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 ...

learn more… | top users | synonyms

0
votes
1answer
20 views

Injections of Markov Processes

In office hours, a professor mentioned that an injective transformation of a Markov process remains Markov. Intuitively, this makes sense to me as you can "recover" the original Markov process from ...
0
votes
0answers
25 views

Transition matrix in left-right hidden semi-Markov model

I'm developing a hidden semi-Markov model left-right . In a left-right model a sequence of $M$ states starts in state $1$ and ends in state $M$, with no repetition of states. Since the model is ...
1
vote
2answers
23 views

Can the ergodic theorem for Markov chains be proved with linear algebra?

This theorem is in my book, let me just say that it is for discrete-time Markov chains, that are time-homogeneous. Ergodic is defined in the book as being positive recurrent and aperiodic. The ...
0
votes
0answers
32 views

Probability of not reaching completion in Markov process

This question is supposed to be easy but is very hard for me. The Norwegian Skating Association has mass produced certain "collectors' cards" with all $N$ speedskaters (Norwegian as well as ...
0
votes
1answer
39 views

Markov chain exercise

Hello i have this Markov chain exercise: Basically we can always move up 1 step, but there is always a possibility that we will go down to the first state 0, the Markov chain consists of N states. ...
0
votes
1answer
34 views

Markov’s inequality

The annual return, R, of a certain stock is a random variable with mean 10. Use Markov’s inequality to obtain a bound for the probability of the stock return being at least 20. Assuming now that R ...
0
votes
1answer
60 views

Tips on how I would find the transition matrix for the following phenomenon?

how would I go about finding the transition matrix for the following phenomenon (which can be modeled as a Markov process)? Any hints or advice is appreciated! During a study break, a student's ...
2
votes
1answer
40 views

Question regarding Notes on Strong Markov Property

I wrote the following notes from a lecture a couple of weeks ago and I don't understand a particular line. Suppose $B_t$ is a Brownian Motion. Now look at $B^x_t = x + B_t$ which is a BM starting ...
0
votes
1answer
50 views

Understanding detailed balance equations

I'm trying to understand how the equilibrium distribution satisfy the detailed balance equation. To my understanding, I only understand that a detailed balance equation would only be satisfied if ...
2
votes
1answer
64 views

Monotonicity and Convexity of Stochastic Matrices

The definition of stochastic monotonicity and convexity is given by "Stochastic Orders and Their Applications" by Moshe Shaked and George Shanthikumar (1994) as: Let $P = \{p_{i,j} \}$ be a ...
0
votes
3answers
89 views

How fast does this Markov chain converge?

Observe the above a Markov chain and limiting matrix of it. Finding the limiting matrix if it exists is easy but I am curious as to how fast this given matrix converges to its limiting matrix. Is ...
0
votes
0answers
85 views

Application of Markov Chain to Game of Life Board Game

I need to calculate the expected outcomes for the Game of Life. I believe that if I multiply the probability of landing on a particular square with the payoff of said square and add up all these ...
0
votes
1answer
18 views

Markov Processes: How to show $\int P(X_t\in B\mid X_0)dPX_0^{-1}=P(X_t\in B)$?

Let $\left\{X_t \right\}_{t\in T}$ be a time homogeneous Markov process with state space $S$. How do I formally demonstrate$$P(X_t\in B)=\int_S P(X_t\in B\mid X_0)dPX_0^{-1}$$(here $PX_0^{-1}$ is the ...
1
vote
1answer
28 views

Markov: Expected time of first visit to a state starting from that state.

Question: Calculate the expected time of first visit to state 2 given we start in state 2. Is the answer to this the mean recurrence time of 2 or simply zero? I at first thought that the answer ...
0
votes
0answers
20 views

The second eigenvalue of a reducible stochastic matrix

The magnitude of the second dominant eigenvalue of a reducible matrix, as I know, is supposed to be 1, why it's not the case for this matrix : $$ \begin{matrix} 0 & 1 & 0 ...
1
vote
1answer
25 views

Markov processes limiting probability questions

I am going over previous mock exams in preparation for an upcoming exam and am having problems with parts (ii) and (iv) and was looking for some guidance. For part (ii), my thinking was that the ...
0
votes
0answers
17 views

How to use symmetry of transition rate matrix in a continuous-time Markov chain?

This is part of a bigger question, so I have to change the question a bit to focus on the point. We have a continuous- time Markov chain with the following transition rate matrix: $$Q= \begin{pmatrix} ...
0
votes
0answers
16 views

Positive probability distribution for Markov Random Fields

While stating results for Markov Random Fields, like the Hammersley-Clifford theorem, why there is emphasis on the requirement that the joint probability distribution should be positive?
0
votes
0answers
55 views

Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail. Just to avoid misunderstanding, ...
1
vote
1answer
129 views

Probability of a trajectory in Markov processes

I need help with a simple formula! (My question is taken from here, pag 26 eq 1.112. ) Consider a Markov Process with associated Master Equation: \begin{equation*} ...
0
votes
0answers
28 views

Poisson process different type of events

Suppose that it arrives people to a store according to a poisson process with rate $\lambda = 6$/hour , females arrive with probability $0.6$ and male with $0.4$. What is the probability that there ...
0
votes
1answer
17 views

Continuous time Markov chain. proportion of time spent in state i

If a question asks for the proportion of time spent in a specific state is this the same as the stationary distribution or something else? For continuous time Markov chain with finite state space.
0
votes
2answers
50 views

Continuous Markov chains, arriving pairs

I have been trying to sort out this exercise but really stuck on this. Preparing myself for exams and found many exercise on continuous Markov chains but I am always stuck when it comes to transition ...
0
votes
1answer
19 views

A problem on Markov process

Suppose, $\Pi_{\theta}$ be the transition probability function of a Markov chain. For any function $f$ define $$\Pi_{\theta}f_{\theta}(x) = \int f(y,\theta)\Pi_{\theta}(x,dy).$$ Is there any ...
1
vote
1answer
33 views

Markov chain property

I would like to make clarification and show my curiosity about markov process. I will show some part of definition related to markov process from here. The Markov property is the dependence ...
0
votes
0answers
17 views

A doubt on markov decision process

Given that a policy is a function from a state action pair to probabilities, the set of policies for a MDP forms a POSET (the partial order is due to value function for a policy). Why there should be ...
1
vote
1answer
32 views

Reflection Principle interpretation

Given a standard Brownian motion $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P},(B_t)_t)$ (the standard filtration $(\mathcal{F}_t)_t$), we define $$\forall t\ge 0: M_t:=\max_{0\le s\le t} B_s$$ ...
4
votes
1answer
85 views

How is the Chapman-Kolmogorov Equation not a fancy name for matrix multiplication?

The Chapman-Kolmogorov Equation: $$p^{m+n}(i,j)=\sum_kp^m(i,k)p^n(k,j)$$ Matrix Multiplication (with $[A]_{i,j}=a_{i,j}$ where $A$ is a linear map "" for B) $$[AB]_{i,j}=\sum_ka_{i,k}b_{k,j}$$ In ...
0
votes
0answers
15 views

Policy Adjustment in Markov Decision Process

I was using MDP on my work to make optimal decision. I used discrete time, finite state MDP. I assumed that I will have an initial parameters, like the Reward/Cost, state transition probabilities and ...
0
votes
2answers
52 views

Prove that something is a Markov chain

Let $\xi_0, \xi_1, \xi_2, ...$be independent, identically distributed, integer valued random variables. Define $Y_n$ = max{$\xi_i: 0 \leq i \leq n$}. Show that $(Y_{n)n\geq0}$ is a Markov chain and ...
2
votes
1answer
77 views

Inverse of a regular stochastic matrix

Is it true that the inverse of a regular stochastic matrix is also regular? Are there any other interesting features that the inverse may have of a regular stochastic matrix? Hope someone could answer ...
1
vote
1answer
63 views

General birth and death process

hi i need some help to understand the following (from the general birth and death process).I'll give some context first , then i ask questions. Consider general birth and death process with birth ...
1
vote
0answers
37 views

References for time-inhomogeneous Markov jump processes?

In some central models in life insurance mathematics, the state of the insured is modeled using a continuous-time time-inhomogeneous Markov process with finitely many states. While many results for ...
0
votes
0answers
36 views

relations between properties of stochastic processess

If we have an integer valued stochastic process, are these implications correct? independent increments $\rightarrow$ Markov property Markov property $\nrightarrow$ independent increments stationary ...
2
votes
1answer
61 views

time-homogeneous continuous time Markov chain

I have a question about the continuous time Markov chain. In the Poisson process we have independent and stationary increments. Do we have this in a continuous time Markov chain that is ...
2
votes
0answers
50 views

Markov Chain with Normal Transition Matrix

Consider a (sub)-stochastic matrix $P$, and the associated Markov chain $X$ with \begin{align*} \mathbf P [X_n =y | X_0 = x] = P_{xy}^n. \end{align*} Suppose we have the condition $P^T P = P P^T$, ...
0
votes
1answer
33 views

Why does markov chains power method converge at the rate of |λ_2/λ_1 |

I'm doing some researches on Markov Chains, and every time I meet this statement, that The rate of convergence of the power method is given by |λ_2/λ_1 |^k→0, when k→inf. And where λ_1 and λ_2 are ...
0
votes
1answer
19 views

Markovian Coefficients Unclear Definitiondtdt

I have come across the following unclear definition: Consider $dS(t) = S(t)[\mu(t)dt + \sigma(t) dW(t)]$ "Assume that the coefficient $\sigma$ is Markovian. That is, (with abuse of notation) ...
1
vote
0answers
26 views

Analysis of Steady State Probability for Markov Process

I have a balance equation, representing a Markov Chain, which yields $$ (K - z) \pi(Z_c = z) = (\lambda_c + (z+1)x) \pi(Z_c = z+1) $$ where K is the maximum state of the server. The term $\lambda_c$ ...
0
votes
1answer
31 views

Markov process which is not martingale

I have seen the examples of a discrete time martingale that is not a Markov Process. Can you construct me an example of discrete time Markov Process that is not a martingale?
1
vote
0answers
21 views

When are the marginals of an extremal invariant measure also extremal invariant?

Let's suppose that $X$ is a compact metric space, and thus as is $X \times X$. If given a Markov process on $X \times X$ with marginals that are Markov processes on $X$, then we know that the ...
0
votes
0answers
12 views

Hidden Markov Model to Markov Model

Can every HMM be converted into some $nth$ order Markov model? My thoughts are that you could just multiply the emission probabilities in each state by the probability of state transitions. For ...
0
votes
1answer
23 views

Computing the PMF for N(t) in a renewal process

I'm in a stochastic processes course, and we just started on renewal theory. Unfortunately, we skipped the section on queuing theory, and nearly example in my textbook for renewal processes uses ...
1
vote
1answer
34 views

$X_n, n> 0$ is a Markov Chain, how to interpret $Z_n = (X_n,X_{n+1}), n > 0$?

Am a newbie to Markov Chain. So, this might be incredibly naive/stupid question. If $X_n, \, n > 0$ is MC, am having difficulty imagining/interpreting process $Z_n = (X_n,X_{n+1}), n > 0$. I ...
1
vote
1answer
31 views

Continuous-time Markov Question

I have a question about a continuous-time Markov process on the discrete space. I am given the generator and asked for find the expected time the Markov process needs to get back to state 3, given ...
0
votes
1answer
36 views

Showing the square of a Markov process is or isn't Markov

Hi I am trying to show that if $X_n$ is a markov process, whether or not $X_n^2$ is a markov process. $X_n$ is a markov process if $P\{X_k = a_k|X_{k-1} = a_{k-1}, X_{k-2} = a_{k-2}, ..., X_k = a_1 ...
0
votes
2answers
59 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 ...
0
votes
1answer
27 views

Steady state of a $4 \times 4$ transition matrix

Normally I just take $q(M_{m\times n} - I_{m\times n})$ to workout the steady state, but here I have: $$\left(\begin{array}{rrrr} 0 & 0 & .8 & .2 \\ .4 & .6 & 0 & 0 \\ .2 ...
0
votes
1answer
23 views

Limiting probability of Markov chain(Terminology)

If I am asked to find the limiting probability of a Markov chain, what does this pertain to? $\lim \limits_{n \to \infty} P^n$? Where $P$ is the stepping matrix and $n$ is the number of steps. "What ...
0
votes
0answers
28 views

Error in thinking: Poisson Process is a Markov Process

I am a bit confused on proving the Markov property for Poisson processes. That is, we want to prove, if $X = (X_t: t \in \mathbb{R})$ is a Poisson process with rate $\lambda$: $P(X_{t_n} = a_n | ...