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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Function Composition over a Markov Chain

Can any one give me an example of how a composition of a function $F\circ(S,P)$ is not a Markov chain? if $(S,P)$ is a finite and discrete Markov Chain. I want to know how to construct a function F ...
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29 views

Markov processes: Hitting times for a point form an i.i.d. sequence

Say that I have a recurrent time-homogenous diffusion process X (continuous strong markov process) and two points $x,y$. If $X_t$ goes from $y$, to $x$ and then back to $y$ again we denote it as a ...
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18 views

a conceptual question on markov chain [duplicate]

Suppose $\{X_n,n\ge 0\}$ and $\{Y_n,n\ge0\}$ are two independent discrete-time markov chains (DTMC) with state space $S=\{0,1,2,\ldots\}$. Prove or give a counterexample to: $\{X_n+Y_n,n\ge 0\}$ is ...
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Showing that $(X_n)$ obeys the Markov Property. [on hold]

Consider a process $(X_n)_{n\geq0}$ where we define $X_0 = 0$ and for $n \geq 1$: $$X_n = X_{n-1} + Z_n$$ where $Z_n$ for $n \geq 1$ are independent random variables on $\{ -1, 1 \}$ with ...
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18 views

Compute distribution in Hidden Markov models

Let $Z_1, Z_2, ..., Z_n$ be the latent variables, and $X_1, X_2, ... X_n$ be the observed ones in a hidden markov models. Let's assume that the parameters of the hidden Markov models are known: the ...
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1answer
46 views

Error in Billingsley?

Problem 8.25 in the third edition of Probability and Measure by Billingsley (1995, p. 142) is as follows: Suppose that an irreducible [Markov] chain of period $t>1$ has a stationary ...
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17 views

Difference between AR model and Markov Chain

We know that Markov Chain can be represented as $$x_t=ax_{t-1}+\epsilon_t,$$ where $\{x_t\}$ are states, $\epsilon$ is noise, and $a\neq 0 \in R$. For the AR model, we know that, the first order ...
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57 views

How many steps would it take to get to the top of this staircase?

There are 26 steps in a staircase. You have a 51% chance to step onto the next step, and a 49% chance to step back down to the step prior. Assuming you are already on the first step, how many steps ...
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1answer
16 views

Find $P(\eta_t=m)$, $m=0,1,2,\dots,$

Let $\epsilon_t$, $t=1,2,\dots$ independent random variables with $P(\epsilon_t=1)=p$ and $P(\epsilon_t=-1)=1-p$. If $\eta_0=0,\eta_t=\eta_{t-1}+\epsilon_t$ , $t=1,2,\dots$ where $\eta_t$ is ...
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2answers
30 views

Ergodic components of Markov chain by transition matrix

I would like to find an algorithm for obtaining all ergodic components of a finite Markov chain with discrete time defined by its transition matrix (i.e. ergodic subchains into which the given chain ...
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24 views

Can an irreducible, recurrent continuous time Markov chain have spontaneous states?

Let $(X_t)_{t\geq0}$ be a continuous time Markov chain on some (possibly countably infinite) state space $S$ with Q-Matrix $q(\cdot,\cdot)$, transition function $p_t(\cdot, \cdot)$ and invariant ...
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1answer
81 views

Combinatorics Statistics Probability of a Letter Chain

What formula could help me quantify the probability of a chain of three letters (English Alphabet) where each letter is based on the previous one (stochastic modeling, Markov-chains, probabilities) ...
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11 views

Metropolis Markov Chain and Mixing Time

I have a statistical mechanical system, which I would like to sample with the Gibbs distribution using a Metropolis-Markov chain. I think the following are standard questions, but I am not sure what ...
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1answer
36 views

Show that if $\{X_n\}$ is a Markov Chain

Show that, if $\{X_n\}$ is a Markov Chain then $$P(X_n=j\mid X_k=l,X_m=i)=P(X_n=j\mid X_m=i),0\leq k<m<n$$ What I did is $$P(X_n=j\mid ...
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11 views

How to compute Generalized Group Inverse?

Given a transition matrix $P \in \mathbb{R}^{n \times n}$, i.e. $\sum_j P_{ij} = 1$ and $P_{ij} \geq 0$ for all $i,j$. One can show that there exists a unique group inverse $B$ of $A:= I - P$ which ...
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15 views

How to write spectral form of probability matrix

I am trying understand Markov chain in genetics process. In book that I am using (Mathematical Population Genetics) (pag 87): (P is matrix transition probability). $E_0$ and $E_M$ are absorbing ...
2
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1answer
20 views

Survey on large deviation bounds of queuing delay in CSMA scheduling

I am trying to do some literature survey on the theoretical guarantees in uplink scheduling algorithms. I found there exist a series of papers from UIUC and UC Berkeley by L.Jiang, J. Walrand, R. ...
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1answer
50 views

Population exercise with Markov chains

I am totally stuck with this exercise on Markov chains. Maybe someone can help me :). Red and green bacteria A growth medium at time $t = 0$ has 500 red bacteria and 500 green bacteria. Each ...
2
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2answers
60 views

Markov chains and conditioning on impossible events

Consider a Markov chain $(X_0,X_1,\ldots)$ with a state space $S\equiv\{s_1,s_2\}$ and the following matrix of “transition probabilities” (I will explain the use of quotation marks below): ...
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2answers
27 views

Markov chain periodicity

Can a Markov chain have 5 states, one open and one closed class and all the states be periodic (e.g. period 2)? I tried the following: https://www.dropbox.com/s/v818oqlizaci23m/Untitled.png?dl=0 but ...
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1answer
33 views

Stationary distribution in continuous-time Markov chain

Consider a barbershop with one barber who can cut hair at rate 4 (people per hour), and three waiting chairs. Customers arrive at rate 5 per hour. Customers who arrive to a fully occupied shop ...
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25 views

Interpolation of random processes

Let $\left(\Omega, \mathcal{F}, \left\{\mathcal{F}_{t}\right\}_{t\geq 0}, P \right)$ be a complete probability space with a nondecreasing family of right continuous sub-$\sigma$-algebras ...
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30 views

Markov Chain- Internet Router Buffer

At each time slot, a router's buffer receives a packet with probability $p$, or releases one with probability $q$, or stays the same with $r$. Initially empty, what is the distribution of the packets ...
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20 views

Gibbs sampling for Hidden Markov Model

I want to understand how to derive the update formula for Gibbs sampling for Hidden Markov Model, for example, in here: $p(z_t | \mathbf{x}, \mathbf{z}_{\setminus t}, \boldsymbol{\alpha}, ...
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The restriction of the green function is non-degenerate

On the context of irreducible Markov chains on a finite graph one defines $$G = \sum_{k=0}^\infty \hat{Q}^k $$ where $\hat{Q}$ is the restriction of a stochastic matrix to a subset (one can think ...
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1answer
24 views

What is the practical meaning of probability vectors?

I have been reading a lot about probability vectors, as a part of "Introduction to Probability" course. Now, whenever it was mentioned, it was defined theoretically as a vector whose entries add up to ...
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53 views

Weak convergence of a sequence of stationary distributions to another stationary distribution

Let $\{X_n(t) \in \mathbb{Z}^+\}$ for each $t \in (0,1)$ denote a discrete time Markov chain (with time index $n$ and parameterized by $t$). For each $t$, the Markov chain $\{X_n(t)\}$ has a unique ...
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1answer
36 views

What is wrong with matrix [[1,.5,0] [0,0,0] [0,.5,1]] steady state?

I know that Markov matrices have steady state since they always have eigenvalue $\lambda = 1$. We just solve the system of equations $A\vec x = 1 \cdot \vec x$ or $$\begin{cases} k_{a\to a} a + ...
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34 views

Transition density of an AR(1) process?

If we have an AR(1) process, i.e: $X_{t+1} = \alpha X_t + e_{t+1}$ with $X_0=0$ then what is its Markov Chain transition density? We know that for a Markov chain, the following holds: $P(X_{t+1}\leq ...
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Relaxation time and Mixing time of Markov chains

The notation is mostly taken from the book "Markov chains and mixing times" by Levin, Peres, and Wilmer. Consider an irreducible, aperiodic, time-reversible, discrete-time Markov chain on a finite ...
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1answer
76 views

Expected Value of a Mosquito

A mosquito is walking at random on the nonnegative number line. She starts at $1$. When she is at $0$, she always takes a step $1$ unit to the right, but, from any positive position on the line, she ...
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Billingsley Exercise 8.8 (Markov Chains)

I am studying from Billingsley and would like some hints on the following exercise. Suppose $S = \{0,1,2,...\}$, $p_{00} = 1,$ and $f_{i0} > 0$ for all $i$. Here, $S$ represents the state ...
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1answer
59 views

Analytic Center of Convex Polytope

I have a convex polytope defined by $Ax \leq b$. I want to know how to find the "analytic center" of my convex polytope, because my goal is to sample from the polytope using Monte-Carlo Markov ...
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2answers
33 views

When to stop checking if a transition matrix is regular?

The definition that I have of a Transition Matrix for a Markov Chain is: A transition matrix is regular if some power of it is positive. Doesn't this mean though that in theory, you could keep ...
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1answer
42 views

Expected time until reaching absorbing state of Markov chain

I currently try to model nucleation as an absorbing Markov chain. I have an idea how to do that but, however, I cannot convince myself that it is correct. The state space consists of the number of ...
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16 views

Reference for General state space Markov chain

What is a good reference for general state space Markov chains? Is there a reference which assumes only familiarity with finite/countable state space Markov chains and then extends the results (e.g., ...
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37 views

Confusion regarding Burke's theorem

Arrivals occur at rate $\lambda$ according to a Poisson process the service time have an exponential distribution with parameter $1/\mu$ in an M/M/1 queue, where $\mu$ is the mean service rate where ...
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101 views

When the sum of Markov chains is a Markov chain: “dumb” algorithm

Suppose I have two (independent) discrete-time and space, preferably non-homogeneous Markov chains $\Gamma^{(i)}=\{\gamma_1^{(i)},\gamma_2^{(i)},...\}, \ i=1,2$ and I want to find a way to check when ...
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1answer
21 views

Closed communicating class

Let $P_{ij}$ a transition matrix, a class $C$ is closed if given two different states $i$ and $j$ $$i\in C, i\rightarrow j\Rightarrow j\in C$$ If a Markov Chain is irreducible the transition matrix ...
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65 views

Probability in a fixed die

I have that transition matrix is ...
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18 views

Chernoff-type bounds for Markov chains

I found the following result adapted from "Chernoff-type bound for finite Markov chains" by Pascal Lezaud, The Annals of Applied Probability, 1998, Vol. 8, No. 3, 849-867. Theorem: Let $P$ be the ...
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2answers
78 views

Probability returning to initial state

Let $P=\begin{bmatrix}0&\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&0&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}&0\end{bmatrix}$ and $P^{(n+1)}=P^{(n)}P.$ I know that if you start in any ...
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23 views

Expected response time of Continuous time Markov chain

I'm studying CTMC (Continuous Time Markov Chains). I came across the following slide I don't understand how they got $M(t+h) = M(t) + \alpha h + M(t)\lambda h - M(t) \mu h +o(h)$ Could anyone ...
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35 views

Markovian systems: Why must controls be independent of state?

I am currently working my way through Probabilistic Robotics by Thrun, Burgard, and Fox. On p. 91, I encountered the following statement: The Markovian assumption implies independence between ...
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Joint Markov Chain (Two Correlated Markov Processes)

I have two Markov Chains, and they exhibit some correlation between them. For instance, when Chain A moves to state i, there is a high likelihood that Chain B moves to state j. How would I go about ...
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21 views

Probability of random walk visit in nonameanable graphs

Consider a vertex-transitive nonameanable graph. Consider a site $x$ having a graph distance $d$ from the origin and let $X(n)$ be a random walk starting from $x$. Is there a general upper bound as a ...
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1answer
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Model Complexity for higher order markov model

I do not understand why is there an increase in parameters when moving from first to second order markov model For example considering a feature space of (a - z) For first order markov model, the ...
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47 views

Tricky Markov Chain

I found this problem a bit tough and was wondering if you could give it a go (especially the last part). This goes as follow : A gambler wins $1$ dollar at each round, with probability $p$, and ...
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41 views

Defining a function over time in terms of a random variable that is undefined at a certain time

Let $X_n$ be a random variable taking on one of three values $a,b$ or $c$ over time. That is, for each $n \in \mathbb{N}$, we have $X_n \in \{a,b,c\}$. Also, for each $n \in \mathbb{N}$, let $F_n$ be ...
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Why is the stationary distribution a distribution?

Suppose we have a time-homogeneous, discrete-time, aperiodic, positive recurrent, irreducible Markov chain $(X_t)_{t \geq 0}$ on a discrete state space $E$. It is known that its stationary ...