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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Markov chains: is “aperiodic + irreducible” equivalent to “regular”?

I have two books on stochastic processes. In one book, it says that the limiting matrix is possible to find if the matrix is regular, that is, if for some $n$ $P^n$ has only positive values. The ...
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Markov Decision Process - Optimal policy invariance to scaling in the Utility Function

The title says it all. If i use a discounted Utility Function, why is the optimal policy invariant with respect tot the scaling of the Utility Function by a positive Factor?
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A theorem in the paper “Noncommuting Random Products” by Furstenberg

I have a question concerning the proof of theorem 2.5 at page 395 of the paper Noncommuting Random Products, by H. Furstenberg, Trans. Amer. Math. Soc., 1963. The statement is as follows: Let $\mu$ ...
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What does stationarity of the point process entail in a Markovian setting?

Let $(\Omega,\mathcal{F},P)$ be a probability space where $\Omega$ is the set of cadlag trajectories from $\mathbb{R}$ to a countable state space $S$. Let $X$ be the coordinate process $X_t(\omega) = ...
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Log-likelihood function

I'm not sure if this could be asked here, or in math overflow... In the following paper Cho, Jin Seo, and Halbert White. "Testing for regime switching." Econometrica 75.6 (2007): 1671-1720. doi: ...
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MDP problem - How is the expected cost calculated?

I have been stuck with a problem for a while regarding Markov Decision Processes for a Policy improvement algorithm. Assume that I have probabilities for certain states to evolve the system into, ...
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182 views

Equivalent defining Markov property

Consider the stochastic process $(X_t)_{t \in \mathbb{R}}$ and show the equivalence of the following two Markov properties: (a) $P(X_t \in A \mid X_u, u \leq s) = P(X_t \in A\mid X_s) \qquad ...
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Is this two dimensional Markov chain correct for this queueing system?

The problem that I have two single server station with no queuing space a customer goes to station 1 if it is available else it goes to station 2 if it is available or it will be lost output from ...
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55 views

Strong Markov property given transition functions

Suppose we are given family of transition functions satisfying Chapman-Kolmogorov equation, what conditions will ensure that there exists a continuous or cadlag Markov process with given transition ...
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State Space Difference Linear Dynamic System

I am interested in finding the DIFFERENCE in the state space distributions for two linear dynamical systems (System A and System B). I am able to solve for this using the matrix exponential. But the ...
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76 views

transition matrix for Markov chain

Can any one help me to solve this home work please? The city of Sacramento recently completed a new light rail system to bring commuters and shoppers into the downtown area and relieve freeway ...
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55 views

Estimate the probability using Markov chains

please consider this question: A study using Markov chains to estimate a patient's prognosis for improving under various treatment plans gives the following transition matrix as an example ...
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63 views

Stationary distribution for a Markov chain which is not irreducible

I have a Markov chain with $K$ states $S$: {$s_1,s_2,...,s_K$}. $s_1$ is reachable from any state in $S$; however not all the states can be reached from $s_1$. What does the stationary distribution ...
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Does time homogeneity imply strong Markov property in a Markovian process

Does a time homogeneous Markovian process necessarily have strong Markovian property? Does continuity in state space, time, or path make a difference? What are the examples if it does not?
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Examples of decreasing-in-some-time-interval variance of a time homogeneous Markovian process

Let $x_t$ be a zero mean, time homogeneous Markovian process over time $t$ starting from $x_0=0$. What are the examples of $x_t$ where the variance at $t$ decrease over some interval of $t$? The ...
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180 views

Proof that Markov Chains converges to the stationary distribution

Let $P$ is a transition matrix of a Markov Chain, which is irreducible, aperiodic and lets assume $\pi$ is its stationary distribution: $\pi = \pi P$. Does anyone knows the proof for the following ...
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Probabilistic event triggered on a Markov Process transition

I would like to assess a system disponibility using a Markov Process. This system has two states : a functionning state 0 and a failure state 1, with a fault rate $\lambda$ and a mean time to repair ...
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Exsistence and uniqueness of stationary density for Markov Chain

Suppose we're given a function $f:\mathbb{R}^2\to\mathbb{R}$. We define a Markov Chain $(X_n)$ by \begin{align} X_0&\sim f_X, \\ X_n&=f(X_{n-1},Y_{n-1}), \end{align} where $(Y_n)$ is a ...
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Asymmetric Markov process

The limiting distribution $\xi(x)$ of a Markov process $$x_0=1\text{ and }x_{i+1}=x_i+\Delta x_i,\tag1$$ where $\Delta x_i=-ax_i$ and $\Delta x_i=a$ occur with equal probability for every $i$, and ...
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Markov operators

Transition probability functions can always be used to generate Markov operators, correct? So is it correct to say that a Markov process is a collection of Markov operators? On the other hand, are ...
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How to prove theorem about consistency of Markov edge process?

How to prove such theorem: Markov edge process $p_E(y_E)$ with respect to DAG $G=(V,E)$ defined as $p_E(y_E) = \prod_{v \in V} p_E\left(y_{E_{\rm out}(v)} \,\big|\, y_{E_{\rm in}(v) } \right) = ...
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Help With Eigenvectors and Dynamical Systems

I have the following system of differential equations: $ \frac{d}{dt} \left[ \begin{array}{c} A(t)\\ N(t)\\\end{array} \right] = \left[ \begin{array}{c c} -(a+b) & 0\\ a & -(a+b)\\ ...
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First and second moments of recurrence time in a finite two-dimensional Markov chain

I have a two dimensional finite Markov chain with $(m+1)^2$ states, and with transition rates: $q_x((x,y)\to (x+1,y))=(m-x)\lambda,\quad 0\leq x< m, 0\leq y \leq m$, $q_x((x,y)\to ...
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94 views

Markov chain and hitting times

I have a Problem about hitting times. That's the following: Let $A\subset E$ and the first passage time $T_A$ and the hitting time $H_A$. Define: $T_A =\inf\{n\geq 0;X_n \in A\}$ and $H_A ...
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62 views

Steady state distribution for Markov pure jump process

Assuming an irreducible, positive recurrent Markov Pure jump markov process with state space, $S={0,1}$ The embedded Markov Chain which is doubly stochastic (i.e) columns and rows of the transition ...
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Calculating cumulative Markov Chain outcomes

I have a Markov process, with 2 possible states (1 or 0) and a transition matrix P. State at time t=n is determined by x0*Pn. As n goes to infinity, xn goes to the steady state vector, q = [q1 q2]. ...
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80 views

transition kernel

I've got some trouble with transition kernels. We look at Markov process with statspace $(S,\mathcal{S})$ and initial distribution $\mu^0$. We have a transition kernel $P:S\times ...
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36 views

Does a Markov Blanket allow connections between Parents of a Node?

In a Markov Blanket, we can connect the childredn of a node between them, as a child can be parent (or spouse) of another child. Does this rule apply as well for Parents of a node? In advance, Thank ...
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Find Markov policy that minimizes(maximizes) the expected discounted cost(reward)

It's an exam problem I found online.Here's a link to the pastpaper. The problem is stated as follows. A repairman who services Q facilities moves between location s and location j according to the ...
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1answer
110 views

What is the difference between positive presistent and null persistent state in a Markov Chain?

I'm not looking for the difference in the mathematical definition, but rather for an intuitive explanation of their differences and possible examples, so that I can have them in my head when ...
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84 views

Markov Chains Proof using Statistics

Source: This came from "Introduction to probability" by Charles Miller Grinstead, and James Aurie Snell. It was located on page 407 and is Theorem 11.1 in the section 11.1 Introduction. Theorem: The ...
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Transition function is a Markov semigroup?

How does the transition function in a Markov process become a Markov semigroup in time homogeneous Markov processes? Thanks a lot.
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Question on the proof of the simple Markov property of a Brownian motion

Today we proofed the (simple) Markov property for the Brownian motion. But I really don't get a crucial step in the proof. The theorem states in particular that for $s\geq0$ fixed, the process ...
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47 views

makov chains and property

let $X_t$ for $t \in \{1,2,...,10\}$ be a sequence of independent tosses of a fair coin. We denote heads with 1 and tails with 0. Define the random variable $S_n=\sum_{t=1}^n X_t$ for $n \in ...
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Markov time $ T= \min\{n : X[n] = 1\}$

Let $T$ is a Markov time such that $T= \min \{ n : X[n] = 1\}$ , $X[n]$ is the number of $h$ (heads) in coin tossing for $n$ times. Let's say I will toss the coin 3 times, so the event collection is ...
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Proof for a Markov process example (using measure theory)

Consider the probability space $(\mathbb{R},B(\mathbb{R}),\delta_x)$ for a given $x\in\mathbb{R}$ (where $\delta_x$ is the Dirac measure) and define the process $X_t(\omega)=\omega - t$, for $t\geq ...
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How to Prove by definition, the given process is a Markov Process?

Define the process Xt by X0 = 1, and for t = 1, 2, . . . Xt = { uXt-1, with probability p, { vXt-1, with probability 1-p where 0 < v < 1 ...
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96 views

How is a simple birth process is time-homogeneous?

Why is it that a simple birth process is time-homogeneous? The incidence of a birth in a small time interval depends on the population size at the time of start of the interval. Doesn't this ...
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Interpretation of potential kernels for Markov processes

One can associate a strongly continuous contraction semi group (SCCSG) to a Markov process with state space $S$ through its transition function, say $P_t$. Now one can interpret $P_t$ as a linear ...
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159 views

Joint density of Markov process as product of conditional densities.

Let $(X(t))_{t\geq0}$ be a Markov proces such that s $X(0)=x_0$. Consider the random vector $(X(t_1),\dots,X(t_n))$ with corresponding joint density $g(x_1,\dots ,x_n)$. Is it then true that ...
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Advanced reference in Markov processes

I am interested in a book which covers the more in depth stuff on continuous time Markov processes e.g. semi-groups, generators... Preferably such book would also contain a list of analysis results ...
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363 views

Why is this infinite-state-space Markov chain positive recurrent?

Given the following transition matrix for a Markov chain, how can I see that the chain is positive recurrent? I want to convince myself that the chain has a limiting distribution, and the chain is ...
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183 views

Why do these expressions tend to zero?

In a chapter on Markov chains, it is claimed that $\binom {2n}{n} p^n(1-p)^n$ (where $p$ is a probability) tends to $0$ as $n$ tends to $\infty$. But why is this so? It is also claimed that for an ...
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115 views

How to understand Markov property?

I'm learning stochastic process in college. How to understand Markov property?I'm curious about what is the power and validity of Markov property ? A stochastic process has the Markov property if ...
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A Continuous-Time Markov Process Taking All Possible Values

Let $\mathbb{N}$ be the set of positive integers. For each $n \in \mathbb{N}$, let $X^{(n)}=\{ X^{(n)}(t): t \geq 0 \}$ be a Markov chain with state-space the two point set $\{0,1\}$ and $Q$-matrix ...
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Hilbert's Barber Shop

Hilbert opens a barber shop with an infinite number of chairs and an infinite number of barbers. Customers arrive via a Poisson random process with an expected 1 person every 10 minutes. Upon arrival, ...
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Markov Chain discarding balls from urn

The following question has me stumped. Any ideas on how to get started? An urn contains $n$ green balls and $n+2$ red balls. A ball is picked at random: if it is green then a red balls is also ...
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For a generator $G$ of a Markov process in continuous time and finite state space, how would one prove that the entries of $e^{tG}$ are non-negative?

I have a generator matrix G for a Markov chain in continuous time and finite state space and I am looking to prove that the entries of $e^{tG} \geq 0 $ By definition $G = P'(0)$ with entries $g_{ij} ...
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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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Inner product space and Reversible Markov chain

I am trying to clarify the following: Suppose P is the $N\times N$ transition matrix of a finite-dimensional Markov Chain, with invariant distribution given by N-vector $\mu$ (i.e. $\mu^T=\mu^T P$). ...