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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Opposite of Absorbing State

This should be fairly standard, but I fail to google it, and nothing on the matter is on Math.SE. How do we call the opposite of an absorbing state? If we think about Markov chains/systems, that ...
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How to Simplify a Markov chain in order to estimate the average number of transitions to reach to a final state?

Is there any approach to approximate the expected number of transitions to complete a Markov chain without knowing the exact transition probabilities? The reason I ask this is because I want to ...
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Probability of going from a set $S$ to its complement on a Markov chain

I need to show that if $\pi$ is the stationary distribution of a Markov chain $M$, then for every set of vertices $S$, the probability to choose a random node in $S$ according to $\pi$ and then going ...
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58 views

Expected number of steps

I play a game as follows. A bucket contains four red balls and three green balls. At each step, a ball is chosen at random from the bucket, with each of the balls there being equally likely to be ...
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15 views

How to estimate a hidden model for an unstationary Markov process?

I have a problem that is very similar to the one solved by the Baum–Welch algorithm. I have a process that is very similar to a hidden Markov process. The only difference is that I have an observable ...
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33 views

Determining probabilities Markov Chain

If I have a Markov Chain $X_0, X_1, X_2 \dots$ that has a transition probability matrix $ \textbf{P} = \matrix{~ & 0 & 1 & 2 \cr 0 & 0.3 & 0.2 & 0.5 \cr ...
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80 views

Determining a transition probability matrix

If I have that $X_n$ is a two-state Markov chain whose transition probability matrix is: $P = \left( \begin{smallmatrix} \alpha & 1-\alpha\\ 1-\beta & \beta \\\end{smallmatrix} \right)$ ...
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68 views

Calculate expected value for a lazy Random Walk

Calculate the mean of time needed for a lazy random walk on $[0,n]$ which starts on $0<k<n$ to hit $0$ or $n$ if in each step the walk stays in probability $\frac 1 3$, goes to the right in ...
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158 views

Random Surfer as a Markov Chain

Consider a random surfer who begins at a web page (a node of the web graph) and executes a random walk on the Web as follows. At each time step, the surfer proceeds from his current page A to a ...
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90 views

In M/M/1 Markov process, why must entering and leaving the zero state be equal?

According to the image below, which I snipped from this article, the rate of leaving State 0 and the rate of arriving into ...
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46 views

An irreducible Markov chain is a martingale

Let $\{X_n\}$ be an irreducible Markov chain. Does exist example of such $\{X_n\}$ which is also a martingale given that: a. $\{X_n\}$ is recurrent with finite number of states (but bigger ...
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37 views

How long does it take two identical hidden Markov models run on same observations to forget their initial distributions (if ever)?

Let $H_1$ and $H_2$ be two instances of a finite Hidden Markov Model (HMM) $H$. That is, $H_1$ and $H_2$ have identical state spaces $Q$ as well as identical transition $A$ and emission probabilities ...
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104 views

Deducing results about continuous time random walks from the corresponding discrete time result

Is there any standard way to prove results about continuous time random walks from the corresponding results for discrete time random walks? Specifically, my problem is that I was reading Lawler and ...
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74 views

Intuition behind Stopping Times

I'm attending a stocahstic processes course. I have some trouble with the intuition behind a stopping time. I will consider the discrete case to make it simpler. a stopping time is given by ...
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47 views

Can two nodes in a Markov chain have transitions that don't total 1?

In all the Markov diagrams I see, the transitions from state A to B always total to one. Just one of many examples, this image ...
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96 views

Using Markov Property in solving PDE/SDE

I am solving the PDE I used Feynman-Kac and eventually arrived at $F(t,x)$ $ = E[X_T^2|X_t = x]$ $ = E[(X_t \pm \sigma (W_T -W_t))^2|X_t = x]$ (iirc) So, I try to evaluate $E[(X_t \pm \sigma ...
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26 views

Does a state which is passed at least 3 times had to be passed 5 times in Markov chain

Prove of disprove: Let $\{X_n\}_n$ be homogenous Markov chain. if we start from state $i$, there is a positive probability that we pass at least 3 times at state $j$. Does it follows that exists ...
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75 views

Markov chain doesn't sum up to 1

Let $\{X_n\}$ be a Markov chain on $S=\{1,2,3,4,5,6\}$ with the matrix suppose we define a new sequence $\{Y_n\}$ by $$Y_n=\cases{1\quad X_n=1\vee X_n=2\\2\quad X_n=3\vee X_n=4\\3\quad ...
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50 views

Operator semigroups (lecture notes)

Can you recommend good lecture notes (or a book) about this topic? Basically I would like something which covers more or less the first chapter of the book "Markov Processes" by Ethier and Kurtz, but ...
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50 views

I have to show that the following stochastic process is a Markov process

I don't understand how to show that some stochastic processes have the Markov property. For example, if I have the following process: $$(\Omega, \mathcal{F}, (X_t)_{t \geq 0}, P^y)$$ where $\Omega = ...
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27 views

Is a Markov Decision Process finite horizon or infinite when the length of horizon depends on actions taken?

I want to model a problem as a Markov decision process but I'm not able to classify it as finite or infinite horizon. In my problem, the time for which the process will continue is a function of ...
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33 views

Uniform convergence of $A^n/n!$

In a proof regarding finite space Markov Jump Processes in which the function $P(t)=e^{tG}$ is a solution to both the backward and forward Chapman-Kolmogrov equations, one of the steps assumes that ...
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67 views

Probability of a time-dependent set of states in Markov chain

Consider a Markov matrix $P$ defining $m$ states. For each time $n$, define a set of states $S_n$ that contains a predefined subset of the states $\left\{ {1,...,m} \right\}$. For time $n=k$, I would ...
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36 views

Construction of pure birth process

I am considering a Markov chain $\lbrace X(t) \rbrace_{t≥0}$ in continunous time on the countable state space $S=\lbrace 0 \rbrace\cup \lbrace (i,j) \mid i \in \mathcal{A} , j \in \mathbb{N} \rbrace, ...
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Invariant Distribution of Two Dependent $\cdot/M/\infty$ Queues Running in Parallel

This is in preparation for an exam I have coming up. We have two $\cdot / M / \infty$ queues with external arrivals occurring according to a Poisson Process of rate $\lambda$. Service occurs with ...
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36 views

First time markov process spends tau units in certain state

Consider a continuous time Markov process $\{X(t)\}_{t≥0}$ on the state space $\{0, 1, 2, . . .\}$ with stationary probabilities $\{π_0, π_1, π_2, . . .\}$. Suppose that, when currently in state $i$, ...
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114 views

Solving recursive integral equation from Markov transition probability

How do I solve something like: $$f(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty e^{\frac{-(y - x/2)^2}{2}}f(y)\:\mathrm{d}y$$ for $f(x)$? Is there also a general formula that this falls under? ...
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27 views

How to prove that the rational thief problem model is monotone.

A thief goes out stealing every day and has a chance of $p_k$ of stealing a sum $k$ with $0\leq k \leq N$. But there's also a chance $q$ of getting caught, in which case he loses everything he got ...
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97 views

Equilibrium distribution of Ehrenfest's urn

(I'll post my own answer to this, but others may be of interest, so post your own if you have one.) (PS: In reply to comments posted below: Stackexchange encourages posting an answer to one's own ...
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30 views

What is the state space of this markov chain?

Consider a system where two persons sit at a table and share three books. At any point in time both are reading a book, and one book is left on the table. When a person finishes reading his/her ...
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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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45 views

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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77 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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31 views

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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45 views

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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30 views

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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50 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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216 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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36 views

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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35 views

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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122 views

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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41 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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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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143 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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76 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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50 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 ...