Stochastic processes (with either discrete or continuous time dependence) on a discrete (finite or countably infinite) state space in which the distribution of the next state depends only on the current state. For Markov processes on continuous state spaces please use (markov-process) instead.

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Memoryless Property of Markov Chains

I'm trying to understand Markov Chains and have across the following in a book: $ \sum\limits_{y=0,1,....m−1}p(x,y)P(T_A<T_B|X_0=x,X_1=y) $ which then becomes the following, under the Markov ...
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1answer
96 views

Probability transition matrix for maximum of iid random variables

I have a homework problem that goes as follows: Let $\xi_i, \ i=0,1,2,\ldots$ be i.i.d. random variables of discrete type. The distribution of $\xi_0$ is given by: $$\mathbb{P}\{\xi_0=i\} = a_i, \ ...
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33 views

“Taking expectation” to yield conditional probability

This argument is taken from Resnicks Adventures in stochastic processes and let $T _{\infty } < \infty $ denote that an infinite number of transitions in a continuous time markov chain has occurd ...
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1answer
93 views

How do I say that an infinite-state Markov chain is positive recurrent? [closed]

I run into this Markov chain while I'm doing my research, and I can't figure out how to find the condition under which this Markov chain is positive recurrent. This is a brief scenario of my ...
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22 views

Equivalent Formulation of Markov Property for Homogeneous Chains

In Shiryaev's Probability (just above the strong Markov property, p.568), the author says that an equivalent formulation of the usual Markov property for homogeneous chains is $$P[\theta_nX\in B\mid ...
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75 views

How to check if a given Markov chain is positive recurrent.

I'm trying to solve a problem which is related to my research, and I have to check whether this infinite-state Markov chain is positive recurrent or not. Suppose the Markov chain I have has state 0, ...
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54 views

Conditional expectation of a Markov-chain - can the conditioning Sigma-algebra be changed?

Let $(X_n)_{n \in \mathbb{N}_0}$ be a Markov-chain and $(\mathfrak{F}_n)_{n \in \mathbb{N}_0}$ the induced filtration $\mathfrak{F}_n := \sigma(X_0, \dots, X_n)$. Is then ...
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15 views

Modelling Transition Between States without Markov Property

I have a data set that I'm trying to model out. My data set tracks an individual items over 20 periods. In each period each item can be in one of four states. There are no restrictions on how items ...
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27 views

Expectation of staying in same state for a simple MC

Consider a simple dicrete-time Markov Chain $X_t$ with finite state $\Omega = \{1,2,3\}$. At time 0 the chain is with probability 1 in state 1 $\mathbb{P}(X_0 = 1) =1$. Then the transition probability ...
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18 views

markov process with extra boundary

In a markov process a random walker has to reach N (absorbing boundary) from $x_o$ on a $[0,N]$ lattice, where $0$ is the reflecting boundary. To find the first exit time of the random walker via N, i ...
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11 views

minimum condition under which time avg. and ensemble avg. are equal for a markov chain

What is the minimum condition under which time avg. and ensemble avg. are equal for a markov chain. Is it ergodicity ?
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Markov Models and Applications

I am looking for resources in Markov models and its applications. I'm looking for tutorials, videos, books etc which provide the following Explain Markov chains in layperson terms and provide ...
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53 views

A question about a Markov Chain

I encountered a question about Markov Chains which looks interesting. Given a homogeneous, irreducible, non cyclic Markov Chain with $K$ possible states and a transition matrix $Q$. We define $T_i$ ...
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26 views

Probabilities in a Markov Model

I am reading a paper on Markov Models and I am trying to figure out how to compute the probabilities for the $\alpha$-pass. I am given an $N\times N$ matrix $A$, that has the probabilities of ...
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35 views

Treatment of Markov process with absolute states

In the standard treatment of a markov process, the state vector is a probability vector, whose elements can be between zero and one. But I have a need to constrain the state vector to zeros or ones. ...
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71 views

Exact probability distribution for hitting time of simple random walk

Consider simple random walk on the line starting from the site $y \in \mathbb{N}$. With probability $p$ the walker moves to the right and with probability $1-p$ to the left. Call $\tau$ the first time ...
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229 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 ...
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28 views

Initializing MCMC walkers with ambiguous direction (-/+)

I'm running a sampler program where there are observations given as sample data which are derived from an equal sized population of parameters that are converted to the observations using a known ...
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72 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 ...
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26 views

Simultaneous multiple perturbations in Markov chain Monte Carlo

I'm coding a McMC algorithm for geophysical applications. Using the Metropolis-Hastings scheme to accept/reject the proposed models is smth that i thought i completely understood, but i don't. To be ...
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1answer
61 views

Control principal eigenvector of a row stochastic matrix

I am just trying to consider the classical discrete-time Markov Chain problem. Consider the transition matrix P, which transforms state vector $x(k)$ to $x(k+1)$, satisfying: $x(k+1)$ = $P*x(k)$ It ...
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1answer
22 views

Return Lemma MC

If a Markov chain is $\phi$-irreducible and has stationary distribution $\pi$, then $\phi\ll \pi$, Proof: We use the irreducibility of the chain to write the state space $E = \bigcup_{n,m \in ...
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54 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 ...
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56 views

How do you find the probability of a certain state in Markov Chain?

This question appears without answer in an old exam I found (not a homework question) Suppose messages that enter a system need to be processed by two servers. They arrive at the system at a ...
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19 views

Solution to linear system around the topic of Markov-chains

Let $(X_n)_{n\geq 0}$ be a Markov-chain with the state space $S$ and transition matrix $P=(p_{xy})_{x, y \in S}$. For $A\subset S$ be $H^A:=\inf\{n = 0, 1, \dots | X_n \in A\}$ the first visit time ...
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74 views

Are random walk variations Markov-Chains?

Let $S_{n}:= S_0 + \sum_{i=1}^{n}X_i$ be a simple random walk, $X_i$ are independent random variables with $P[X_i=1] = p, P[X_i = -1] = 1-p$. Let $M_n:=\max\{S_0, \dots, S_n\}$. The task at hand is ...
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24 views

Skew and Kurtosis of Absorbing Markov Chains

An absorbing Markov chain $P$ can be put in canonical form: $$ P = \left( \begin{array}{cc} Q & R\\ \mathbf{0} & I_r \end{array} \right), $$ where $Q$ is a t-by-t matrix, $R$ is a nonzero ...
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38 views

Proof of mean recurrence time theorem in a Markov chain?

How can this formula been proven? $$\lim_{n\to \infty} p_{i,i}^{[n]} = {1\over \mu_{i,i}}$$ where $p_{i,i}^{[n]}$ is the probability that we've returned to state $j$ after $n$ steps in the Markov ...
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41 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 ...
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47 views

four state Markov chain

If there are four states:A,B,C,D. Probability of moving to the left is b and prob of moving to the right is a. If starting at state B, what is probability of arriving at state D? The hit says to ...
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Does Markov property imply $\mathbb P (X_n=i \ | \ X_0=j)= \mathbb P (X_{n+1}=i \ | \ X_1=j)$?

If the future depends only on the present and not on the past (aka Markov property), one could expect $$\mathbb P (X_n=i \ | \ X_0=j)= \mathbb P (X_{n+1}=i \ | \ X_1=j)$$ to hold. Is that true? I've ...
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38 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 ...
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1answer
64 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. ...
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1answer
128 views

The expected number of visits before hitting zero in simple random walk

I am learning Markov chains and encounter the following problem: Suppose in simple random walk, we start from state k. What's the expected number of visits to k before we hit 0? The book does not ...
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16 views

action of transition operator on function

Let $P$ be the transition operator of a markov chain with discrete time and discrete state space $X$. The action of the transition operator on a function $X \to \mathbb{R}$ is defined by $Pf(x) = ...
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55 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 ...
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56 views

Markov Chains and Return Times

Let $(X_n)_{n≥0}$ be a Markov chain with transition kernel $p$ on a countable state space $S$, starting at $x∈S$ $T^{(1)}=\inf\{n≥1:X_n=x\} \quad \quad$ first return time to $x$ ...
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32 views

Markov decision processes with action space only revealed at point of decision.

I have a problem which looks like a finite horizon Markov decision process, except the actions space at each time is revealed at the decision making point. There is no way to know before hand the ...
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2answers
70 views

Proof that steady state is not affected by initial distribution in Markov chain.

I was following a proof provided in Gilbert Strang's book "Introduction to Linear Algebra". And I am confused by one step of the proof. Suppose we have a $n$ by $n$ stochastic matrix $A$, where all ...
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49 views

Perron Frobenius Theorem and Markov chains and more

I came across few ways of calculating convergence rates of Markov chains but I am a bit confused as to how these differ from each other and what may be the best way to calculate. The second ...
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171 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 ...
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26 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 ...
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1answer
59 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} ...
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1answer
50 views

Calculating probability from Markov Chain

I have a Markov Chain with states {1,2,3,4,5} which has the following transition matrix below: $$P= \begin{bmatrix} 0.3 & 0 & 0.7 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0.5 ...
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2answers
42 views

Proof of Markov Property

I'm trying to understand a simple proof for the markov property which states that: "$A_1$, $A_3$ are conditionally independent given $A_2$ iff $P(A_3 | A_1 \cap A_2)=P(A_3|A_2)$" The Proof begins as ...
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1answer
346 views

Best martingale for sequence of “dozen” bets at roulette game

Jim goes the Casino to play roulette. He only makes “dozen” bets at each spin ; his probability of winning is therefore $\frac{1}{3}$ every time (to simplify, we neglect the effect of the zeros in ...
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119 views

Markov chain notation

In a book of stochastic approximation, in the convergence of the ODE method chapter I see the following notation : the state vector of a system $X_n$ has a dynamic representation controlled by ...
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107 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 ...
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72 views

How to prove stationary distribution of a particular MC

I am reading a paper since afternoon and having searched extensively im not yet clear how does the author derive the equation. The paper is [1] Its 'Learning Random Walk Models for Inducing Word ...
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42 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 ...