The tag has no usage guidance.

learn more… | top users | synonyms

6
votes
2answers
554 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. ...
2
votes
0answers
90 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 ...
4
votes
3answers
2k 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 ...
0
votes
1answer
121 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 ...
0
votes
1answer
145 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 ...
4
votes
3answers
337 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 ...
2
votes
1answer
176 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 ...
1
vote
1answer
190 views

forward algorithm Hidden Markov Model

I am studying the the forward-backward algorithm used in Hidden Markov Models. I understand that that you are trying to propagate through a sequence (and the available states) to find the most ...
3
votes
0answers
104 views

Nonlinear Markov process

Consider the following nonlinear $\mathbb{R}$-valued stochastic recursive sequence: $ X_{n+1} = F(X_n) + W_{n+1}, \quad (W_n)_{n\ge1} \stackrel{ \scriptsize \mathrm{i.i.d.} }{ \sim } \phi. $ How can ...
3
votes
2answers
212 views

Markov Partitions for toral automorphisms

I know that my question is more practical than theoretical. But, I do know where to look for the theoretical sources. I want to find a program in the case that it exists (does it?), or to program it. ...
1
vote
0answers
147 views

segmental k-means hmm algorithm

I'm trying to determine the state parameters for an hmm with a minimum of 9 states. I'm running chunks of my data through hmmtrain on MATLAB which I think uses the Baum/Welch aka forward/backward ...
2
votes
0answers
437 views

Closed-form solution to a system of linear equations

Consider the following $n \times n$ matrix with a particularly nice structure: \begin{equation}\mathbf{P}=\begin{pmatrix} 0 & 0& \dots&0 & 0 &1\\ 0 & 0& \dots&0 & ...
0
votes
1answer
336 views

Generating independent random variable from two correlated random variables

Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...
14
votes
4answers
791 views

Eigenvectors of a particular transition matrix

I am considering a Markov chain with $n$ states with a particularly nice structure. The transition matrix is as follows: \begin{equation}\mathbf{P}=\begin{pmatrix} 0 & 0& \dots&0 & 0 ...
1
vote
0answers
77 views

CLT for a Markov Renewal Process

Suppose $(X,T)=\{(X_n,T_n)\}_{n\geq0}$ is a Markov renewal process, where $X$ is a finite-state, discrete-time Markov chain with state space $\{1,2,...,R\}$. $T$ is the additive component, more ...
4
votes
1answer
278 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 ...
2
votes
1answer
183 views

Transition probabilities in coupled Markov chains

I know that for a continuous-time Markov chain, the probability of transition from time $0$ to $t$ is given by $P(t)=e^{Q(t)t}$. I have a system of $N$ interdependent continuous-time Markov chains ...
1
vote
1answer
78 views

Minimal variance for phase-type distributions?

Let $\mathcal{D}(m)$ be the set of phase-type distributions constructed from $m+1$-state Markov chains. Recall that the coefficient of variation of a distribution $D$ is the ratio of the standard ...
3
votes
0answers
257 views

Maximization of a total variation distance subject to another total variation distance in Markov chain

Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
5
votes
1answer
133 views

Relative vulnerabilities in SIS epidemic model

Consider the SIS model of epidemic spreading. There is a finite graph $G(V,E)$, link infection rates $\lambda_{ij}$ and node recovery rates $\mu_i$. There are a few initial nodes which are infected at ...
4
votes
1answer
155 views

Log-likelihood in regime switching

Not sure if this question is too simple to be asked here... In the following paper Cho, Jin Seo, and Halbert White. "Testing for regime switching." Econometrica 75.6 (2007): 1671-1720. doi: ...
6
votes
1answer
172 views

A family of skew-symmetric matrices corresponding to cycles in graphs

When investigating loops in Markov chains I ran into the following observation. A cycle in a graph $G$ with $n$ vertices may be represented by a matrix $\Gamma \in \mathbb R^{n \times n}$ having the ...
2
votes
0answers
139 views

An optimization in Markov Chain

We are given two correlated random variables $V$ and $X$ supported over a finite alphabets $\mathcal{V}$ and $\mathcal{X}$. Suppose the marginal $P_V$ and conditional distribution $P_{X|V}$ are ...
0
votes
1answer
507 views

Stationary distribution in general Markov Chains

This is just a reference request for a result which is very general, useful and should be well-known, but I've failed to find a good reference to cite. The problem is to define the "most natural" ...
6
votes
1answer
184 views

Finding cohesive (low exit probability) sets in a Markov process

The following is a fact about Markov chains that came up in a game theory paper. The purpose of this question is to ask if related notions or similar results are found elsewhere in probability, or are ...
0
votes
0answers
115 views

Is it unique when a irreducible and aperiodic markov chain on general space has an invariant measure?

Recently I'm reading Markov chains and Stochastics Stability(sencond edition 2009) written by Meyn and Tweedie. And in the proof of Theorem 10.4.5 on page 243, it says "if $\pi_m$ is invariant for the ...
0
votes
1answer
299 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 ...
3
votes
4answers
372 views

Does the variance of a continuous time, time homogeneous, Markov process starting from one point necessarily not decrease?

Let $x_t$ be a zero mean, time homogeneous Markovian process (chiefly look at the case where the value is in $1$ dimension) over time $t$ starting from $x_0=0$. Is it necessary that, in continuous ...
1
vote
0answers
138 views

The problem of the drunkard in a valley [closed]

We consider a Markov chain on a subset of positive integers S = {0, 1, 2, 3, .......N}, with transition probabilities defined as follows: The chain jumps only one unit to the left or right. p(i, j) ...
4
votes
2answers
236 views

Anticoncentration of the convolution of two characteristic functions

Edit: This is a question related to my other post, stated in a much more concrete way I think. I am interested in anything (ideas, references) related to the following problem: Suppose that $A ...
3
votes
0answers
84 views

Best convergence rate for convolutions on $\mathbb{Z}_p$

Suppose, that we have sequence of i.i.d variables $X_1,\ldots,X_n$ taking values in $\mathbb{Z}_p$ such that $d_{TV}(X_1,U) < \delta$. How fast, in terms of $\delta$ and $n$ does the sum ...
7
votes
2answers
323 views

Convergence rate of the convolution of almost uniform measures on $\mathbb{Z}_p$

Statement Given a finite abelian group $G$ and two independent random variables $X,Y$ taking values in $G$ and satisfying $d_{TV}(X,U_G)\leqslant \delta$ and $d_{TV}(Y,U_G)\leqslant \delta$ (where ...
4
votes
0answers
203 views

Generalized Markov Processes on CW complexes of dimension > 1

Markov processes have a large variety of applications to physics and chemistry (as well as many other fields). Such processes are formulated on graphs, i.e., CW complexes of dimension one. It is ...
0
votes
3answers
902 views

Stationary distribution for bipartite graph

I was wondering if there is any stationary distribution for bipartite graph? Can we apply random walks on bipartite graph? since we know the stationary distribution can be found from Markov chain, but ...
2
votes
0answers
154 views

Kullback-Leibler Divergence of Stationary Distributions of Markov chains

Consider two finite Markov chains on the same state space, both assumed to be irreducible, with transition matrices $P$ and $Q$ and associated stationary distributions $\pi$ and $\tilde \pi$. Is it ...
1
vote
1answer
141 views

Optimum control of a probabilistic automaton

Suppose we have a probabilistic automaton and we assign a weight to each state. An "interaction strategy" would be a fixed map from states to inputs. Any interaction strategy could be used to ...
4
votes
0answers
430 views

Coin Toss Probabilities like Penney's Game

Generate a binary number, using coin toss. Until you receive a predefined sequence. What is the probability that the number is a multiple of some k. For example, the terminating sequence could be ...
4
votes
1answer
283 views

Uniqueness of Gibbs Measure on Ising model

If I understood this correctly, the Gibbs Specification for the Ising model on $ℤ^d$ dos not have a unique Gibbs Measure for β above the critical level. But what about the Ising model on a finite ...
1
vote
1answer
170 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
1answer
346 views

Markov Chain: state reduction

Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following: Firstly we have a Markov chain $\{Y_k\}$ with finite ...
3
votes
0answers
172 views

Markov operators and existence of ergodic measures

My question refers to the yesterday's question (see here) of John Learner and goes as follows: Can we deduce the existence of an ergodic measure if we know that an invariant measure exists, but the ...
2
votes
0answers
64 views

iterated mutinomial and hitting time

Let $N,k \geq 1$ be two integers and consider the following Markov chain on $[0,k\times N]^k$. It starts at $X^0=(N, N, \ldots, N)$ and $X^{n+1}$ is the realisation of a multinomial distribution with ...
0
votes
0answers
108 views

Inequality relating stationary probabilities and transition probabilities

Let $P$ be the transition probability matrix of a aperiodic irreducible DTMC and let $\pi$ be its stationary distribution. I would like to know if there is any literature on types of Markov chains ...
8
votes
0answers
109 views

Functions between Markov chains that preserve local harmonicity

Given two Markov chains with respective state-spaces $S$ and $T$, say that a function $\phi$ from $S$ to $T$ is holomorphic iff for all states $t \in T$, every real-valued function $f$ on $T$ that is ...
3
votes
1answer
145 views

Markov Chains based on sampled transition probabilities [closed]

If I have a process that transitions between states with some set, unknown probability, I can sample to find the transition probability. This probability is a sample average, with a well understood ...
1
vote
1answer
154 views

Is anything known about Large Deviation Principle for non additive functionals on Markov chains?

Let $\Sigma$ be a finite set of cardinality $|\Sigma |$ and $$\Pi = \{ \pi(i,j)\}_{i,j = 1}^{|\Sigma|}$$ a stochastic matrix (ie a matrix whose elements are non negative and such that each row sum ...
0
votes
1answer
164 views

A basic question on necessary and sufficient condition for positive recurrence

If state $j$ is recurrent and the following holds can it be called as positive recurrent ? $$\lim_{n -> \infty}\frac{1}{n}\sum_{k=1}^{n}p_{jj}^{(k)} > 0$$ I know that this a necessary ...
4
votes
0answers
138 views

Relaxation = absorption?

Let $A$ be a stochastic matrix, that is, the entries are non-negative and each row adds to $1$. Assume that it is primitive, that is, $A^n$ has only positive entries for sufficiently large $n$. We ...
2
votes
1answer
237 views

The first eigenvalue of a branching process matrix

Let $M$ be the real square matrix of a typed branching process, such that $M_{ij}$ is the expected value of offspring of type $j$ emanating from type $i$. We know that if the first eigenvalue if $M$ ...
1
vote
0answers
199 views

Stationary distribution of a countable state Markov chain

We assume the Markov chain to be countable state space, time-homogeneous. Does it necessarily have a stationary distribution? I found a paper on arXiv.org (http://arxiv.org/abs/math/0610707) that ...