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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Finding Hitting probability from Markov Chain

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

Joint Probability from Marginal Probabilities

$X, Y_1, Y_2$ are random variables with (possibly) different finite alphabets. For given conditional probability mass functions $\mathbb{P}(Y_1|X)$ and $\mathbb{P}(Y_2|X)$, is it always possible to ...
4
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2answers
116 views

The problem of the drunkard in a valley.

We consider a Markov chain on a subset of positive integers $S =$ {$0, 1, 2, 3, .......N$}, with transition probabilities defined as follows: The chain jumps only one unit to the left or right. ...
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2k views

Why Markov matrices always have 1 as an eigenvalue

Also called stochastic matrix. Let $A=[a_{ij}]$ - matrix over $\mathbb{R}$ $0\le a_{ij} \le 1 \forall i,j$ $\sum_{j}a_{ij}=1 \forall i$ i.e the sum along each column of $A$ is 1. I ...
4
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280 views

question involving Markov chain

Let $S_{2m}$ be the group of all permutations $\pi$ of $\{1, 2, \ldots, 2m\}$. The following transition kernel $S$ generates the random transposition walk $$ Ch(\pi, \pi')= \begin{cases} \frac{1}{2m} ...
4
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1answer
216 views

Does every continuous time minimal Markov chain have the Feller property?

Consider a Q-matrix on a countable state space. (A Q-matrix is a matrix whose rows sum up to $0$, with nonpositive finite diagonal entries and nonnegative offdiagonal entries). As explained for ...
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109 views

random walk along edges of tetrahedron — which face gets hit last?

Suppose we have a tetrahedron $abcd$, and start at edge $ab$. Now walk to any "adjacent" edge (i.e. in this case any edge other than $cd$), each with equal probability $1/4$. This gives a stationary ...
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0answers
48 views

Markov chains and natural filtration

I have the following problem Consider a homogeneous Markov chain $(X_n)$ with countable state set $E$. Suppose that $A$ is a proper subset of $E$ and consider the stopping times $\tau^0=0 $ and ...
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45 views

Standard deviation of a quantum walk?

The standard deviation of a classical random walk with $n$ steps is $\sqrt n$ - Standard deviation of a random walk. I have read in many places that the standard deviation of a quantum walk $n$ with a ...
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76 views

A Continuous-Time Markov Process Taking All Possible Values

Let $\mathbb{N}$ be the set of positive integers. For each $n \in \mathbb{N}$, let $X^{(n)}=\{ X^{(n)}(t): t \geq 0 \}$ be a Markov chain with state-space the two point set $\{0,1\}$ and $Q$-matrix ...
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129 views

Conditional probability and integrating out part of a random walk

Suppose that I have a random walk process defined by $\alpha_{t+1}$ ~ N$(\alpha_t, \omega^2)$. Given $\alpha_t$ and $\alpha_{t+2}$, I understand why the conditional formula for ...
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185 views

Lower bound for multivariate recurrence

I have a recurrence that looks like $$p(i,j,k) = \frac{j}{n}p(i-1,j-1,k-1) + \frac{i-j}{n}p(i-1,j,k-1)$$ $$p(i,0,k) = 1$$ $$p(i,j,0) = 0$$ $$p(0,j,k) = 0$$ The base cases are to be considered in ...
4
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1answer
232 views

Intuition on Harris recurrence

I am trying to get some intuition on Harris recurrence in Markov chains. Define state space $\mathcal S$ comprising a single communication class, $f_{ii}^{(n)}=P(X_n=i, X_{n-1}\ne i,\ldots X_1\ne ...
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181 views

estimation of transition probabilities from aggregate data

Please, O mathematicians, help me understand the approach to the problem of estimating transition probabilities given only aggregate data in Kalbfleisch & Lawless' 1984 paper "Least-Squares ...
4
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135 views

Behavior of explosive random process

Inspired somewhat by this problem, I've been investigating the behavior under iteration of the following discrete random process: Given $n\in\mathbb{N}$, choose an integer from $\{0,1,\ldots,n\}$ ...
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1answer
2k views

Markov process vs. markov chain vs. random process vs. stochastic process vs. collection of random variables

I'm trying to understand each of the above terms, and I'm having a lot of trouble deciphering the difference between them (note, my mathematics training isn't very strong - so please go easy on the ...
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3answers
704 views

Probability of absorption in a discrete Markov chain

Let $\{X_{n}\}$ be a Markov Chain on the state space $S=\{1,...,100\}$ with $X_{0}=30$, and transition probabilities given by $p_{1,1}=p_{100,100}=1$, $p_{99,100}=p_{99,98}=1/2$ and for $2\leq ...
3
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1answer
268 views

Expected number of runs

Let $S[16]$ be a binary array i.e, elements of $S$ are 0/1 with elements $S[i]$ are taken uniformly and independently form $\{0,1\}$. Let $k$ be a random element taken uniformly from $\{0,1\}$. I have ...
3
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2answers
414 views

Irreducible and aperiodic Markov chain : $P^t(x,y)>0$

Consider a Markov chain $X$ with transition probability $P$ and finite state space $\Omega$. Which of the following statement is true? If $X$ is irreducible then $\exists t>0 \ni P^t(x,y)>0, ...
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2answers
167 views

Markov Chains and Linear Transformations

I just have a quick question about Markov Chain and linear algebra. Background. Let $\{M_n: n= 0, 1, 2, \dots \}$ be a Markov Chain. We can represent the transition probabilities $_{n}Q^{(i,j)}$ in a ...
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1k views

Null-recurrence of a random walk

In a random walk on $\mathbb{Z}$ starting at $0$, with probability 1/3 we go +2, with probability 2/3 we go -1. Please prove that all states in this Markov Chain are null-recurrent. Thoughts: it is ...
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304 views

How can a Markov chain be written as a measure-preserving dynamic system

From http://masi.cscs.lsa.umich.edu/~crshalizi/notabene/ergodic-theory.html irreducible Markov chains with finite state spaces are ergodic processes, since they have a unique invariant ...
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1answer
1k views

Calculating stationary distribution of markov chain

I am asked to compute the stationary distribution of the markov chain with state space $E=\{0\dots,n\}$ and transition matrix below: \begin{bmatrix} 0 & 1 \\ \frac{1}{n} ...
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2answers
435 views

Simple proof that stationary birth-death chains are reversible

A Markov chain with state space $\mathbb{Z}$ is a birth-death chain if the transition probabilities satisfy $p(x,y) = 0$ for $|x-y| > 1$. That is, the only possible transitions are to move one ...
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4answers
73 views

Determining vector equations

Let $A\in \Bbb R^{n\times n}$ be a matrix such that $\mathrm{rank}(A) = n-1$ and consider the equation $$ Ax = 0. $$ Clearly, its solutions span a $1$-dimensional space, thus an additional ...
3
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2answers
362 views

How can I compare two Markov processes?

There is a discrete-time irreductible Markov process with $r$ possible states. $k$ observations were performed. At each observation a state of process was determined. $T_0 = \lbrace 0,1,\dots ...
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463 views

Markov chain with uncountable state space

I'm self-studying probability theory and struggling with understanding Markov chains on uncountable state spaces, notably I would like to solve the following exercise from this book. ...
3
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1answer
84 views

For finite Markov Chain, time average distribution is always a stationary distribution?

Given some finite state space $\Omega\equiv\{\omega_1,\ldots,\omega_n\}$ be given, and let any Markov chain $\{X_t\}$ with $n\times n$ transition matrix $A$ on $\Omega$ be given. I would like to know ...
3
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1answer
35 views

Coupling between two CTMCs

Suppose I have two random processes $X(t)$ and $Y(t)$ starting at time $t=0$ and $X(0)=Y(0)=0$. The processes obey the following transition rates: $$ X(t):\begin{cases} 0\to 1,\text{at rate } A\\ ...
3
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1answer
130 views

Markov Chain Initial Distribution

Suppose $\{X_0,X_1,X_2,\dots\}$ is a discrete-time Markov chain taking values in a finite set $\{1,\dots,N\}$ with initial distribution $p_i(0) = P(X_0 = i)$ for $i\in\{1,\dots,N\}$ and transition ...
3
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2answers
72 views

What's the probability that A wins finally

Suppose A has \$2 and B has \$3. They play a game, each game gives the winner \$1 from other. A has a probability $\frac{3}{5}$ to win each game. They play this game until one of them is bankrupt. ...
3
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1answer
112 views

How to find the limit of a markov chain

Given a markov chain where the next state is related to the previous state by the following matrix: $$\begin{array}{c|ccc} & A & B & C\\ \hline A & p_1 & q_1 & r_1\\ B & ...
3
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1answer
199 views

Conditional probability of a general Markov process given by its running process

I have a question as follow: "Let $X$ be a general Markov process, $M$ is a running maximum process of $X$ and $T$ be an exponential distribution, independent of $X$. I learned that there is the ...
3
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1answer
72 views

When are stable continuous time Markov chains Feller? Always?

This is a question is similar to this 2 year-old one that never got answered (truthfully it's pretty much the same question except that I'm adding a bit more detail and the assumption that the $Q$ ...
3
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1answer
200 views

Elementary proof of geometric / negative binomial distribution in birth-death processes

The birth-death process concerns a population of $n_0$ individuals, each of which reproduce and die at a constant rate as time $t$ increases from $t=0$. Each individual splits into two individuals ...
3
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1answer
71 views

Question about Markov chain

We know that if $\{X_n\}$ is a Markov chain, then $X_{n+1}$ is independent with the past states $X_0,\ldots,X_{n-1}$ given current state $X_n$, that is ...
3
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1answer
105 views

Markov chain problem with finite states

Consider any Markov chain on a state space with exactly $r$ states. We want to find the largest $N>0$ such that there exists states $i,j$ for every $n < N$ we have $p_{ij}^{(N)} > 0$ and ...
3
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1answer
70 views

Random walk where increment depend on current position

Consider the following stochastic process, $$b(i+1) = b(i) + \xi_i (b_i),$$ where $\xi_i(b_i) \in \{-1, k \}$ are the independent increments having the following distribution: $$\begin{align} P (\xi ...
3
votes
1answer
152 views

recurrence criterion for random-walk like (simple) inhomogeneous Markov chain

This question is to some degree a follow-up of this question. Suppose we have a random-walk like Markov chain, i.e. state space is the set of all integers from $-\infty$ to $\infty$, the transition ...
3
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2answers
149 views

Given that $(X_n)_{n\geq 0}$ is a Markov Chain, prove that $(X_{kn})_{n\geq 0}$ is a Markov Chain

Given that $(X_n)_{n\geq 0}$ is a Markov Chain, prove that $(X_{kn})_{n\geq 0}$ is a Markov Chain. I don't know what this exercise has been so difficult for me, I've been playing around with the ...
3
votes
1answer
90 views

Sojourn time of a CTMC

Soujourn time of a CTMC at time $t$ is defined as : $$T(t)= \inf\{ s > 0 : X(t+s) \neq X(t)\}$$ My question is why "inf", not min ? Here $T(t)$ belongs to the set $\{ s > 0 : X(t+s) \neq ...
3
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1answer
112 views

Probability with Markov chains

I need some hint about Markov chains. So here is my homework. Let $\{ X_t : t = 0,1, 2, 3, \ldots, n\}? $ be a Markov chain. What is $P(X_0 =i\mid X_n=j)$? So I need to calculate if it's $j$ ...
3
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1answer
105 views

Limit of a probability regarding a random walk

Consider a random walk on the integers starting at 0, where in each step you move either 1 or 2 meters (back or forth alike). As soon as you reach either the $N$ or $N+1$ meter mark, you stop. What is ...
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196 views

a theorem on transient and recurrent state in a DTMC

Is the following statement true: In a finite Markov chain, if $i$ is a transient state then there is at least one recurrent state $j$ such that $j$ is reachable from $i$.
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2answers
213 views

Martingale associated to Markov chain

$X$ is a (continuous time) Markov chain with generator matrix $\Lambda$ and finite state space $G$. I know that for $g\colon G \to R$ $$ M_t = g(X_t) - g(X_0) - \int_0^t (\Lambda g)(X_s)\, ds $$ is a ...
3
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1answer
2k views

Finding the stationary distribution of a markov chain

I am asked to compute the stationary distribution of the markov chain with state space $E=\mathbb{N}_0$, $q_n >0$ for all $n \in \mathbb{N}_0$ and transition matrix below: \begin{bmatrix} q_0 ...
3
votes
2answers
370 views

Expectation of a stopping time uniquely determined by a function

Let $(X_t)_{t\ge0}$ be a Markov chain on a finite state space $\Omega$, with transition probability $P$. Let $T$ be a stopping time such that $T=\min \{t\ge 0;X_t \in A \subset \Omega \}$.  If ...
3
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1answer
31 views

Mean exit time / first passage time for a general symmetric Markov chain

Suppose I have a Markov chain as depicted in the following figure: where $N$ is even. State 0 and $N$ are the two sinks of the chain. The transition probabilities have the following property: ...
3
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1answer
59 views

A linear growth model with immigration

Ill give some background first before asking questions.(the text below is straight out of the book) Each individual in the population is assumed give birth at an exponential rate of $\lambda$ in ...
3
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1answer
124 views

Sum of two Markov processes another Markov process?

Let $dX_{t} = m_1(l_1-X_{t})dt+\sigma_1 dW_{t}$ and $dY_{t} = m_2(l_2-Y_{t})dt+\sigma_2(\rho dW_{t}+\sqrt{1-\rho^2}dW_{t}^{1})$ where the $m_i$'s, $l_i$'s and $\sigma_i$'s are constants, $\rho \in ...