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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What is the probability that Markovian pure birth process is state $k$ at time $n$? [on hold]

Suppose $(X_t)_{t \in \mathbb{R}_{\geq 0}}$ is a Markovian pure birth process with values in $\{0,1,2,\dots\}$ and $X_0 = 0$. The holding times $S_1,S_2,\dots$ are independent exponential random ...
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1answer
30 views

Proving specific formula for stationary markov process [on hold]

In my probability class, right now we are dealing with Markov chains and I was stumbled by parts of this problem: Given a $ \{ X_n \}_{n=0}^{\infty} $ be a homogeneous Markov chain (the transition ...
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24 views

Statement of the strong Markov property in Norris' book

In J.R.Norris' Markov chains book, the strong Markov property for discrete-time, Markov chains is stated and proved as follows: Let $(X_n)_{n \geqslant 0}$ be a Markov chain with transition ...
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13 views

Transition functions and Markov processes

I am wondering whether there is a one-to-one correspondence between transition functions and homogeneous Markov processes? We say that $(X_t,\mathcal{F}_t)_{t\geq 0}$ is a Markov process if ...
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19 views

Show there exist $N $ such that $p_{s,s}(n) > 0$ for all $n \geq N$ for an aperiodic state $s$ of a Markov chain

Show there exist $N \geq 1$ such that $p_{s,s}(n) > 0$ for all $n \geq N$ for an aperiodic state $s$ of a Markov chain. Where $p_{s,s}(n)$ is a transition probability. My approach: because $s$ is ...
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1answer
53 views

Prove Markov Chain - help me to start!

I came across this problem in homework: $U_n$ are i.i.d random variables with $P[Un=1]=P[Un=−1]=0.5$. a) Show that $X_n=U_nU_{n+1}$ is a Markov Chain. b) Show that $X_n=(U_n+U_{n+1})/2$ is not a ...
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29 views

Is the process Markov or not?

Consider the stochastic process with $X_0=0$ and $$ X_t= \begin{cases} 0 & \text{ for } \ \ t<\tau_1 \\ 1 & \text{ for } \ \ \tau_1\leq t < \tau_1+\tau_2 \\ 2 & \text{ for } \ \ ...
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PTM using Hastings-metropolis [on hold]

[Compute the 4 × 4 PTM (pij ) under the T = 2 dynamics of Hastings–Metropolis][1]
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1answer
26 views

Do Markov generators form a linear space?

Let $G_1$ and $G_2$ be generators for two distinct continuous-time Markov processes $X^{(1)}$ and $X^{(2)}$ on a common probability space $\Omega$ (with Markov semigroups $S^{(1)}$ and $S^{(2)}$) so ...
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1answer
31 views

About the expected transitions in Markov Chain

The problem is here: The given answer is here: K = $2+ X_1 + X_2$, where $X_1$ and $X_2$ are independent exponential random variables with parameters $2/3$ and $3/5$. $$ E[K] = 2=2+1/p_1 +1/p_2 = ...
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61 views

Crossing of Brownian Motion Sample Paths

I would like to ask for a more rigorous statement and proof of Lemma on page 5 of this paper. In essence, it states that two distinct sample paths of a Brownian motion does not strictly cross (meaning ...
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13 views

How to handle Finite-state-machine with correlated inputs?

My system can be represented by the following state-diagram. The inputs to this FSM are correlated. This implies that I can no longer make "independent input" assumption. My question is: How ...
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1answer
23 views

$\sum_{i=1}^{n-k}\frac{i}{n-k}\binom{2n-2k}{n-k+i}\frac{1}{2}^{2(n-k)}=\frac{1}{2}^{2n-2k}\binom{2(n-k)-1}{n-k}$

2$\sum_{i=1}^{n-k}\frac{i}{n-k}\binom{2n-2k}{n-k+i}\frac{1}{2}^{2(n-k)}=2\frac{1}{2}^{2n-2k}\binom{2(n-k)-1}{n-k}$. This is an identity in a note for a class in Markov Processes, but I can't ...
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1answer
21 views

Strong Markov Property for Markov Chains - Statement Verification

I suspect that my handwritten lecture notes for the Strong Markov Property are wrong. I'd appreciate corrections to them. We first define the following: A random variable $\tau$ is called a ...
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Exploiting the Markov property

I've encountered the following problem when dealing with short-rate models in finance and applying the Feynman-Kac theorem to relate conditional expectations to PDEs. Let ...
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1answer
22 views

Finding Initial state vector with given values

I am not sure how to use a given values to form a initial state vector. There are 30% of customers from Company A switch to company B every month, and 35% customers from Company B switch to company A ...
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21 views

how to define probability in Markov chain [on hold]

I am wondering if probability of the state transition in Markov chain is a pre-defined known number. if so, does it mean that to sovle every problem using Markov process decision, the probability of ...
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2answers
30 views

$\mathcal{P}$ stochastic matrix. If there is $k > 0$ st $\mathcal{P}^k(j, i) > 0$, then there is $r \leq (n-1)$ st $\mathcal{P}^r(j, i) > 0$

Let $\mathcal{P}$ be stochastic matrix of order n. If there is $k > 0$ such that $\mathcal{P}^k(j, i) > 0$, then there is $r \leq (n-1)$ such that $\mathcal{P}^r(j, i) > 0$. My attempt: ...
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1answer
40 views

Probability that two random walks on $\mathbb{Z}^2$ meet at the origin

Suppose $X,Y$ are symmetric, independent random walks on the lattice $\mathbb{Z}^2$. I am trying to find the probability: $$\mathbb{P}\big(X_n=Y_n=(0,0)\;\text{for ...
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1answer
30 views

Why do we have these probability functions for this Markov Chain?

The following shows one of the questions we were given in lectures a while back: We have been given the following solutions to this question: I'm rather confused by these. Take, for example, the ...
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2answers
23 views

What is an example of a second-order markov chain? [closed]

I'd like to see an example of a second-order markov chain. Haven't found one over google or in any of my textbooks
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1answer
28 views

Sufficient condition for a measure to be invariant

Given a Polish metric space $H$ and a Borel probability measure $\pi$. Let $\mathcal B_b(H)$ be the set of bounded measurable functions on $H$, and $L^2(H, \pi)$ be the set of square integrable ...
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1answer
35 views

Expectation in reversible Markov chain

Let $X$ be a Markov chain with transition matrix: $$\mathbf{P}=\begin{pmatrix} 0 & \frac{3}{5} & \frac{2}{5} \\ \frac{3}{4} & 0 & \frac{1}{4} \\ \frac{2}{3} & \frac{1}{3} & ...
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Spectral Density of an ARMA process.

For an upcoming Stochastic Processes exam, we have had a sudden brief email about Spectral Density as the lecturer had forgotten to mention it in classes. He states, For an ARMA process with ...
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21 views

Transition functions induced by Markov processes

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and denote by $(X_t,\mathcal{F}_t)_{t\geq 0}$ a time-continuous Markov process with values in $(E,\mathcal{E})$. For $s<t\in ...
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22 views

Strategy for selling/buying a stock by average reward value iteration

At beginning of any day $t$, I may own $0$ or $1$ share. The price of the share follows the Markov chain in the table below. At the beginning of a day where I own a share, I may either sell at today’s ...
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5 views

Recurrent states - proof of claim

I want to prove: If $x↔y$, then $x$ is recurrent iff $y$ is recurrent. $i\in S$ is recurrent if $P(T_i<\infty)=1$ How can I properly prove this? I don't know where to start from. Thanks
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20 views

How to Prove that a (Centered) Gaussian Process is Markov if and only if this Equation Holds?

A centered Gaussian process is Markov if and only if its covariance function $\Gamma: \mathbb{R}\times\mathbb{R} \to \mathbb{R}$ satisfies the equality: ...
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1answer
76 views

Distribution of a process dependent on a Markov chain's states

Consider a Markov chain $X_t$ with state space $\{0,1\}$, initial distribution $$ \begin{array}{l} \mathbf{P}(X_0=1)=\lambda \\ \mathbf{P}(X_0=0)=1-\lambda \end{array} $$ and transition ...
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9 views

Inference on deterministic HMM

I have a Discrete HMM with hidden Markovian signals of the form $\{X_n\} \in \{ 1,2,3,4\}$ and observed outputs of the form $\{Y_n\} \in \{ 1,2\}$. I have a transition probability matrix ...
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2answers
30 views

Markovian Gaussian stationary process with continuous paths

Could you, please, help me figure out the following problem. We call a stationary Gaussian process $\xi_t$ (with continuous paths) an Ornstein-Uhlenbeck process if its correlation function ...
2
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1answer
22 views

Building a hidden markov model with an absorbing state.

I'm working on trying to implement a hidden markov model to model the affect of a specific protein that can cut an RNA when the ribosome is translating the RNA slowly. Some brief background: The ...
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37 views

Write out the explicit Kolmogorov forward differential equation

Let $(X_t)$ be a continuous-time Markov process with two states, as shown below. Assume that there are two positive numbers $a$ and $b$ such that for all times $t\geq 0$ and $h>0$, $P(X_{t+h} = 2 ...
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Kolmogorov forward and Backwards equation interpretation

Let $\lambda_i$ be the sojourn rate of state i, $q_{ij}$ be the transition rate form i to j, and $p_{ij}$ be the transition probability from i to j. The Kolmogorov Forward and backwards equation are ...
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2answers
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Simple Random Walk - Why are these two events the same?

Let $S = (S_n)_{n \geq 1}$ be a simple random walk. We denote the hitting time of a point $b$ by $\tau_b = \min \{i \geq 1 : S_i \geq b\}$. My text says that the events $\displaystyle\{\max_{k \leq ...
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1answer
24 views

Are queues CTMC?

The $M/M/1$ queue have all the properties of the countable state continuous time markov chain. Is any general queue also a countable state CTMC?
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Markovian Model: scheduling jobs to servers

I have the following problem. I tried to look at queuing theory, but it probably fits better as a scheduling problem. I have a set of $C$ servers: each one can perform 1 job. Processes arrive ...
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10 views

Identification of Infinite Dimensional State in Hidden Markov Model

Consider a hidden markov model (HMM) where the state, $X_t(\alpha)$, is a stochastic distribution over $\alpha \in \mathbb{R}_+$ and one observes a signal $Y_t$, which is simply a moment of this ...
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6 views

How to make Markov Chain model from sequence of data in MATLAB?

Markov Chain model considers only 1-step transition probabilities i.e. probability distribution of next state depends only on current state and not on previous state. I have a sequence and from that I ...
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9 views

Predicting Nash equilibrium after one player enters or leaves

Suppose I have a game with $N$ players, and that the Nash equilibrium can be calculated. If one player enters or leaves the game, is it possible to predict or quickly calculate the resulting Nash ...
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11 views

Markov Chains that preserve an ordering of the state space

Suppose $X = (X_k)_{k=0}^\infty$ is a homogeneous Markov chain/process (for example on the state space $E = \lbrace 1, \dots, m\rbrace$). We can interpret the elements of the state space as "values". ...
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49 views

Dice Game with 1 die and Payoff Function

Imagine a dice game where you may repeatedly roll a die until you either decide to stop, or roll a 1, with the following payoff function (where k is the number on the die), $f(k) = 0$ when $k=1$ ...
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1answer
14 views

Does an embedded discrete-time Markov chain preserve its properties in continuous time?

Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a continuous ...
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1answer
55 views

Example of a Continuous-Time Markov Process which does NOT have Independent Increments

1. Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a ...
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33 views

Markov chain - Stationnary distribution - Unique

Consider the following respective infinitesimal generators of Markov chains in continuous time: \begin{equation} A=\begin{bmatrix} -4 & 1 & 3 \\ 3 & -5 & 2 \\\ 0 & 3 ...
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1answer
26 views

Show that a matrix satisfying certain conditions is non-singular

I have a square matrix $A$ satisfying the following conditions: The elements on the diagonal are negative; All other elements are non-negative; All row sums are less than or equal to $0$; There is ...
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7 views

What is the probability of this Markov Jump process remaining in this state?

Suppose you had a time homogeneous Markov jump processed defined by the following transition diagram I'm assuming that this means that the process remains in state $0$ for time $t$ with probability ...
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1answer
21 views

Period of an irreducible Markov Chain is given by the number of eigenvalues with unit modulus

Suppose $\{X_n\}$ is an irreducible Markov Chain on finite state space $S$. Then, the number of eigenvalues of the transition matrix with unit modulus is precisely equal to the period of the chain. ...
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What does the notation $P_{\overline{MM}}(t)$ mean in this context?

The notation $P_{\overline{MM}}(t)$ is used in part (iii) of the following question: I'm unsure of exactly what this notation represents. My guess would be that it represents the probability that a ...
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Pure jump process

I'm having touble understand the pat of the solution that I have underlined in green for b)