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.

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41 views

Question about random walk markov chain

For a random walk, let $a$ denote the probability that the markov chain will ever return to state $0$ given that it is currently in state $1$. Because the markov chain will always increase by $1$ with ...
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1answer
36 views

Independence of random variables derived from a Random walk

Let $w=(w_x)_{x \in \mathbb Z}$ be i.i.d random variables taking values in $(0,1)$. Let $(X_n)_{n \in \mathbb{N}_0} (\mathbb{N} \cup {0})$ be a Markov chain (more specifically a simple random walk ...
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22 views

Bayes: markov chain, serial connection, marginalization

Goal is to check if p(a) is unconditionally independent to p(c) in the markov chain - serial connection. $$ p(a,b,c) = p(a) p(b|a) p(c|b) $$ $$ p(a,c) = \sum_b p(a) p(b|a) p(c|b) = p(a) p(c|a) \neq ...
0
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1answer
21 views

Probability that Markov chain process has particular state after n steps

If we have a Markov chain X with four discrete states, and we want to find the probability the process is in a certain state (one of the four) n iterations later, would we raise X to the nth power and ...
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1answer
24 views

Conditional Probabilities Poisson Process

If I let ${X(t); t>=0}$ be a Poisson process having rate parameter $\lambda = 2$. I'm supposed to determine the probability: Pr{${X(1)>=2 | X(1) >=1}$} My solution: I looked at this as ...
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1answer
22 views

Conditional Distribution Poisson Process

In class, our professor told us to verify this solution on our own time. The problem is: Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the ...
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1answer
38 views

Modelling a continious-time queue which behaves differently when there are more or less people being served.

For my research I am trying to model a continuous-time queue which behaves differently when there are more or less people being served. The arrival rate in the queue is constant, however the departure ...
2
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0answers
22 views

$T_n$ stopping time, is $\{X_{T_n}\}$ markov chain

Let $\{X_n\}$ be a Markov Chain with finite state space $S$. Let $T_n$ be the $n$-th hitting time of $A \subset S$ i.e. $n$-th time it hits some state from the set $A$. Is $\{X_{T_n}\}$ a Markov chain ...
4
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1answer
38 views

How do stochastic matrices really converge?

We are given the matrix $A=\begin{bmatrix}0.9&0.5\\0.1&0.5\end{bmatrix}$ and any initial vector $X^{(0)}=\begin{bmatrix}a\\b\end{bmatrix}$. The matrix $A$ has the following eigensystem: ...
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0answers
29 views

mean recurrence time

$E\left[T_j |X_0=i,X_1=k\right]$ \left\ space{\begin{matrix} 1+U_{kj} \space\ k\neq j & \\ 1 \space\ k=j &\end{matrix}\right. Does this mean that the number of steps it takes to get back to ...
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2answers
23 views

Something about Markov chains

We check $P(X_{n+1}\in B|\mathcal{F}_n)=P(X_{n+1}\in B|X_n)$ when we want to prove $X_n,n=1,2,\dots$ is a Markov chain. Through this equation it seems that $X_n$ is a Markov chain if $X_{n+1}$ is ...
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1answer
32 views

Proving that the Markov chain is recurrent - Confusion/Help

Giving the following transition matrix [ 0.9 0.1 ] [ 0.8 .2 ] Classify the states From drawing the graph I know that both stats are recurrent. However I'm really failing to prove mathematically ...
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8 views

Is the steady state of a uniform markov chain always a vector of proportions?

Given that all edges in a markov chain are bi-directional (though not necessarily equally weighted), and each edge for a given node has equal probability, does the steady state always converge to a ...
1
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1answer
17 views

Definition of limiting distribution in a Markov chain — why do we condition on the initial state?

Given a Markov chain $\{X_n \mid n \in \{0, 1, \ldots\}\}$ with states $\{0, \ldots, N\}$, define the limiting distribution as $$ \pi = (\pi_0, \ldots, \pi_N) $$ where $$ \pi_j = \lim_{n \to +\infty} ...
0
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1answer
26 views

Expected time to absorption

I have been trying to solve the following problem for quite a while now, but not with much luck. The Question Let $P$ be the TPM(Transition Probability Matrix) of a DTMC with state space ...
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0answers
37 views

I am stuck on this Probability Question. Please help.

The Problem: Let a Markov Chain have R states. Show that if j is recurrent, then there exists $0\leq x\leq 1$ such that for $n > r$ the probability that the first return from state j occurs after ...
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1answer
28 views

Infinite$-$state absorbing Markov chains

Could someone provide a good reference/book about infinite$-$state absorbing Markov chains? Most of what I've found so far deals only with the finite$-$state case.
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36 views

Help with this Markov Chain Proof please

Problem: Consider a finite Markov Chain with N states $(1,2,...,N)$. Let $P(n) = [P_{i,j} (n)]$, be an n-step transition matrix. Suppose that $lim_{n\to\infty} P_{i,j} (n) = \pi_{j} $ for any $1 ...
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11 views

Conditional independence for a dynamic random field

Let $X = \left\{ {{X^{\left( \alpha \right)}}:\alpha \in {\mathbb{N}_0}} \right\}$ be a dynamic random field with a set of places $V$ and a phase space $\Lambda $ such that $\left| V \right| < ...
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0answers
57 views

When is $A^{+} P^{\top} A$ non-negative?

$P$ is a $n \times n$ stochastic matrix (non-negative, rows sum to one). $A \in \mathbb{R}^{n \times k}$ with $k < n$ has non-negative entries and independent columns. Denote by $A^+ \in ...
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1answer
25 views

Understanding the proof of stationary distribution of a markov chain

I am reading the proof of existence of stationary distribution in an irreducible markov chain from the book Markov Chains and Mixing Times by P. D. A. Levin, Y. Peres, E. L. Wilmer, and I have the ...
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0answers
15 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 ...
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16 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
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1answer
27 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 ...
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1answer
25 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, $P \geq \beta I$. ...
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0answers
30 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 ...
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1answer
17 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
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1answer
88 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. ...
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0answers
9 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, ...
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18 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 ...
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1answer
35 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 ...
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2answers
65 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
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1answer
44 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
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0answers
19 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. ...
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1answer
30 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 ...
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20 views

Markov chains for a two players with a coin, how can I find pij and transition matrix?

There are two players: A and B and there are N bill of one dollar, and at any stage of the game A has k and B has N-k of them. In turn either of them flips a fair coin, starting with A. If it comes up ...
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34 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 ...
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0answers
35 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 ...
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0answers
22 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 ...
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0answers
15 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 ...
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1answer
34 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 ...
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2answers
42 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
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0answers
49 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
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1answer
49 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 ...
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1answer
76 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 ...
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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!
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0answers
12 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 ?
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25 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 ...
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1answer
73 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 ...
4
votes
3answers
100 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 ...