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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1answer
39 views

How to get transition matrix of markov process?

I am monitoring a Markov process with ~21 states. I know all the states, initial state and what states transitions can/cannot be, so that zero elements of transition matrix are known. I know the ...
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2answers
57 views

Weather Transition Matrix

Based on observation, I've gathered some data: Day 1 2 3 4 5 6 7 8 9 10 S R S F S R F F R S (S) = Sunny (R) = Rainy (F) = Foggy How do I construct this into a transition matrix, for markov ...
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0answers
56 views

Markov Transition Matrix

I have some data, shown below. How do I construct a transition matrix, for Markov Chain ? I need the formula to calculate observation data into transition matrix. Thanks! Accumulative ...
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0answers
174 views

How to solve this simple 2D Markov chain

I have 2D Markov chain like below : I have already solved for $p_{b,0}$ and $p_{0,d}$ like this : $$p_{b,0} = \rho^bp_{0,0}$$ $$p_{0,d} = \rho^dp_{0,0}$$ But I don't know how to get $p_{b,d}$. So ...
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1answer
30 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 ...
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1answer
45 views

Strict proof of markovity of queing system of type $M/M/n/\infty$

I have a queing system of type $M/M/n/\infty$. The service time is exponential, and the arrival process is poisson. I do understand that because of these two facts the future of the system in ...
1
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1answer
80 views

Markov Chain Ergodic Theorem (Proof references)

Where can I find a proof of the erogidc theorem for Markov chains that doesn't use Birkhoff? The theorem states the following : Let $(X_n)_{n\in \mathbb{N}}$ be an irreducible and positively ...
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1answer
56 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 ...
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1answer
53 views

Recurrence of infinite markov chain

I have a Markov chain with state space $S=\{0,1,2...\}$ and a sequence of positive numbers $p_1,p_2,...$ where $\sum p_i=1$. The transition probabilities are based on these where $p(x,x-1)=1, ...
0
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1answer
12 views

First property of discrete time homogenous markov chain

I'm trying to understand the properties of a DTHMC. I am having trouble understanding with the first one. My textbook says - "$X_t$ takes values in $X$ for all $t$ (i.e. $X_t$ is a random variable ...
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0answers
47 views

markov chain computation

I consider a 2 state Markov chain: $X = \{1,2\}$, transitions are $M(i,j)$ and the matrix has a unique stationnary distribution $\pi$: $$ \pi(1) = \frac{M(2,1)}{2-M(1,1) - M(2,2)} \\ ...
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1answer
41 views

Markov chain probability that a state changes

For the Markov chain given below what is the best way to find the probability P{$x_n = x_{n-1}$} and P{$x_n \neq x_{n-1}$} The transition matrix of the chain is \begin{array}{} 1/3 & 2/3 & ...
0
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1answer
82 views

M/M/1 queue with probability of new client leaving

I'm looking at a M/M/1 queue system and trying to show that $\{M_t\}_{t\geq}0$, the number of clients in the system, is a birth-death process. In the simplest of cases this is true if $\lambda_i = ...
0
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1answer
22 views

radius of symmetric random walk on $\mathbb{Z}$

How to calculate the radius of the symmetric random walk on $\mathbb{Z}$, i.e. $\limsup_k (p^{(k)}(0,0))^\frac{1}{k}$? ($p^{(k)}(0,0)$ denotes the probability to get from $0$ to $0$ in $k$ steps and ...
2
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1answer
110 views

Proving that Markov Chain Monte Carlo converges

I am trying to understand how the very basic Markov Chain Monte Carlo approach works: We try to approximately calculate the expected value $E_{\pi(x)}[X]$ by drawing sequential samples from a Markov ...
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0answers
15 views

Are there any models that have mean $\sqrt{t\log(t)}$?

R. Arratia (The Motion of a Tagged Particle in the Simple Symmetric Exclusion System on Z) shows in theorem 2 that for step initial condition in the SSEP, the position of the lead particle, $x_1(t)$ ...
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0answers
74 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 ...
0
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1answer
52 views

Conditional mutual information and Markov chain.

If we have the Markov chain $X \to Y \to Z$, or equivalently $$I(X;Z| Y)=0, \tag{1}$$ where $I(\cdot)$ denotes the mutual information. Does the Markov chain $X \to (Y,W) \to Z$ also hold? Or ...
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1answer
52 views

Stickers in a box Markov chain problem

I'm revising for exams in June and my university, very irritatingly, doesn't provide mark schemes for past questions. I'm stuck a few parts into a question and am not totally confident of my preceding ...
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32 views

Markov chain definition: should the conditional probability also hold if $P(X_{n} = i_{n}, \ldots, X_0 = i_0) = 0$? Is $S$ a set of real numbers?

Definition of Markov Chain (as it is stated in my textbook): Let $S$ be a set of states and $\mathbb P = \{p_{i,j}\}$, $i,j \in S$ a transistion matrix . Then the sequence of RV's $(X_{n})_{n \ge 0}$ ...
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1answer
101 views

rate of convergence of absorbing markov chain

Let $G$ be a biconnected and non-bipartite graph. I can simulate a random walk on this graph with a markov chain. The stochastic matrix is $M = AD^{-1}$, where $A$ is the adjacency matrix of $G$ and ...
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1answer
57 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 ...
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1answer
33 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 ...
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0answers
110 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 ...
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0answers
26 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 ...
0
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1answer
28 views

limiting probability - what matrix to write

To find the limiting probability you solve the systems of equations: $\vec{\pi}=P\vec{\pi}$ $\Sigma \pi_j = 1$ and my teacher told us "you could rewrite this as matrices". Having just completed a ...
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0answers
91 views

Markov Process Feller Construction

I have this assignment question and I am stumped on how to complete a Feller construction: A system consisting of two components is subject to a series of shocks . The time be- tween consecutive ...
2
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2answers
207 views

Transition Matrix eigenvalues constraints

I have a Transition Matrix, i.e. a matrix whose items are bounded between 0 and 1 and either rows or columns sum to one. I would like to know if it is possible that in any such matrices the ...
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1answer
380 views

Mean time spent in transient states/Markov chain

I dont get this in my book: For transient states $i$ and $j$ , let $s_{ij}$ denote the expected number of time periods that the markov chain is in state $j$ , given that it starts in state $i$. Let ...
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56 views

Markov chains and natural filtration

I have the following problem Consider a homogeneous Markov chain $(X_n)$ with countable state set $E$. Suppose that $A$ is a proper subset of $E$ and consider the stopping times $\tau^0=0 $ and ...
0
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1answer
41 views

How many iterations are generally required when using the power iteration method?

Suppose I have an n x n matrix and I want to find the dominant eigenvalue and its associated eigenvector. Given these dimensions, what is the minimum number of iterations of the power iteration ...
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24 views

How to get from this probability formula to the one I need?

I'm working on a gambler's ruin problem where a player starts out with $i$ money, and 'winning' is when their total money reaches $N$ (ie they will keep playing until they reach N or run out of money, ...
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58 views

Standard deviation of a quantum walk?

The standard deviation of a classical random walk with $n$ steps is $\sqrt n$ - Standard deviation of a random walk. I have read in many places that the standard deviation of a quantum walk $n$ with a ...
0
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1answer
40 views

Markov chains for group decision making

I am new to Markov chains since I am doing my own studying on it recently. I was doing some questions and came across this one that got me stuck. Suppose there are four employees and they need to ...
2
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1answer
106 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 ...
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2answers
77 views

A Card game problem related to Markov chain

This card game problem originates from the killer game Sanguosha. We assume that all cards drawn in the game procedures below are with replacement, in order to keep the probabilities fixed when a card ...
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0answers
15 views

condition for recurrence of the chain

Let $\{X_n\}$ be an irreducible Markov chain with transition probability $P=(p_{ij})$ on a countable state space $S=\{0,1,2,\dots\}$.Suppose $s\in S$.Show that $s$ is a recurrent state if the there is ...
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1answer
64 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!
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2answers
263 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
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2answers
127 views

How do you prove that the second largest number $Y_n$ shown among the first $n$ rolls is not a Markov Chain?

How do you prove that the second largest number $Y_n$ shown among the first $n$ rolls is not a Markov Chain? My attempt: Consider the case, $P(Y_{n+1}=3|Y_n=1)=\frac{1}{6}$ if the current ...
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2answers
155 views

Find steps to reach absorbing markov chain state

How can I find the steps it takes or days or whatever the time variable is till the matrix reaches the absorbing state. e.g. take the matrix (The probability of each column adds to 1) $$ \left[ ...
2
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2answers
311 views

Professor has 4 umbrellas, Markov chain and Probability

OK this problem is making me tear my hair out. I need someone to walk me through this in baby-steps method like 1 + 1 = 2. I am trying to figure out what I don't understand. I know this is going to be ...
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2answers
668 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 ...
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0answers
58 views

Parametric transition matrix in Markov Chains

I am trying to model a discrete-time MC with transition probabilities that depend on some function of parameters i.e $p_{ij} = f(X_0,X_1)$. Suppose we take a log-linear model where $p_{ij} = ...
2
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1answer
68 views

Probability distribution of Poisson process

Let $X_t$ and $Y_t$ be two independent Poisson process with rate parameter $\lambda_1$ and $\lambda_2$, respectively, measuring the number of customers arriving in stores $1$ and $2$, respectively. ...
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65 views

Markov chain question

Consider an irreducible, recurrent Markov Chain ($X_n$) on a countable state space $S$ with transition probability $p(x,y).$ Pick a sigma-algebra $A \subset S$ and let $T_k=\inf\{n>T_{k-1}:X_n \in ...
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1answer
126 views

Markov chain problem, Help!

I am stuck on this question for a long time Question: Consider 4 balls, labelled from 1 to 4 and distributed amongst two urns (Urn 1 and Urn 2). At each time $n>1$, a number from 1 to 4 is chosen ...
0
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1answer
29 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 ...
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0answers
16 views

$\psi$-irreducibility of m-skeletons.

In Proposition 5.4.5 of Meyn and Tweedie's Markov Chains and Stochastic Stability, it is said that if a chain $\Phi$ is $\psi$-irreducible and aperiodic, then every $m$-skeleton of it is also ...
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2answers
49 views

Looking for good literature on Markov Chains with explicit calculations

I am currently starting my thesis on Markov Chains and am looking for good books and papers that include explicit calculations. I have taken a small course on Markov Chains so the subject is not ...