The tag has no usage guidance.

learn more… | top users | synonyms

4
votes
1answer
191 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 ...
0
votes
0answers
9 views

How can I efficiently approximate the stationary distribution of an infinite CTMC with a sparce 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. ...
1
vote
1answer
45 views

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 ...
12
votes
3answers
6k views

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 ...
4
votes
3answers
343 views

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 ...
6
votes
2answers
294 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 ...
1
vote
0answers
89 views

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 ...
4
votes
1answer
204 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' ...
2
votes
2answers
142 views

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 ...
0
votes
1answer
399 views

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 ...
-1
votes
1answer
36 views

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 ...
6
votes
2answers
590 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 ...
5
votes
1answer
313 views

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 ...
1
vote
1answer
119 views

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? ...
1
vote
1answer
58 views

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 $[0,n]\cap\mathbb{Z}$ ...
2
votes
0answers
151 views

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 ...
0
votes
0answers
32 views

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 ...
3
votes
0answers
61 views

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) ...
3
votes
1answer
90 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 ...
0
votes
0answers
29 views

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 $E[\sup_n |X_n|]< \infty$
1
vote
0answers
46 views

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 ...
0
votes
1answer
29 views

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 ...
2
votes
1answer
98 views

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) ...
36
votes
7answers
3k 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 ...
0
votes
0answers
53 views

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 $E[X_n], 0 \leq n \leq k$
-2
votes
1answer
33 views

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 ...
0
votes
0answers
18 views

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 ...
1
vote
0answers
63 views

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 ...
2
votes
1answer
67 views

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 ...
1
vote
0answers
23 views

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 ...
0
votes
0answers
30 views

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 $X_{t}=s$ ...
0
votes
0answers
56 views

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:= ...
0
votes
1answer
147 views

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 ...
1
vote
0answers
16 views

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 ...
0
votes
0answers
48 views

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 ...
4
votes
2answers
214 views

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: ...
0
votes
0answers
62 views

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 ...
1
vote
0answers
30 views

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 ...
3
votes
1answer
88 views

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, ...
0
votes
0answers
26 views

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 ...
2
votes
0answers
42 views

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 $Y_1,Y_2,\ldots,$ ...
6
votes
1answer
318 views

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 ...
0
votes
0answers
29 views

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 & ...
0
votes
1answer
87 views

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, ...
0
votes
1answer
110 views

Markov chain with Feller property

Does anybody know whether there is an analysis of when the monotone decreasing chain has the Feller-property? The monotone decreasing is defined as a chain on $\mathbb{N}$ and the rate of going down ...
0
votes
0answers
156 views

Calculate the KL divergence between two transition matrices

I want to calculate how different two markov transition matrices are. For example: $\begin{pmatrix} .2 & .8 \\ .1 & .9 \end{pmatrix}$ and $\begin{pmatrix} .3 & .7 \\ .1 & .9 ...
3
votes
0answers
39 views

Memorylessness of residence times for a Markov process

I'm stuck on the trivial problem of showing memorylessness of holding (residence) times for a continuous time homogeneous Markov chain on finite state space. I have a homogeneous Markov process ...
8
votes
1answer
154 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
0answers
43 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
vote
1answer
131 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 ...