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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Semigroup of operators: weak continuity at 0+ implies weak continuity at any t > 0

Let ($E$, $d$) be a metric space. Consider the semigroup $\{P(t)\}_{t\geq 0}$ of bounded linear operators on the Banach space $\hat{C}(E)$ of continuous real functions on ($E$, $d$) vanishing at ...
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Solving simple decision-making model over multiple periods

Consider the following model. Each period t=0,1,..., an agent makes an effort $x\in R_+$ to solve a problem. The value from solving the problem is $V>0$. The relationship between effort and ...
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38 views

Markov chain steady state existence

Is it possible for a Markov chain to have no steady state solution ? What is an example of such system ?
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Understanding the proof of an Ergodic theorem for Markov chains

An ergodic theorem for Markov chains is as follows. If a Markov chain $(X_n)_{n \ge 0}$ is irreducible and has an invariant distribution $\pi$, then $$\frac{1}{n} \sum_{k=0}^{n-1} f(X_k) \to ...
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Robustness of Markov Chains

A Markov Chain on a measurable space $X$ is uniquely determined by a stochastic kernel $P$ on $X$. Let $\mathsf P_x$ denote the probability on paths generated by $P$ and the initial condition $x\in ...
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probability of hitting state $i$ in random walk

We have a random walk on the integers with probability of going to the right is $\lambda$ and to the left is $\mu$. Suppose we start at 0. I want to find the probability of ever hitting a fixed state ...
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markov property in Durrett's textbook

Assume $B_t(\omega)=\omega(t),\omega\in (C,\mathcal{C},\mathbb{P}^x)$ is a B.M.(C is the continuous function space ,$\mathcal{C}$ is generated by the coordinate maps) In Durrett's textbook,he proved ...
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Computing world states from uncomprehensive sensor readings

I have a real world system, which consists of items assuming different locations at different times. The state transitions are controlled by machinery in the real world, which is well understood. ...
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40 views

What is a convolution kernel?

What is a convolution kernel? (in measure theory, probability theory) In which book can I read about kernels on measurable spaces and convolution kernels? Thank you!
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17 views

Markov Chain Steady State 3x3

I have been learning markov chains for a while now and understand how to produce the steady state given a 2x2 matrix. For example given the matrix, ...
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Generator of a Feller semigroup on a coutable space

Let $E$ be a countable set in the discrete topology. Let $(T_t)_{t \geq 0}$ be a Feller semigroup on $E$, i.e. a strongly continuous semigroup of operators on $\mathcal{C}_0(E)$ (in the topology of ...
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1answer
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Is $(\int_0^t W_s ds, W_t)$ Markov?

Approximating $I_t = \int_0^t W_s ds$ by Riemann sums I have convinced myself that it is not Markov, but I have been met by the claim that $(I,W)$ is and I cannot figure out why. Do you guys have any ...
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Poison Process Dance Marathon Question

The Dance Marathon is a 30 hour event during which people can make online or cash donations. Assume that 80 percent of the donations are made online and all other donations are made by cash. Donations ...
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Conditions for birth and death process having only finitely many deaths.

Consider a birth and death process on $\mathbb{N}=\left\{0,1,2,\ldots\right\}$, given by the transition probabilities $p(n,n+1)=\lambda_n$ and $p(n,n-1)=\mu_n$ (those are the birth and death rates, ...
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29 views

Mean time for the renewal process

The system is as below. The energy arrival process is $Y_{k}$ with a constant rate of $\rho$. Node has files of size exponential(λ) to be transmitted with fixed rate of transmission $r$. Hence the ...
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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
25 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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63 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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36 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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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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15 views

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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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
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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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63 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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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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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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1answer
40 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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1answer
37 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
32 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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36 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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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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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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1answer
15 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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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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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
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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
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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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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1answer
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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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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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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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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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215 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 ...