0
votes
0answers
10 views

Computing the PMF for N(t) in a renewal process

I'm in a stochastic processes course, and we just started on renewal theory. Unfortunately, we skipped the section on queuing theory, and nearly example in my textbook for renewal processes uses ...
1
vote
0answers
13 views

$X_n, n> 0$ is a Markov Chain, how to interpret $Z_n = (X_n,X_{n+1}), n > 0$?

Am a newbie to Markov Chain. So, this might be incredibly naive/stupid question. If $X_n, \, n > 0$ is MC, am having difficulty imagining/interpreting process $Z_n = (X_n,X_{n+1}), n > 0$. I ...
0
votes
1answer
20 views

Steady state of a $4 \times 4$ transition matrix

Normally I just take $q(M_{m\times n} - I_{m\times n})$ to workout the steady state, but here I have: $$\left(\begin{array}{rrrr} 0 & 0 & .8 & .2 \\ .4 & .6 & 0 & 0 \\ .2 ...
0
votes
1answer
17 views

Limiting probability of Markov chain(Terminology)

If I am asked to find the limiting probability of a Markov chain, what does this pertain to? $\lim \limits_{n \to \infty} P^n$? Where $P$ is the stepping matrix and $n$ is the number of steps. "What ...
1
vote
0answers
26 views

How's the damping factor in Google PageRank algorithm calculated

I'm doing some researches about Google's PageRank algorithm for my thesis, I've found that the damping factor x (for example), where x is in : P` = x.P + (1-x)Q where P is the original ...
1
vote
0answers
25 views

showing a condition implies convergence to invariant distribution

For an arbitrary Markov chain, I'm trying to show that $\lVert{ P_{\mu}\circ \theta_n^{-1} - \nu \rVert}\to 0$ iff $\sum_j \lvert{ P_{ij}^n - \nu(j)\rvert}\to 0$, where $\theta_n^{-1}$ is a shift ...
2
votes
1answer
51 views

continuous time Markov chain, something the book does not explain

I have a problem with something in my book, under the chapter of continuous time Markov chains. I will post a link to what the book does. They do something which they seem to take for granted, but I ...
0
votes
0answers
11 views

radius of markov chain

For an irreducible markov chain one can show, that $\limsup \sqrt[n]{p_{ij}^{(n)}}$ is independent of the choice of the states $i$ and $j$ where $p_{ij}^{(n)}$ is the probability to get from $x$ to ...
0
votes
1answer
47 views

Markov chain with infinite number of transient and positive recurrent states?

Is it possible to have a markov chain with an infinite number of transient states, and an infinite number of positive recurrent states? Thank you!
0
votes
2answers
221 views

Prove markov chain is null recurrent

Two fair coins are tossed repeatedly. Let Xn denote (Total Number of Heads from Coin 1)-(Total Number of Heads from Coin 2) after n tosses. Thus the state space is {0, ±1, ±2, .... }. Show that the ...
2
votes
2answers
59 views

Mean recurrence time and stationary distribution of a Markov chain?

In a Markov chain is there a theorem relating the existence of the stationary distribution and the mean recurrence time? E.g. impossible for stationary distribution to exist therefore mean recurrence ...
0
votes
0answers
61 views

proving null recurrence of random walk (Markov chain)

How would I prove that the zero state of a random walk with a positive probability of staying in the same state is null recurrent. (sorry if this isn't a random walk and just a Markov chain.) eg. ...
0
votes
1answer
26 views

Show $P(S_{2n}=x|S_0=x) \ge \frac{1}{N}$

Let $X_n$ be an aperiodic, discrete-time Markov chain so $S=\{1,...,N\}$ whose transition probability is symmetric. How can I show that for all $x \in S$ and all integers $n$, $P(S_{2n}=x|S_0=x) \ge ...
0
votes
1answer
19 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
0answers
28 views

Continuous time Markov chains, how would this definition be expanded from time-homogeneous to time-inhomogeneous.

Below I have a picture of how we can view a continuous time Markov chain that is time-homogeneous. Now, I am wondering what happens when we have a inhomogeneous continuous Markov chain. I have ...
0
votes
1answer
33 views

What is the Deterministic Traffic Generation Model?

I am studying Markov chains and queuing theory. I was curious about traffic generation models and actually happened to see the Deterministic Traffic Model, referred to as $D$ in Kendall's notation. ...
0
votes
1answer
20 views

Queue system with queue-triggered input process

I have a queue system, a classic system with an input generator, a queue and a servant. The servant is a $M$-servant with a certain serving rate $\mu$. The queue can contain an infinite number of ...
0
votes
1answer
19 views

Average time of permanence in a state of a Markov-chain

I know that in a Markov-chain the mean permanence time in a state is a random variable distributed accordingly to: Geometric distribution for Time Discrete Markov Chains Exponential distribution for ...
0
votes
0answers
26 views

Markov renewal process vs Markov Jump process

The venerable wikipedia for "Markov renewal process" says that: "a Markov renewal process is a random process that generalizes the notion of Markov jump processes". So what's the definition of a ...
1
vote
1answer
87 views

Elementary proof of geometric / negative binomial distribution in birth-death processes

The birth-death process concerns a population of $n_0$ individuals, each of which reproduce and die at a constant rate as time $t$ increases from $t=0$. Each individual splits into two individuals ...
1
vote
0answers
18 views

Finite state Markov chains and expectations

Let $X_t$ be a finite state Markov chain with generator matrix $Q$. For a give function $f(x)>0$ define: $$ u(t, i) = \mathbb{E}\left[\int_t^T f(X_s) \mathrm{d} s| X_t = i\right] $$ Are there any ...
1
vote
2answers
64 views

Expected number of steps between states in a Markov Chain

Suppose I am given a state space $S=\{0,1,2,3\}$ with transition probability matrix $\mathbf{P}= \begin{bmatrix} \frac{2}{3} & \frac{1}{3} & 0 & 0 \\[0.3em] \frac{2}{3} ...
0
votes
0answers
19 views

Poisson processes are the only renewal processes which are Markov Chains.

How would one prove the Proposition: "Poisson processes are the only renewal processes which are Markov Chains." A renewal process $N=(N(t))$ is a process for which $$N(t)=\max\{n : T_n \leq t\}$$ ...
0
votes
1answer
64 views

Exercise on Markov chain

Prove, or give an explicit counterexample to refute, the following assertion: if $\{X_n\}$ is a Markov chain, then $\{X_n^2\}$ is also a Markov chain. It's easy to show that ...
3
votes
1answer
51 views

Question about Markov chain

We know that if $\{X_n\}$ is a Markov chain, then $X_{n+1}$ is independent with the past states $X_0,\ldots,X_{n-1}$ given current state $X_n$, that is ...
2
votes
1answer
94 views

Markov chain problem with finite states

Consider any Markov chain on a state space with exactly $r$ states. We want to find the largest $N>0$ such that there exists states $i,j$ for every $n < N$ we have $p_{ij}^{(N)} > 0$ and ...
1
vote
0answers
32 views

Markov Chain depicting unruly customer behavior

A store has 2 bins of balls. 1 bin is red, and contains 3 red balls. The other bin is gray and contains 2 gray balls. Every minute, on the minute, exactly one customer comes by the bins, picks up ...
0
votes
0answers
22 views

Calculation of the Gallavotti-Cohen fluctuation theorem made by Lebowitz

I have a problem understanding a calculation in this paper (another form of the theorem an be found here at equation 11). For those who want to read the paper, I have difficulties with formula 2.14 in ...
1
vote
1answer
30 views

Consider a Stochastic process X which moves between the corners of a regular tetrahedron. Find P(Xn = 1)

Consider a Stochastic process $X$ which moves between the corners of a regular tetrahedron. The process starts at time 0 in node 1. At each time step, the process chooses an one of the connected edges ...
3
votes
1answer
59 views

Random walk where increment depend on current position

Consider the following stochastic process, $$b(i+1) = b(i) + \xi_i (b_i),$$ where $\xi_i(b_i) \in \{-1, k \}$ are the independent increments having the following distribution: $$\begin{align} P (\xi ...
1
vote
1answer
57 views

Library chain stationary distribution

This is an exercise 1.47 from Richard Durrett's Essentials of Stochastic Processes p.85 (doi: 10.1007/978-1-4614-3615-7_1 or Google Books). On each request the ith of the $n$ possible books is the ...
1
vote
2answers
47 views

Markov chain simulation

I'm wonder if there is an algorithm to simulate a discrete Markov chains with a specific number of occurence of state knowing the transition matrixway. For example, how to simualte in $\mathbb R$ a ...
1
vote
1answer
50 views

Equivalence Classes of a Markov Chain with Transition Matrix

I have the following transition probability matrix for a markov chain with state space S={0,1,2,3,4,5,6}: $\begin{bmatrix} \frac13 & \frac13 &0 & 0 & \frac16 & 0 & \frac16\\ ...
4
votes
1answer
76 views

random walk in a certain environment

Consider the following random walk in one dimension, starting from $r(0)=0$. $$ r(i+1) = r(i) + \xi, $$ where $\xi(i, r(i))$ is an increment with distribution $P(\xi=1) = \frac{c^{r(i)}}{i-r(i)+1}$ ...
1
vote
0answers
22 views

Prove equilibrium theorem without irreducibility and aperiodicity

I have to solve the following question: Consider a random walk Markov chain on $S = \{1, 2, \ldots, 100\}$. If the chain is between 2 and 99, it selects one of the adjacent states with equal ...
0
votes
1answer
97 views

Markov chains: is “aperiodic + irreducible” equivalent to “regular”?

I have two books on stochastic processes. In one book, it says that the limiting matrix is possible to find if the matrix is regular, that is, if for some $n$ $P^n$ has only positive values. The ...
5
votes
1answer
111 views

Log-likelihood function

I'm not sure if this could be asked here, or in math overflow... In the following paper Cho, Jin Seo, and Halbert White. "Testing for regime switching." Econometrica 75.6 (2007): 1671-1720. doi: ...
0
votes
0answers
37 views

Is this two dimensional Markov chain correct for this queueing system?

The problem that I have two single server station with no queuing space a customer goes to station 1 if it is available else it goes to station 2 if it is available or it will be lost output from ...
0
votes
0answers
58 views

transition matrix for Markov chain

Can any one help me to solve this home work please? The city of Sacramento recently completed a new light rail system to bring commuters and shoppers into the downtown area and relieve freeway ...
1
vote
1answer
44 views

Estimate the probability using Markov chains

please consider this question: A study using Markov chains to estimate a patient's prognosis for improving under various treatment plans gives the following transition matrix as an example ...
0
votes
1answer
41 views

Stationary distribution for a Markov chain which is not irreducible

I have a Markov chain with $K$ states $S$: {$s_1,s_2,...,s_K$}. $s_1$ is reachable from any state in $S$; however not all the states can be reached from $s_1$. What does the stationary distribution ...
0
votes
0answers
31 views

Does time homogeneity imply strong Markov property in a Markovian process

Does a time homogeneous Markovian process necessarily have strong Markovian property? Does continuity in state space, time, or path make a difference? What are the examples if it does not?
1
vote
0answers
64 views

A matrix of size $n\times n$ with several properties like Markov matrices

Could you find a square matrix $A=[a_{ij}]$ of size $n$ such that satisfies to following properties 1) For all $1\le i\le n$, $\sum_{j=1}^n a_{ij}=0$ 2) For all $i$, $a_{ii}<0$ and for $1\le i\ne ...
2
votes
1answer
75 views

Proof that Markov Chains converges to the stationary distribution

Let $P$ is a transition matrix of a Markov Chain, which is irreducible, aperiodic and lets assume $\pi$ is its stationary distribution: $\pi = \pi P$. Does anyone knows the proof for the following ...
0
votes
0answers
44 views

Continuous-Time Markov Chain conditioned on not visiting a part of the state space

Let $(X_t)_{t\geq 0}$ be a homogeneous continuous-time Markov chain with state space $\Psi =\{1,\dots,N\}$. Consider $S=\{1,\dots,M\} \subset \Psi$. Define $T = \inf \{ t \geq 0 : (X_t \not \in S ...
1
vote
0answers
68 views

First and second moments of recurrence time in a finite two-dimensional Markov chain

I have a two dimensional finite Markov chain with $(m+1)^2$ states, and with transition rates: $q_x((x,y)\to (x+1,y))=(m-x)\lambda,\quad 0\leq x< m, 0\leq y \leq m$, $q_x((x,y)\to ...
2
votes
1answer
66 views

Markov chain and hitting times

I have a Problem about hitting times. That's the following: Let $A\subset E$ and the first passage time $T_A$ and the hitting time $H_A$. Define: $T_A =\inf\{n\geq 0;X_n \in A\}$ and $H_A ...
1
vote
1answer
47 views

Calculating cumulative Markov Chain outcomes

I have a Markov process, with 2 possible states (1 or 0) and a transition matrix P. State at time t=n is determined by x0*Pn. As n goes to infinity, xn goes to the steady state vector, q = [q1 q2]. ...
0
votes
1answer
29 views

Does a Markov Blanket allow connections between Parents of a Node?

In a Markov Blanket, we can connect the childredn of a node between them, as a child can be parent (or spouse) of another child. Does this rule apply as well for Parents of a node? In advance, Thank ...
0
votes
0answers
33 views

Under what circumstance should I use a Continuous time Markov Chain instead of a discrete time Markov Chain?

Why should I use one over the other, if I can basically reduce the small time-interval $h$ to be small enough that it simulates continuity? I guess this question is somewhat analogous to control ...