0
votes
0answers
22 views

Continuous time markov chains, is this step by step example correct

I have some questions regarding CTMC... and most importantly whether the step-by-step example I provide below is correct. My main sources about CTMC are: ([1], and [2]). Let's assume 3 possible ...
0
votes
0answers
25 views

Can the transition probabilities of an inhomogeneous Markov chain be written as an exponential?

If $Z_t$ is a homogeneous continuous-time Markov chain with finite state space $E=\{1,\ldots,p\}$, transition matrices $(P(t))$ and intensity matrix $Q$, it holds that $$ P(t) = \exp(tQ), $$ see for ...
0
votes
1answer
29 views

Board Game Markov Process - Transient Probabilities

I need to write an essay on the Game of Life board game, and so I studied up on Markov Chains to help me calculate the probabilities and average payoffs for the spaces; however I'm not sure whether ...
0
votes
1answer
25 views

Q-matrix vs. P-matrix description of a Markov chain

Consider a continuous time Markov chain $(X_t)_{t \geq 0}$ on some state space $S$ with transition matrix (P-matrix) $p_t(x,y)$, the probability density of jumping from $x$ to $y$ in time $t$. The ...
2
votes
2answers
23 views

Probability of returning to a given state after n transitions-Markov chains

Let us denote $f_j^{(n)}$ denote the probability of the first return to state $j $after n transitions. Let $p_{jj}^{(n)}$ be the probability of returning to the state $j$ after $n$ transitions when ...
1
vote
1answer
28 views

Symmetric random walk and the distribution of the visits of some state

I need help with this problem: Let $(S_n)_n$ a symmetric random walk in $\mathbb{Z}$ i.e $S_n=X_1 + \cdots + X_n$ with $(X_n)$ iid $\mathbb{P}(X_n=1)=\mathbb{P}(X_n=-1)=\frac{1}{2}$. Let $m \in ...
0
votes
0answers
39 views

Can ergodic Markov chains be periodic?

I found a statement in one of my notes which said If a state is persistent, aperiodic and not null the it is said to be ergodic Is it necessary that it should be aperiodic? This statement ...
1
vote
1answer
21 views

Markov chains: An issue in classification of states

I recently came across a lemma which goes as follows. Suppose a Markov chain has N states. Let i and j be pair of states. Then j can be reached from i iff there is an integer $ 0 ≤n< N$ such ...
0
votes
2answers
52 views

Expected number of steps to absorbtion - Markov chain

I want to calculate the expected number of steps (transitions) needed for absorbtion. So the transition probability matrix $P$ has exactly one (lets say it is the first one) column with a $1$ and the ...
0
votes
1answer
18 views

Writing down the transition matrix of a discrete Markov chain

Please consider the following scenario: One person is walking along a discrete circle induced by $\mathbb{Z}/n\mathbb{Z}$ In each round we roll a dice with $w\in\left\{2,\ldots n\right\}$ sides If ...
0
votes
1answer
29 views

Inferring transition rates from continuous markov chain question

A house has 2 rooms of similar sizes with identical air conditioners equipped with thermostats which turn on and off as needed to maintain the temperature in each room to a desired level of 22 ...
1
vote
1answer
56 views

random walk with sticky barriers

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are ...
0
votes
0answers
19 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
22 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
29 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
35 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
33 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
48 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
61 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
85 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
59 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 ...
1
vote
1answer
25 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
16 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
21 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
14 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} ...
1
vote
1answer
100 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
1answer
12 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
42 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
18 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
27 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 ...
4
votes
1answer
73 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
14 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
48 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
51 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
55 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
29 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
55 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
40 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
31 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 ...
1
vote
0answers
22 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
0answers
11 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
29 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 ...
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 ...
1
vote
0answers
47 views

How's the damping factor in Google PageRank algorithm calculated

I'm doing some researches about Google's PageRank algorithm for my thesis, I've found that the damping factor x (for example), where x is in : P` = x.P + (1-x)Q where P is the original ...
1
vote
0answers
25 views

showing a condition implies convergence to invariant distribution

For an arbitrary Markov chain, I'm trying to show that $\lVert{ P_{\mu}\circ \theta_n^{-1} - \nu \rVert}\to 0$ iff $\sum_j \lvert{ P_{ij}^n - \nu(j)\rvert}\to 0$, where $\theta_n^{-1}$ is a shift ...
2
votes
1answer
63 views

continuous time Markov chain, something the book does not explain

I have a problem with something in my book, under the chapter of continuous time Markov chains. I will post a link to what the book does. They do something which they seem to take for granted, but I ...
0
votes
0answers
14 views

radius of markov chain

For an irreducible markov chain one can show, that $\limsup \sqrt[n]{p_{ij}^{(n)}}$ is independent of the choice of the states $i$ and $j$ where $p_{ij}^{(n)}$ is the probability to get from $x$ to ...