A stochastic process satisfying the Markov property: the distribution of the future states given the value of the current state does not depend on the past states. Use this tag for general state space processes (both discrete and continuous times); use (markov-chains) for countable state space ...

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Mean time for the renewal process

The system is as below. Energy keeps coming at a node with a constant rate ρ. Node has files of size exponential(λ) to be transmitted (with fixed rate of transmission $r$, time for transmission would ...
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15 views

What is the limiting distribution of this Markov Chain?

Take a Markov Chain with state space $\left\{ 0, 1, \dots, 20 \right\}$. Then we have the rule that given $X_n$: Compute $Z = X_n + 1$ or $Z = X_n - 1$ with probability $\frac{1}{2}$ each (if the ...
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Autocorrelation of a Markov Chain?

Is there a general characterization of the autocorrelation metric of a Markov chain? There are some tangential issues as well: do $n$-state transition probabilities obtained through Chapman-Kolmogorov ...
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1answer
21 views

Markov property of Brownian motion

There are two statements about Markov property: $B_t $ is Brownian motion and $\mathcal{F}$ is generated by $B$ If $s>0$ and $Y$ is bounded and measuable, then ...
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1answer
49 views

Gambler's ruin and Markov Chain, coin toss and stakes

I'm considering a classical problem about Markov Chains: A gambler has $£8$ and wishes to get to $£10$. A coin is tossed repeatedly : if it comes down tails, the gambler loses his stake, and if it ...
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28 views

A die whose score cannot be as before (Markov chains)

A die is "fixed" so that each time it is rolled the score cannot be the same as the preceding score, all other scores having probability $1/5$. Given that the first score is 6, what is the probability ...
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1answer
50 views

Application of Lévy–Khinchine formula

How can we express the characteristic functions of Wiener and Poisson processes by using the Lévy–Khinchine formula? I don't know how to find the characteristic functions of particular Levy ...
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19 views

Setting up and Solving Kolmogorov Forward Equations

Consider a Markov Chain with $3\times 3$ generator matrix: $$ G = \begin{bmatrix} -1 & 1/2 & 1/2 \\ 1/2 & -1 & 1/2 \\ 1/2 & 1/2 & -1 \end{bmatrix} $$ What are the ...
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HMM - forward algorithm (Part-of-Speech Tagging)

In order to understand the Forward algorithm for Hidden Markov Models, I created a Little example of Part-of-Speech Tagging. Consider the Hidden Markov Model with states $N$ (Noun), $V$ ...
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2answers
64 views

Find the generator of Markov Process

Homework question: Consider the Markov process $X_t=B_t-t^2+t$ where $B_t$ is the Brownian motion. Find the generator $Q$ of this process. I am completely confused how to find the generator for ...
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1answer
10 views

Bayesian Network vs Markov Decision Process

I am wondering if somebody can tell me anything about the practical differences between using Markov Decision Processes and and Bayesian Networks in reasoning about probabilistic processes?
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2answers
60 views

Resolvent operators and inverses proof

I am trying to prove for myself that $A(R_{\alpha}g)=\alpha R_{\alpha}g-g$ which is proving problematic. The definition of $A$, the generator, is $\displaystyle Af(x)= \lim_{t \rightarrow 0} ...
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Proportion of arrivals taking a particular path in a Routing Matrix

I have a routing matrix with Node-0 being the source/sink (outside world) and there are service Nodes 1,2..k in the system. The matrix has entries R_ij = Probability of an arrival at Node-i moving to ...
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3answers
38 views

an exercise about mean and probability

Let $(\Omega, \mathcal{F}, P)$ be a probability space, $X : \Omega \rightarrow \mathbb{R}$ be a discrete random variable and $$\phi : [0, \infty) \rightarrow (0, \infty)$$ an increasing function (so ...
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1answer
61 views

Finite state Markov chain

Under what conditions a Markov chain can be considered as finite (and not infinite)? Thank you!
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How can I calculate value of theta(state-probability distribution) for this markov game with one-sided information? [closed]

It's an infinite horizon markov game, where player 1 observes realization of each state and player 2 doesn't. They both observe each other's actions (typical markov game). I am trying to simulate this ...
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20 views

Stochastic scheduling to maximize the expected number of customers arrived at the root of a Jackson tree

In a Jackson network, organized as a tree rooted at queue r, several customers are queued at time 0 and there is no new customer arrival. The service time of each customer in queue i is geometrically ...
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23 views

proof of Bernstein inequality using Markov

I have to solve the following problem: Use the Markov's inequality to prove the Bernstein's inequality: $$P(X\geq x) \leq e^{-tx}E[e^tX],$$ for any t>0. Suppose $X_{1}$,$X_{2}$,.... are independent ...
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1answer
44 views

Connections between SDE and PDE

I have encountered a number of situations where the solution of a PDE and a certain expectation associated to a Markov process are equal. Two examples include: The heat equation $u_t = \frac{1}{2} ...
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1answer
68 views

How to fix the embedded discrete time Markov chain of a continuous time Markov chain …

Suppose $X$ is a continuous time Markov chain with three states $\{0,1,2\}$ and suppose that $\lambda_0<\lambda_1<\lambda_2$ be all positive such that $\lambda_i$ is the rate of transitions out ...
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1answer
33 views

How to connect the deterministic and probabilistic descriptions of the SIR model

I am a 17 year old student and I was reading up on epidemic modelling for a math project, specifically the SIR model and I came across this: "This" refers to the assumptions to which the Markov Chain ...
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1answer
38 views

convergence of nullrecurrent markov chain

Hi guys! At the moment I'm working on this proof. It's in a german book so hopefully you understand everything. I understand everything in the picture without the use of the markov property at ...
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Given initial state and steady state, how do I find the transition matrix?

The initial vector is $x_0 = \{.5, .5\}$ and the steady state is $x_\infty = \{1/9, 8/9\}$. How do I get the transition matrix, such that $$\lim_{p\to\infty} x_0 A^p = x_\infty $$ where $A$ is a ...
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1answer
25 views

Computer failure with Markov chains and n-step transition matrix

Hi I am struggling with a Markov Chain question: A computer network has two servers, only one of which is in operation at any given time. A server may break down on any given day with probability p. ...
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30 views

A Counter Example of Doeblin Condition

The question is to prove that the following Markov process doesn't satisfy the Doeblin Condtion. Let $X=\{\ldots,-n,\ldots,-1,0,1,\ldots,n,\ldots \} $, The Markov Transition Matrix $P$ is defined as ...
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27 views

Kolmogorov backward equations for Birth-Death

I'm trying to solve the Kolmogorov backward equations for a Birth-Death Markov chain with three states. I have 2 equations: $$P_{00}'(t) = \lambda_0 (P_{10}(t)-P_{00}(t))$$ $$P_{10}'(t) = ...
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1answer
49 views

Is $(B_t^2)$ Markov where $(B_t)$ is Brownian motion?

I am pretty sure $(B_{t}^{2})$ not Markov because the squared random walk is not. Showing the square of a Markov process is or isn't Markov I guess I can repeat the method since to be Markov it ...
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33 views

Chapman-Kolmogorov equations of time inhomogenous Markov chains

Let us assume that we are given a time inhomogenous Markov chain in continuous time (ICTMC) $(X(t))_{t \geq0}$ with a finite state space $\{1,\ldots,n\}$. Set $P(t)_{i,j} := \mathbb{P}(X(t) = j \mid ...
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2answers
46 views

Verifying the Markov property

We throw a dice infinitely often. Define $U_n$ to be the maximal number shown up to time $n$. How can I verify that $$ ...
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1answer
48 views

Why is the Stochastic Process in the HJM model non-Markovian?

I want to understand exactly what my title asks "Why is the Stochastic Process for the short rate in the HJM model of interest rates non-Markovian?" That process is the following: ...
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2answers
447 views

Expected state of a Markov chain

Let's start with a slightly trivial Markov chain defined as follows: the beginning state is called $1$ and the set of states is $\mathbb{N}$. At each step, when the current state is $n$, the ...
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3answers
434 views

Estimating the transition matrix given the stationary distribution

Let's say we are given a Markov chain for variable $X = [x_1, ..., x_n]$; also we are given a desired stationary distribution for this graph $P_\infty = [p_1, ..., p_n]^\top$. How can we design an ...
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Why Markov matrices always have 1 as an eigenvalue

Also called stochastic matrix. Let $A=[a_{ij}]$ - matrix over $\mathbb{R}$ $0\le a_{ij} \le 1 \forall i,j$ $\sum_{j}a_{ij}=1 \forall i$ i.e the sum along each column of $A$ is 1. I ...
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1answer
102 views

Brownian Motion inequality (related to Dvoretzky-Erdoes test)

i have the following question: Let $B(t)$ be a d-dimeansional Brownian motion $d\ge 3$, and $f$ be a monoton increasing function from the positive reals to the positive reals. Let $A_n=(\exists t\in ...
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1answer
19 views

Brownian Motion and Progressive Process

Let $B_t$ be a Brownian motion. Define sign function as follows. $sign(0) = 0$ and $sign(x) = \frac{x}{|x|}, \forall x \neq 0$. I do not know how to show the following two questions, especially on the ...
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What is the importance of the infinitesimal generator of Brownian motion?

I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have ...
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4answers
261 views

Discover where Bob is sleeping using hidden Markov chains

Bob lives in four different houses $A, B, C$ and $D$ that are connected like the following graph shows: Bob likes to sleep in any of his houses, but they are far apart so he only sleeps in a house ...
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Can we find a correlation between states of a Markov chain?

I have a fair bit of knowledge on Markov chains but I recently wondered if there is a way to find out a correlation between the states of a finite Markov chain. I could not find any material on this. ...
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Application of Strong Markov Property

Theorem SMP (Strong Markov Property) Let $X$ be a time homogenous Markov process with $T=\mathbb R_+$ or $\mathbb Z_+$ and let $\tau$ be a stopping time taking countably many values. Then ...
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Linear Filtering Problem (Keynman Fac/Particle Model)

$lienar Filtering Problem $$X_n^1 = X_{n-1}^1 + \epsilon_n *W_n $$ $$X_n^2 = (1-\alpha* \delta) X_{n-1}^2 + \beta*\delta X_n^1 $$ $$X_n^3 = X_{n-1}^3 + \delta*X_n^2$$ above is $$\approx$$ $$dX_n^1 ...
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1answer
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Ergodic behaviour for bounded random dynamical system

Considering an iterated system described by $$ X_n =\gamma_nX_{n-1} , $$ where $\gamma_i$ are non-negative i.i.d. variables. It is easy to show that the expectation will grow unbounded for $\mathbb ...
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23 views

Stationary VS. limiting probability

I'm just wondering what the difference between stationary probability and limiting probability is. And, if any of you know: What does it mean that some elements exist and some elements doesn't, when ...
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Stochastic process using Markov chain (thief on the run!!)

I'm given an exercise where we are to simulate a thief escaping from an officer. The thief (let's call him/her T for simplicity) and an officer (O) have four cities to be in. Let's call the cities A, ...
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Covariance of states of a finite Markov chain

I know it is possible to construct a covariance matrix for states of a Markov chain but I cannot seem to find a proper way to compute it. I will attach some theories I found from Kemeny and Snell's ...
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210 views

Hidden Markov Model, transition probabilities which are modeled with an exponential distribution

I'm looking at implementing an algorithm described in a paper, but I'm having trouble understanding how the transition probabilities for a Hidden Markov Model are defined. In the first sections, I ...
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27 views

Strong Markov property of continous time Markov process

In the book "Applied probability and queues" which is available here http://books.google.de/books?id=BeYaTxesKy0C&pg=PA32&hl=de&source=gbs_toc_r&cad=3#v=onepage&q&f=false , ...
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1answer
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Including non-markovian processes in a birth-death process

Current model I want to model a stochastic system with a birth-death (Markovian) model. I therefore have this kind of $n$ times $n$ (where $n$ is the number of possible states) transition matrix: ...
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A simple question on the inverse z-transformation of $\frac{z}{1-z}\mathscr{T}(z)\mathbf{q}$

I'm wondering if anyone who is familiar with the book Dynamic Programming and Markov Processes by Ronald Howard or simply z-transform can help me figure out an inverse z-transformation on page 23 of ...
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2answers
51 views

Interpretation for the determinant of a stochastic matrix?

Is there a probabilistic interpretation for the determinant of a stochastic matrix (i.e. an $n \times n$ matrix whose columns sum to unity)?
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1answer
23 views

Recurrence of states in a function of a Markov chain

Suppose $X$ is a Markov chain (or process, for that matter) and suppose further $f(X)$ is also a Markov chain. Let $s$ be a recurrent state in $X$. Is there a general way to determine the recurrence ...