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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Transformation of state-space that preserves Markov property

I am solving a problem in Mathematical Statistics by Jun Shao Let $\{X_n \}$ be a Markov chain. Show that if $g$ is a one-to-one Borel function, then $\{g(X_n )\}$ is also a Markov chain. Give an ...
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1answer
707 views

Eigenvalues for $3\times 3$ stochastic matrices

This is a plot of the non-real eigenvalues of 10000 randomly generated $3\times3$ stochastic matrices. It's pretty clear that they lie in the convex hull of the three cube roots of unity. The ...
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2answers
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Expected number of turns for a rook to move to top right-most corner?

Suppose a rook starts on the lower left-most square of a standard $8 \times 8$ chess board. The board contains no other pieces. The rook randomly makes a legal chess move with every turn (directly ...
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1answer
776 views

When the sum of independent Markov chains is a Markov chain?

I try to find as much as possible cases, when the chain $Z(t) = |X_1(t)-X_2(t)|$ is Markov, where $X_1(t)$ and $X_2(t)$ are independent, discrete-time and space, preferably non-homogeneous Markov ...
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0answers
246 views

6-digit password - a special decoding method

Consider the situation of decoding a 6-digit password that consists of the symbols A to Z and 0 to 9, where all possible combinations are tried randomly and uniformly. Consider the following ...
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3answers
406 views

Characterize stochastic matrices such that max singular value is less or equal one.

By a stochastic matrix, I mean any non-negative square real matrix with rows summing to one. It is well-known that singular values of stochastic matrices can be more than one. Is there a ...
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1answer
776 views

Probability distribution for the position of a biased random walker on the positive integers

I initialize a biased one-dimensional random walk on the positive integers at the origin, $x = 0$, which also serves as a reflecting boundary blocking steps onto the negative integers. Let's say that ...
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655 views

Finite State Markov Chain Stationary Distribution

How does one show that any finite-state time homogenous Markov Chain has at least one stationary distribution in the sense of $\pi = \pi Q$ where $Q$ is the transition matrix and $\pi$ is the ...
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What is the difference between all types of Markov Chains?

I have been looking for some good material covering Markov Chains but everything seems so difficult to me... After reading about the subject, I figured out that there is basically three kinds of ...
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308 views

Conditional return time of simple random walk

Consider a simple symmetric random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$. Call $\tau_k = \min\{t \in \mathbb{N}\, : \, \, S_t =k \}$, the hitting time of $k \in \mathbb{N}$. Call $\tau^* = \min\{t &...
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Equilibrium distributions of Markov Chains

I often get confused about when a Markov chain has an equilibrium distribution; when this equilibrium distribution is unique; which starting states converge to the equilibrium distribution; and ...
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1D random walk-probability to go back to origin

Suppose There is a random walk starting in origin while probability to move right is 1/3 and probability to move left 2/3.What is the probability to return to the origin. Thank you
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643 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 ,k-1\...
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1answer
2k 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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1answer
664 views

If $P$ is a regular transition probability matrix then $P^{n^2}$ has no zero element

A transition probability matrix $P\in M_{n\times n}$ is regular if for some $k$ the matrix $P^k$ has all of its elements strictly positive. I read that this can be tested by using the following result....
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Perron-Frobenius theorem

In the proof of the Perron-Frobenius theorem why can we take a strictly positive eigenvector corresponding to the eigenvalue $1$? Before that, why can we even take a non-negative eigenvector? Books ...
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1answer
77 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 \...
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2answers
224 views

Expected number of random binary vectors to make matrix of order n

I have the following problem: I pick random vectors from $\mathrm{F}_2^n$. The chance that position $i$ is $1$ equals $p_i$, $0$ otherwise (each position is picked independently). Let $X$ be a random ...
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1answer
4k views

Irreducible, finite Markov chains are positive recurrent

I am under the impression that an irreducible, finite Markov chain is necessarily positive recurrent. How might I show this? Regards, Jon
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766 views

Why is every irreducible matrix with period 1 primitive?

In a certain text on Perron-Frobenius theory, it is postulated that every irreducible nonnegative matrix with period $1$ is primitive and this proposition is said to be obvious. However, when I tried ...
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1answer
104 views

Flea on a triangle

"A flea hops randomly on the vertices of a triangle with vertices labeled 1,2 and 3, hopping to each of the other vertices with equal probability. If the flea starts at vertex 1, find the probability ...
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1answer
83 views

Computational methods for the limiting distribution of a finite ergodic Markov chain

We wish to show what can be discovered about the limit of a finite, homogeneous, ergodic Markov Chain $X_1, X_2, \dots,$ using simple methods of computation and simulation. Specifically, consider the ...
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Under what conditions does $(I-N)^{-1}$ exist?

Given an nxn matrix N and $I=I_n$, under what conditions does $(I-N)^{-1}$ exist? On one hand $(I-N)(I + N + N^2 + ...) = (I + N + N^2 + ...) - (N + N^2 + ...) = I?$ On the other hand, $(I-N)(I + N +...
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2answers
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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 $f(x)=...
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$(S_0+\ldots + S_n)_{n\geq 0}$ not a Markov chain

Assume that $Y_0,\ldots , Y_n$ are independent random variables with the following identical distribution: $Y_i=1$ with propability $p$ and $Y_i=0$ with propability $1-p$. Also set $S_0=0$ and $S_n=...
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1answer
134 views

Markov Chain transitional probability query.

Say I have the transitional probability matrix P= $\begin{bmatrix}.8 & .2\\.6 & .4\end{bmatrix}$ And the entry (1,1) denotes the probability that I stay in state 0, (1,2) I move from state 0 ...
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1answer
133 views

Determining computational complexity of stochastic processes

I have an program which implements a Markov chain Monte Carlo process on a system of N bits, stopping when the process converges. Let's use T to denote the average number of steps made by the Markov ...
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1answer
146 views

Markov chain: join states in Transition Matrix

I need to merge two states in the Transition Matrix: For example: I have the matrix below ...
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1answer
113 views

probability, random walk, Markov chain question

Let $P$ be a transition matrix for a regular Markov chain and let $w$ be it’s equilibrium vector. Show that $w$ has no zero entries.
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1answer
65 views

2 Questions about Markov chain

The first question: $ (X_n : n = 1, 2, ...) $ is a Markov chain with state space $(-1, 0, 1)$. $(sin(X_n) : n = 1, 2, ...$) is a Markov chain. $(cos(X_n) : n = 1, 2, ...$) is a Markov chain. $(|X_n| ...
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Nice references on Markov chains/processes?

I am currently learning about Markov chains and Markov processes, as part of my study on stochastic processes. I feel there are so many properties about Markov chain, but the book that I have makes ...
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Good introductory book for Markov processes

Which is a good introductory book for Markov chains and Markov processes? Thank you.
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Knight returning to corner on chessboard — average number of steps

Context: My friend gave me a problem at breakfast some time ago. It is supposed to have an easy, trick-involving solution. I can't figure it out. Problem: Let there be a knight (horse) at a ...
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1answer
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Strong Markov property - Durrett

I recently had great success with my first question here so I will boldly go on to a second. Here goes: I'm studying Markov Chains in Rick Durrett - Probability: Theory and example and I'm stuck with ...
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2answers
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Is ergodic markov chain both irreducible and aperiodic or just irreducible?

As I find some definition says: Ergodic = irreducible. And then Irreducible + aperiodic + positive gives Regular Markov chain. A Markov chain is called an ergodic chain if it is possible to go ...
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299 views

Generalization of the Jordan form for infinite matrices

Under what conditions is it the case that for a matrix $M$ whose rows and columns are indexed by a countably infinite set $S$ one has a Hamel basis consisting of generalized eigenvectors (i.e. $v \in \...
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1answer
418 views

Probability of a substring occurring in a string

Consider a random string of length $n<\infty$ where each digit can be between 0-9 with equal probability and a substring of length $k<n$ consisting of only zeros. What is the probability of ...
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2answers
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Can Markov Chain state space be continuous?

I looked for a formal definition of Markov chain and was confused that all definitions I found restrict chain's state space to be countable. I don't understand purpose of such a restriction and I have ...
5
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1answer
181 views

recurrence criterion for random-walk like Markov chain

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 probability $P_{ij}$ is nonzero only when $j=i+1$ or $j=i-1$. ...
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1answer
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How can I compare two matrices?

I have a matrice A. It is model probability matrice for some process (Markov chain). Then, I have estimated matrice B. I have to somehow compare these two matrices to tell whether process that gave ...
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1answer
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Expected number of steps/probability in a Markov Chain?

Can anyone give an example of a Markov Chain and how to calculate the expected number of steps to reach a particular state? Or the probability of reaching a particular state after T transitions? I ...
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2answers
145 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 $Q$ ...
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3answers
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Estimating the transition matrix given the stationary distribution

Let's say we are given a Markov chain for variable $X = [x_1, ..., x_n]$; also we are given a desired stationary distribution for this graph $P_\infty = [p_1, ..., p_n]^\top$. How can we design an ...
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What does it mean to observe a Markov Chain after a certain kind of transition?

I'm working on a problem concerning censoring of transitions in a Markov Chain. For example, take a Markov Chain that models a counter, it goes up or down but does not stay in position. A possible ...
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0answers
115 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where $b,\...
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1answer
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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 off-diagonal entries.) As explained for ...
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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
386 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 ...
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1answer
300 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} &...
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1answer
300 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 ...