $\{X(1), ..., X(t), ..., X(n)\}$, generated from a discrete time finite state (time homogeneous) Markov chain over states $\{1,...,m\}$, with transition ...

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### steady state of a continuous-time birth-death process

we consider a continuous-time birth-death process $\{X(t),t\geq 0\}$ with discrete state space taking non0negative integer values $\{0,1,2,3,...\}$. The transition rates of the process $\{X(t)\}$ are ...

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### Similarity transformation of transition matrix of reversible Markov chain (reference request)

If $P$ is the transition matrix of a reversible Markov chain, and $\pi$ is its stationary distribution, and let $R$ be defined by:
$$R_{ij} = \sqrt{\frac{\pi_i}{\pi_j}}P_{ij}~.$$
By reversibility, ...

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### steady state distribution of the following infinite-state Markov chain

Given the following state transition equation:
$P_0(n+1)=P_0(n)(1-\lambda \Delta t)+ P_1(n)\mu \Delta t$
$P_j(n+1)=P_{j}(n)(1-\lambda \Delta t-\mu \Delta t)+\lambda \Delta t P_{j-1}(n)+ \mu \Delta t(...

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### Spectral gap of two step Markov chain

Suppose $X_1,X_2,\ldots$ is a reversible Markov chain with state space of size $k$ and absolute spectral gap $\gamma_*.$ What is the spectral gap of the (non-reversible) Markov chain $Y_1,Y_2,\ldots,$ ...

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### Quasi-stationary distribution for a death process

In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived.
Consider a ...

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### steady state distribution for a jump Markov chain

Consider a queueing process with the following transition matrix:
$\mathbf{P}=\left( \begin{smallmatrix}
1-\lambda & \lambda & & & & & & &\\
\mu & 1-\...

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### Finite hitting time implies hits at any finite time?

I was wondering about the following problem:
Assume we have a state space $S:=\mathbb{Z}$ and a Markov chain, such that we can go from any state $x$ to some state $y$ with positive probabilities, i.e....