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

0
votes
0answers
28 views

Repair Chain (Markov Chain Sample Model)

A machine has $3$ critical parts that are subject to failure, but can function as long as two of these parts are working. When two are broken, they are replaced and the machine is back to working ...
0
votes
0answers
18 views

Show if $p^n(i,j)\rightarrow \pi(j)$ as $n\rightarrow\infty$ then $\pi(j)$ is a stationary measure..

Suppose $p(i,j)$ is a transition kernel on $S$ for a countable state markov chain $X_n$ with $$p^n(i,j)\rightarrow \pi(j)$$ as $n\rightarrow\infty$ for all $i,j\in S$. want to verify that $\pi$ is a ...
0
votes
1answer
31 views

Proof of aperiodic Markov Convergence Theorem for null recurrent case.

Status quo: We consider a irreducible, aperiodic Markov chain $(X_n)_{n\in\mathbb{N}}$ on a countable set $S$ with tranistion function $p(\cdot,\cdot)$. Now we want to examine ...
1
vote
1answer
28 views

The lower bound for the smallest eigenvalue given the condition

In a paper, i saw a statement that the smallest eigenvalue of $P$($P$ is reversible Markov chain with stationary distribution $\pi$) is greater than $2 \beta - 1$ with the condition, $P \geq \beta I$. ...
1
vote
0answers
32 views

Does Markov Chain converge in Variance Norm?

Assume the chain $\{X_n\}_{n\in\mathbb{N}}$ on the statespace $(S,\mathcal{F})$ is aperiodic, irreducible and positive recurrent. We denote with $\pi$ its (unique) stationary distribution. Is it true ...
0
votes
1answer
31 views

How would I solve for long run average profit?

I was looking at a problem, and I was wondering how I would set this up. Any help would be welcome. Thank you! A store stocks a particular item. The demand for the product each day is 1 item with ...
0
votes
1answer
98 views

How do I compute the variance of expected number of fair coin flips for HTH sequence using linear system of equations?

Assuming fair coin flips, I know how to compute the expected number of coin flips to see HTH sequence by writing out the linear system of equations from the state transition diagram below. ...
0
votes
0answers
11 views

Proof of the “Markovian property” for the LERW?

I'm trying to understand this proof by Werner of the Markovian property of the Loop-erased random walk http://arxiv.org/pdf/math/0303354v1.pdf (page 10). The first part I see but the second "again, ...
2
votes
0answers
22 views

Formal argument on independence of consecutive hitting times of a Markov chain.

I'm refering to the question: Differences of consecutive hitting times. I'm interested in the independence of consequtive hitting times of certain values of a Markov chain. And I do "understand" the ...
0
votes
1answer
60 views

Can a reducible Markov chain have an unique stationary distribution? [closed]

I know for irreducible and positive recurrent Markov Chain there exists an unique stationary distribution. For Markov Chain with several communication classes (example C1, C2) there exist stationary ...
1
vote
2answers
66 views

How to compute the variance of number of coin flips to see HTH sequence using linear system of equations.

Assuming fair coin flips, I know how to compute the expected number of coin flips to see HTH sequence by writing out the linear system of equations from the state transition diagram below. Define ...
1
vote
1answer
47 views

What is the probability there will be no failures?

"A machine has 4 components and the machine cannot operate when any one of these components fail. At the beginning of each day, the machine starts running. During any day component $i$ fails with ...
3
votes
0answers
28 views

$ X_n = 2 Y_n + Y_{n+1} $ (non)Markov Chain

Let $Y_1,Y_2,\dots$ be iid random variables with $P(Y_n=0)=1-p,\; P(Y_n=1)=p$ where $p\in(0,1)$. Define $$ X_n = 2 Y_n + Y_{n+1} $$ The question is, whether $\{X_n\}$ is a Markov chain or not. ...
0
votes
1answer
38 views

Calculating the information per symbol of a markov chain source

I have a 4-state 2nd order markov chain source with symbols 0 and 1. I have all the transition probabilities and have worked out the probabilities of each state. How do I go about finding the amount ...
1
vote
0answers
38 views

The value of a stochastic game

I understand why a stochastic game with discounted payoff has a value $v$ and optimal strategies over the set of stationary strategies. But why is $v$ also the game's value over the set of behavioral ...
0
votes
0answers
52 views

Transition probability matrix for $X_1 = \# heads$, *flip heads* $X_2 = \# tails$ * flip tails* $X_3 = \# heads$

Three fair coins are tossed, and we let $X_1$ denote the number of heads that appear. Those coins that were heads on the first trial (there were $X_1$ of them) we pick up and toss again, and now we ...
0
votes
0answers
27 views

Decide whether a class is recurrent or transient (Example)

Consider the Markov chain $(X_n)_{n\geqslant 0}$ with state space $E=\left\{1,2,3,4\right\}$ and transition matrix $$ T=\begin{pmatrix}0 & 1/3 & 1/3 & 1/3\\0 & 0 & 1 ...
0
votes
0answers
20 views

Explicit Probability for Markov Chain on Power Set

A have a Markov chain $F_t$ in discrete time on the power set of a finite totally ordered set $A$. Its probably easiest to explain the transition probabilities in a small example, since they are easy ...
1
vote
1answer
39 views

Transience in a simple Markov chain

Consider the following simple game from a textbook called "Competitive Markov Processes" by Filar & Vrieze (Springer 1996). This is a two player game with two states. In the first state (the ...
1
vote
2answers
54 views

If a Markov chains converges then the limit is a stationary distribution

Let $p$ be a transition function of a Markov Chain on a countable state $S$ and $i \in S$. Assume for every $j \in S$, $$ \lim_{n\to \infty} p^n(i,j) = \pi(j)$$ Show that $\pi$ is a stationary ...
2
votes
0answers
75 views

Why are inter arrival times in the continuous version of discrete-time Markov chains always exponentially distributed?

I am curious whether there exist continuous time Markov processes for which the times between jumping times (which I call inter arrival times) are not exponentially distributed, but have some other ...
3
votes
1answer
63 views

Stochastic matrices with orthogonal eigenvectors

I stumbled across an odd fact while thinking about Markov chains, and I have an odd proof for it. Claim: Let $P$ be the transition matrix of a finite irreducible aperiodic Markov chain, and assume ...
1
vote
1answer
104 views

Expectation and Variance of random walks

Consider random walks of fixed length (e.g. $5$) starting at node $u$ in an undirected and connected graph with $N$ vertices. If a node $k$ has $N_k$ edges, the probability of the walk reaching any of ...
1
vote
1answer
20 views

Markov chain knowing future

I was wondering whether or not P(X1 = S1 | X0 = S0) and P(X1 = S1 | X0 = S0 and X2 = S2) are the same? What I mean is can we get some information from the future states? Thanks!
0
votes
0answers
13 views

Is Markov Chain sampled at stopping times a Markov chain?

Given a Markov hain $\{X_n\}$ and $T_n$ be an increasing sequence of stopping times, is $\{X_{T_n}\}$ Markov chain ?
1
vote
0answers
26 views

Prove that $E_x[T_y]<\infty$ for irreducible positive recurrent Markov chains.

This is an exercise from Durret's "Probability: Theory and Examples". Suppose $p$ is a irreducible and positive recurrent Markov chain. Then $E_x[T_y]<\infty$ for all $x,y$. I had the following ...
2
votes
1answer
75 views

Maximal Principle: Why using the new transition matrix $\tilde{P}$?

First some notation: Let $(X,E,P)$ denote a finite, irreducible Markov chain with finite state space $E$ and transition matrix $P$. Choose and fix a subset $E^°$ of $E$, which will be called ...
5
votes
3answers
133 views

A recursive formula to approximate $e$. Prove or disprove.

Let the sequence $\{x_n, n=1,2,...\}$ be defined as follows: Let $x_2=x_1=1$ and for $n>2$ let $$x_{n+1}=x_n-\frac{1}{n}x_{n-1}.$$ This sequence, generated by the recursion above, tends to zero ...
0
votes
0answers
53 views

emails arriving in a Poisson process

Emails arrive according to a Poisson process with rate $λ=2/hour$. You check your inbox (instantly reading all new emails) at time $t=5$ hours and also at some uniformly distributed random time ...
0
votes
0answers
29 views

Mean and Variance of an offspring

If I have that the number of offspring of an individual in a population is $0$, $1$, or $2$ with respective probabilities $a>0$, $b>0$ and $c>0$, where $a+b+c=1$, how would I express the mean ...
2
votes
1answer
36 views

Finding mean and variance of a population problem

A population beings with a single individual. In each generation, each individual in the population dies with probability $1/2$ or doubles with probability $1/2$. If I let $X_n$ denote the number of ...
4
votes
1answer
95 views

Find the Stationary Distribution of an infinite state Markov chain

A Markov Chain on states 0,1,..... has transition probabilities $P_{ij}=1/(i+2)$ for j=0,1,....,i,i+1. I'm supposed to find the stationary distribution. So do I take the limit as n goes to ...
1
vote
2answers
44 views

Limiting Distribution of a Markov Chain

I'm having trouble understanding how to find a limiting distribution. If I have a Markov Chain whose transition probability matrix is: $$ \mathbf{P} = \matrix{~ & 0 & 1 & 2 & 3 & ...
1
vote
1answer
40 views

Electrical components failing in a Poisson process

A machine has infinitely many identical components. They fail according to a Poisson process with rate $\lambda = 4$/hour. A repairman arrives at time $t$ and instantly repairs all of the broken ...
2
votes
1answer
71 views

Prove that the absolute value of the difference of two invariant distributions on a Markov chain is invariant

If we have $a(x)$, $b(x)$ which are invariant distributions on a Markov chain $X_n$ with state space $S$, how can I prove that $|a(x)-b(x)|$ is also invariant? I know that I must show that: ...
0
votes
0answers
34 views

Multiclass Markov process

There are two car M/M/1 queues Q1 and Q2. Arrival rate of Red car and Green car in Q1 is $\lambda_{1R}$ and $\lambda_{1G}$ respectively. Similarly arrival rate of red car and green car in Q2 is ...
1
vote
0answers
21 views

Converse of Perron Frobenius Theorem: Necessary and Sufficient Conditions for positivity (or non negativity)

As I understand, Perron Frobenius theorem asserts only in one direction, i.e. if Matrix A is positive then there is a perron eigenvalue, eigenvector etc. What I wanted to know is what are the ...
0
votes
1answer
36 views

Measurability of a stopping time in a Markov chain

Suppose you have a finite-state continuous-time inhomogeneous Markov chain with transition rate $Q(t)$. Further, let us suppose that $Q(t)$ is a piecewise continuous function of $t$. Two questions: ...
3
votes
1answer
73 views

Markov Property Confusion

I feel like I'm being very dense/employing some sort of circular reasoning, but I'm having trouble understanding the Markov Property. According to Durrett (ISBN-10:1461436141), $X_n$ is a Markov chain ...
0
votes
1answer
25 views

Strategy for Unbalnaced Gamber Ruin

A gambler plays the following game: A fair coin is tossed until getting three times continuously head. When that happens the Gambler gets 20$\$$. Each round costs the gambler 1$ (even if he won the ...
0
votes
0answers
51 views

Practical Way to Detect a Markov Chain is Regular Given the Transition Matrix

I understand that a Markov Chain is reducible if, given its transition matrix $P$, there exists $n$ such that every element of $P^n$ is greater than 0. However, I am wondering that if there is an ...
1
vote
1answer
71 views

Eigenvalue range of $P+P^T$ (P is a transition matrix)

$P$ is a transition matrix of dimension $N\times N$. I know $\lambda_1=1$ and $|\lambda_i|<1, 2\leq i \leq N$. I want to know the eigenvalue range of $P+P^T$. Because $P$ is not symmetric, so I ...
2
votes
2answers
32 views

why are the recurrent classes closed?

i've recently started studying about markov chain, we call a communication class a recurrent one in a markov chain if by starting from that class we infinitely return to it with probability 1,with ...
0
votes
1answer
41 views

Definition of Stationary Distributions of a Markov Chain

I'm having a lot of trouble understanding the definition of the stationary distribution of a Markov Chain from Hoel, Port, Stone's Introduction to Stochastic Processes. They define the stationary ...
1
vote
0answers
16 views

Estimating Markov transition matrix for regularization

Suppose that I have a sequence of discrete distributions: $$ p_j = (p_{1j},...,p_{Cj}), \: j=1...D,\\ p_{ij}>0 \:\: \forall i,j,\: \sum_{k=1}^Cp_{kj}=1\:\:\forall j. $$ I suppose that these ...
2
votes
0answers
19 views

Showing which classes are recurrent and which are transient

If I have a Markov chain on states {0,1,2,3,4,5} $$ \mathbf{a} = \matrix{~ & 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & 1/3 & 0 & 2/3 & 0 & 0 & 0 \\ ...
0
votes
1answer
68 views

Markov chain for two players with two coins [closed]

Two players A and B toss two fair coins independently. Whoever gets the smaller number of heads will pay that many dollars to the other player. For example, if player A tosses two coins and gets 2 ...
0
votes
0answers
49 views

Show that this Markov chain is recurrent or transient

Consider the Markov chain $(X_n)_{n\geq 0}$ with state space $E=\left\{1,2,3,4,5\right\}$ and transition matrix $$ T=\frac{1}{2}\begin{pmatrix}0 & 1 & 1 & 0 & 0\\0 & 0 ...
1
vote
1answer
53 views

Markov chain - Can anyone explain me why this is the solution?

Customers arrive according to a Poisson process at a rate of four customers per hour. A customer who finds four other customers in already waiting gives up and leaves. Some clients in the 3rd ...
0
votes
0answers
33 views

diffusion- stuck

In a round room of radius R, a large number of coins N of diameter d are randomly dispersed upon the floor. A ladybird starts from the centre of the room, crawling at speed v. Suppose that every time ...