All Questions
Tagged with pr.probability markov-chains
362 questions
7
votes
1
answer
337
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 $Z_{t} = (X_{t}[1], \ldots, X_{t}[k])$ ...
- 71
6
votes
2
answers
912
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. ...
- 161
4
votes
0
answers
153
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 ...
- 19.7k
19
votes
5
answers
18k
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 ...
- 191
4
votes
1
answer
1k
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 ...
- 1,322
8
votes
4
answers
1k
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,\...
- 203
3
votes
1
answer
403
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 ...
- 33
2
votes
1
answer
1k
views
forward algorithm Hidden Markov Model
I am studying the the forward-backward algorithm used in Hidden Markov Models. I understand that you are trying to propagate through a sequence (and the available states) to find the most probable ...
- 123
3
votes
0
answers
305
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 ...
- 61
0
votes
1
answer
408
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 $Z$ ...
- 1,109
2
votes
0
answers
199
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 ...
- 41
6
votes
1
answer
387
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 ...
- 519
2
votes
1
answer
665
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 ...
- 519
1
vote
1
answer
149
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 ...
- 3,979
3
votes
0
answers
494
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 ...
- 1,109
6
votes
1
answer
171
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 ...
- 519
4
votes
1
answer
207
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: 10.1111/...
2
votes
0
answers
162
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 ...
- 1,109
0
votes
1
answer
2k
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" ...
- 1,341
6
votes
1
answer
218
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 ...
- 1,068
1
vote
1
answer
687
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 ...
- 21
3
votes
4
answers
681
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 ...
- 2,239
1
vote
0
answers
196
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) =...
5
votes
2
answers
516
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 \...
- 696
3
votes
0
answers
95
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 $X_1+\...
- 696
7
votes
2
answers
698
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 $...
- 696
4
votes
0
answers
264
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 ...
- 18.8k
4
votes
3
answers
4k
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 ...
- 51
3
votes
0
answers
770
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 ...
- 418
1
vote
1
answer
160
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 ...
6
votes
0
answers
1k
views
Coin Toss Probabilities like Penney's Game
Generate a binary number, using coin toss. Until you receive a predefined terminating sequence. What is the probability that the number is a multiple of some $k$.
For example, the terminating ...
- 91
6
votes
3
answers
460
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: ...
- 1,731
0
votes
1
answer
2k
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 ...
- 23
4
votes
0
answers
282
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 ...
- 141
2
votes
0
answers
74
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 $...
- 2,133
0
votes
0
answers
151
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 ...
- 23
8
votes
0
answers
130
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 ...
- 19.7k
3
votes
1
answer
188
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 ...
- 191
1
vote
1
answer
228
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 ...
- 3,245
0
votes
1
answer
599
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 ...
- 209
2
votes
1
answer
356
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
1
answer
293
views
Empirical distribution of a collection of iid Markov chains
Suppose we have $N$ independent 2-point Markov chains each having a rate matrix $Q = [-1,1;1,-1]$ and stationary distribution $\pi = [0.5,0.5]$. At time $t=0$, we initiate the chains so that the ...
- 1,034
3
votes
1
answer
2k
views
Hitting time probability in a Random Walk with possibility to die.
A Random Walker can move of one unit to the right with probability $p$, to the left with probability $q$ and it can jump again to the starting point with probability $r$ and die. Naturally $p+q+r=1$. ...
- 295
1
vote
1
answer
295
views
Equivalent Markov Random Fields
Hi,
Is it possible to have topologically different Markov Random Fields (few different edges) and yet yielding the same inference results ?
Thanks!
- 167
1
vote
2
answers
489
views
The limiting behavior of geometric random walk
I would like to know what the asymptotic limiting behavior is for the following random walk on $\mathbb Z^d$. By Donsker's invariance principle, I suspect that its behavior is diffusive, i.e., the ...
- 8,512
2
votes
1
answer
2k
views
How to prove ergodic property from aperiodicity and positive recurrence
How to prove that in case of an irreducible, aperiodic and positive recurrent Markov Chain time average along sample paths is equal to the ensemble average ? i.e.
$$\lim_{n\to \infty }\frac{1}{n}\...
- 209
1
vote
1
answer
238
views
Hidden Markov: representing joint probability for set of observations as a product of two subset probabilities
Good day to everyone!
My question concerns Hidden Markov Models and is pretty basic. In one of the books ("Introduction to Machine Learning" by Ethem Alpaydin, 2nd Edition, p.373), I get the ...
- 13
4
votes
1
answer
555
views
Approximating a hitting time for some state using the stationary distribution?
Provided a random walk on a bounded interval, with step probabilities, $p$ and $q$ and a stationary distribution $\pi$, how "bad" of an approximation is to assume that the hitting time for a position $...
- 41
0
votes
1
answer
529
views
Estimates for the mixing time of a Markov Chain with biased initiation
Imagine I have some Markov process consisting of a biased random walk on the integers, over some interval $[0, L]$, with $+1$ and $-1$ step probabilities of $p$ and $q$, respectively, s.t. $(p + q) = ...
1
vote
0
answers
1k
views
a problem on DTMC
For a Markov chain $\lbrace X_n, n\ge0\rbrace$ with transition probabilities $P_{i,j}$, consider the conditional probability that $X_n = m$ given that the chain started at time $0$ in state $i$ and ...
- 209