Stochastic processes (with either discrete or continuous time dependence) on a discrete (finite or countably infinite) state space in which the distribution of the next state depends only on the current state. For Markov processes on continuous state spaces please use (markov-process) instead.

learn more… | top users | synonyms

1
vote
2answers
33 views

Expected time between successive visits in a Markov Chain?

This is a pretty basic question and I know the answer is probably really obvious, but I am having trouble reasoning as to why the following is true: (From my lecture notes): """ Expected time ...
-1
votes
0answers
11 views

Markov chain for closed queueing network [on hold]

closed queueing network diagram Im trying to figure out the arrival rates to these individual queues, but Im not quite sure if the Markov chain I drew is even correct. Could someone tell/show me what ...
0
votes
1answer
39 views

How to understand this kind of Markov chain?

There are two unidirectionally coupled processes $X_t$ and $Y_t$. The coupling is $Y_t=g(X_{t-\delta},\dots)$. The Bayesian network of the two process is described in the following figure: Now this ...
0
votes
0answers
24 views

Finding the stationary distribution of specific homogeneous Markov chain and determining its uniqueness

I am presented with $P =\begin{bmatrix} 0.5 & \alpha & \beta \\ \alpha & \beta & 0.5 \\ \beta & 0.5 & \alpha \end{bmatrix}$ where $\alpha+\beta=0.5$ and $\alpha,\beta \in ...
0
votes
0answers
18 views

Underlying sample space in a markov chain

I am studying discrete-time Markov chain and I am confused about the very first example. The example is the Gambler's Ruin: Consider a gambling game in which on any turn you win $\$1$ with ...
0
votes
0answers
18 views

A problem on random q-colourings of a graph for randomly chosen vertex

Here is an exercise from Olle Haggstrom's "Finite Markov Chains and Algorithmic Applications" from the chapter "Fast Convergence of MCMC Algorithms". The exercise is based on random $q$-colorings of ...
1
vote
0answers
18 views

Reducing sequential correlations in Metropolis Algorithm

In our last lab, we use MCMC method to simulate a walker walking in the phase space. Using the Metropolis method, a walker at its currect position will sample another point inside a cube (centered at ...
0
votes
0answers
17 views

Pure Birth Question. Find the probability that the population at time $t$ is an odd # given it starts at $0$.

Here is the question. Consider a pure birth process $\{X(t) : t ≥ 0\}$ with birth parameters $\lambda_{2n} = α>0$ and $\lambda_{2n+1} =β>0$ for $n∈N$. Compute $Pr\{X(t) \text{ is odd } \mid ...
2
votes
1answer
419 views

Gradient-descent and Hidden Markov Models

I would like to use gradient-descent to fit the parameters of a simple 2-state HMM. This paper Levinson, S. E., Rabiner, L. R. and Sondhi, M. M. (1983), An Introduction to the Application of the ...
1
vote
1answer
25 views

Symmetric Simple Random Walk - Definition Clarification

I'm finding conflicting answers everywhere, including in my own notes. In the phrase "symmetric simple random walk", which part, "symmetric" or "simple" refers to having a probability of $0.5$ to go ...
0
votes
0answers
10 views

Markovian Model: scheduling jobs to servers

I have the following problem. I tried to look at queuing theory, but it probably fits better as a scheduling problem. I have a set of $C$ servers: each one can perform 1 job. Processes arrive ...
0
votes
1answer
12 views

Does an embedded discrete-time Markov chain preserve its properties in continuous time?

Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a continuous ...
1
vote
0answers
42 views

For a finite state irreducible aperiodic MC, show that $P^{d^2}$ has all coordinates positive.

Suppose $X_n$ is an irreducible aperiodic finite state MC, with $P$ being the transition matrix. Then we know that $P^n$ has all positive entries for some $n\in\mathbb N$. If the state space $S$ of ...
0
votes
0answers
31 views

Question regarding regular stochastic matrix

We say that a stochastic matrix is regular iff $\exists n\in \mathbb N$ such that $p_{ij}(n)>0$ for all states $i,j$ How many powers of a matrix do we need to compute at most in order to verify ...
1
vote
1answer
36 views

Show that a function of a markov chain is not a markov chain [on hold]

How do you show that a function, $Y(n)=g(X(n))$ if some Markov chain $X(n)$ cannot be a Markov chain?
-1
votes
0answers
31 views

A few questions related to a Markov chain.

In the picture I have a Markov chain, call it $X(n)$. I have a few questions about this Markov chain. First, is it aperiodic? Second, what is the value of $P[X(1)=1,X(2)=0,X(3)=0,X(4)=1\mid ...
0
votes
0answers
5 views

How to find transition matrix in cascaded FSMs?

I am considering a 4-state system Equivalently, I can use a cascade approach to represent the same system as In cascade approach, If I consider independent condition on input on both ...
1
vote
1answer
81 views

Markov Chains where Time Spent in State Matters

I have done a good bit of research on the subject, and cannot seem to find many materials. I was just wondering if you all knew of a good resource regarding chains which are Markov excepting the fact ...
0
votes
0answers
3 views

How to make Markov Chain model from sequence of data in MATLAB?

Markov Chain model considers only 1-step transition probabilities i.e. probability distribution of next state depends only on current state and not on previous state. I have a sequence and from that I ...
3
votes
2answers
39 views

Example of a markov chain that has a distribution that converges to some limit.

Can someone give me an example of a Markov chain that has a distribution that converges to some limit which depends on the initial distribution?
0
votes
1answer
43 views

Markov Chain that isn't Irreducible

What is an example of a Markov chain that isn't irreducible but has a unique distribution, such that its distribution converges to that unique invariant distribution for any initial distribution.
-1
votes
0answers
8 views

Counter-example for Markov property preservation [on hold]

Suppose $X$ is a $(\mathbb{P}, \mathbb{F})$-Markov chain. I need to construct an example where $\overline{\mathbb{F}} := \{\overline{\mathcal{F}_n}\}$ is a super-filtration, i.e., $\overline{F}_n ...
1
vote
1answer
23 views

How to test quality of probability estimates?

I have a Markov chain model which produces a probability distribution for absorption in 4 possible absorbing states. I.e. the model estimates the probability distribution for a discrete random ...
3
votes
1answer
40 views

Example of a Continuous-Time Markov Process which does NOT have Independent Increments

1. Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a ...
1
vote
1answer
29 views

Markov dynamic programming recursion

I'm learning Markov dynamic programming problem and it is said that we must use backward recursion to solve MDP problems. My thought is that since in a Markov process, the only existing dependence is ...
1
vote
0answers
55 views

Prediction Interval from Markov Chains

Thank you for taking the time to look at my question. Short, less involved question: How do you find Prediction Intervals with non-Gaussian residuals? Here is the situation: I have made a model that ...
-1
votes
0answers
33 views

Exercise about Markov Chain

Let be ($S_{n}$) a Markov chain on $\mathbb{Z}$ such that $P(S_{n+1} = x+1| S_{n} = x)=\alpha_{x}$ and $P(S_{n+1} = x-1| S_{n} = x)=1-\alpha_{x}=\beta_{x}$ for all $x \in \mathbb{Z}$ . Let ...
0
votes
1answer
25 views

What does it mean for an object to not be following a definition based on some implication

I want to get a deeper understanding of what being an object that doesn't follow a definition means in terms of predicates and logical operators. Suppose the following definition of the closedness ...
1
vote
1answer
29 views

probability of which alarm clock goes off first

I'm learning about continuous time markov chains, in the text I am reading, they are setting up the discussion by talking about a series of alarm clocks that are set: Suppose $T_1,...,T_n$ are ...
1
vote
1answer
25 views

distance from the origin in a simple random walk on $\mathbb Z^2$

let $S_{n}= \sum_{i=1}^{n}X_i$ be a simple random walk on $\mathbb{Z}$, with $S_0 = 0$. $X_i = 1$ with probability $p$ and $X_i = -1$ with probability $1-p$. It can be shown that ...
3
votes
1answer
26 views

Show that a matrix satisfying certain conditions is non-singular

I have a square matrix $A$ satisfying the following conditions: The elements on the diagonal are negative; All other elements are non-negative; All row sums are less than or equal to $0$; There is ...
-1
votes
1answer
17 views

Initial point and initial distribution of the Markov chains

I am reading about Markov chains on a general state space and the ergodicity theory. Some of the ergodic theorems are presented when we consider n-step transition probability conditional on initial ...
0
votes
1answer
19 views

Period of an irreducible Markov Chain is given by the number of eigenvalues with unit modulus

Suppose $\{X_n\}$ is an irreducible Markov Chain on finite state space $S$. Then, the number of eigenvalues of the transition matrix with unit modulus is precisely equal to the period of the chain. ...
-1
votes
1answer
23 views

Markov Chain Market Share Model [closed]

Given that, $$ \begin{matrix} & Coke & Pepsi \\ Coke & 0.8 & 0.2 \\ Pepsi & 0.3 & 0.7 \\ \end{matrix} $$ What is the current market ...
1
vote
1answer
21 views

Markov chain - Expectation $+1$?

Let the transition matrix of a markov chain with states $\{0,1,2\}$ : \begin{equation} A=\begin{bmatrix} \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{2} & ...
-2
votes
0answers
25 views

Markov chain with a PMF [closed]

I don't know how to attack the following problem. I don't get how can I link the value K with the value in Pij that are supposed to be part in a markov chain. The times between successive customer ...
0
votes
1answer
24 views

Stationary distribution vs invariant distribution of a Markov chain

Lets $p$ be a distribution on a finite sample space with $n$ points. I wish to find a transition matrix that is invariant with respect to $p$, that is $$p^T T = p^T$$. The problem is clearly ...
0
votes
0answers
23 views

Mixing time for lazy random walk on hypercube.

I am studying for a probability exam and am having trouble with the following exercise: Let $X_n$ be a lazy random walk on $\{0,1\}^d$ starting at $(0,\ldots,0)$ (stays put with probability ...
-1
votes
0answers
55 views

Probability of returning to the origin

Suppose I have a markov chain which models a random walk on the integers $\mathbb{Z}$, such that $X_{i}$ takes on 1 with probability q and -q with probability 1-q. And $\{X_{n}: n \in \mathbb{N}\}$ I ...
2
votes
1answer
38 views

Stability of a dimer on a square grid after $n$ random steps

On a white square grid there are two black cells. Each step consists of each of the cells 'moving' in one of the four directions with equal probability $p_0=1/4$ (a cell can't stay in the same place). ...
1
vote
0answers
21 views

When can an embedded Markov chain X for a Markov process Y be reducible?

It's pretty widely documented that a Markov process Y is reducible/irreducible if and only if the embedded Markov chain X is reducible/irreducible. However I'm not sure this works in reverse. I'm ...
0
votes
0answers
29 views

Modeling the joint distribution of stream statistics

I have a question regarding computing the joint discrete probability distribution of statistics in a number stream. I tried searching all previous posts to look at different forms of this problem ...
13
votes
1answer
407 views

What happens to a random walk when we increase the probabilities of going right?

Consider a random walk on the integers where the probability of transitioning from $n$ to $n+1$ is $p_n$ (and of course, the probability of transitioning from $n$ to $n-1$ is $1-p_n$); we assume all ...
0
votes
2answers
475 views

what are “filtering” and “smoothing” mentioned in hidden Markov model wikipedia article?

the article mentions "filtering" and "smoothing" tasks, see here http://en.wikipedia.org/wiki/Hidden_Markov_model#Filtering . It gives brief explanation but no motivating examples and no references to ...
1
vote
1answer
450 views

Transition rate matrix from transition probability matrix

If I have a $2 \times 2$ continuous time Markov chain transition probability matrix (generated from a financial time series data), is it possible to get the transition rate matrix from this and if ...
0
votes
1answer
8 views

emmission probabilities in a hidden markov model with 2 states and an alphabet of 4 characters

I'm reading through a text that is describing how to use use hidden markov models to identify areas of biological sequences that correspond to specific biological features. It starts with a simple ...
1
vote
1answer
39 views

Expected steps in a Markov chain

I have two jars; initially one is empty while the other holds $n$ red and $n$ blue marbles. Every minute, I do the following procedure: take any pair of marbles of different colors but from the same ...
1
vote
1answer
724 views

Expected state of a Markov chain

Let's start with a slightly trivial Markov chain defined as follows: the beginning state is called $1$ and the set of states is $\mathbb{N}$. At each step, when the current state is $n$, the ...
-1
votes
0answers
28 views

Markov chains- construction of the stationary distribution of an irreducible, aperiodic and positive recurrent Markov chain

Proof In the excerpt of a proof (above) that proves that If markov chains are irreducible, aperiodic and positive recurrent. Then the distribution $\mathbf{\pi}$ with the entries of $\mathbf{\pi}$: ...
0
votes
0answers
41 views

Random walk with reflection and skips in linear system

Let's take a case of simple and linear Random walk (0, 1...n) with only one absorbing state n and reflecting state -1, which we can define as: P (move right at state i) = 1/2 and P (moving left at ...