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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Transition probability matrix for $X_1 = \# heads$, *flip heads* $X_2 = \# tails$ * flip tails* $X_3 = \# heads$

Three fair coins are tossed, and we let $X_1$ denote the number of heads that appear. Those coins that were heads on the first trial (there were $X_1$ of them) we pick up and toss again, and now we ...
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10 views

Decide whether a class is recurrent or transient (Example)

Consider the Markov chain $(X_n)_{n\geqslant 0}$ with state space $E=\left\{1,2,3,4\right\}$ and transition matrix $$ T=\begin{pmatrix}0 & 1/3 & 1/3 & 1/3\\0 & 0 & 1 ...
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13 views

Explicit Probability for Markov Chain on Power Set

A have a Markov chain $F_t$ in discrete time on the power set of a finite totally ordered set $A$. Its probably easiest to explain the transition probabilities in a small example, since they are easy ...
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22 views

Transience in a simple Markov chain

Consider the following simple game from a textbook called "Competitive Markov Processes" by Filar & Vrieze (Springer 1996). This is a two player game with two states. In the first state (the ...
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22 views

If a Markov chains converges then the limit is a stationary distribution

Let $p$ be a transition function of a Markov Chain on a countable state $S$ and $i \in S$. Assume for every $j \in S$, $$ \lim_{n\to \infty} p^n(i,j) = \pi(j)$$ Show that $\pi$ is a stationary ...
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24 views

Why are inter arrival times in the continuous version of discrete-time Markov chains always exponentially distributed?

I am curious whether there exist continuous time Markov processes for which the times between jumping times (which I call inter arrival times) are not exponentially distributed, but have some other ...
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35 views

Stochastic matrices with orthogonal eigenvectors

I stumbled across an odd fact while thinking about Markov chains, and I have an odd proof for it. Claim: Let $P$ be the transition matrix of a finite irreducible aperiodic Markov chain, and assume ...
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36 views

Expectation and Variance of random walks

Consider random walks of fixed length (e.g. $5$) starting at node $u$ in an undirected and connected graph with $N$ vertices. If a node $k$ has $N_k$ edges, the probability of the walk reaching any of ...
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19 views

Markov chain knowing future

I was wondering whether or not P(X1 = S1 | X0 = S0) and P(X1 = S1 | X0 = S0 and X2 = S2) are the same? What I mean is can we get some information from the future states? Thanks!
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6 views

Is Markov Chain sampled at stopping times a Markov chain?

Given a Markov hain $\{X_n\}$ and $T_n$ be an increasing sequence of stopping times, is $\{X_{T_n}\}$ Markov chain ?
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Prove that $E_x[T_y]<\infty$ for irreducible positive recurrent Markov chains.

This is an exercise from Durret's "Probability: Theory and Examples". Suppose $p$ is a irreducible and positive recurrent Markov chain. Then $E_x[T_y]<\infty$ for all $x,y$. I had the following ...
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Maximal Principle: Why using the new transition matrix $\tilde{P}$?

First some notation: Let $(X,E,P)$ denote a finite, irreducible Markov chain with finite state space $E$ and transition matrix $P$. Choose and fix a subset $E^°$ of $E$, which will be called ...
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83 views

A recursive formula to approximate $e$. Prove or disprove.

Let the sequence $\{x_n, n=1,2,...\}$ be defined as follows: Let $x_2=x_1=1$ and for $n>2$ let $$x_{n+1}=x_n-\frac{1}{n}x_{n-1}.$$ This sequence, generated by the recursion above, tends to zero ...
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30 views

emails arriving in a Poisson process

Emails arrive according to a Poisson process with rate $λ=2/hour$. You check your inbox (instantly reading all new emails) at time $t=5$ hours and also at some uniformly distributed random time ...
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10 views

a reference for problems and exercises in markov chains with solutions

i've started studying markov chains from the book of Lawler,i've solved its exercise but i want a solution for Lawler in order to check my answers. can any one help me finding? any other resource ...
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18 views

Mean and Variance of an offspring

If I have that the number of offspring of an individual in a population is $0$, $1$, or $2$ with respective probabilities $a>0$, $b>0$ and $c>0$, where $a+b+c=1$, how would I express the mean ...
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20 views

Finding mean and variance of a population problem

A population beings with a single individual. In each generation, each individual in the population dies with probability $1/2$ or doubles with probability $1/2$. If I let $X_n$ denote the number of ...
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42 views

Find the Stationary Distribution of an infinite state Markov chain

A Markov Chain on states 0,1,..... has transition probabilities $P_{ij}=1/(i+2)$ for j=0,1,....,i,i+1. I'm supposed to find the stationary distribution. So do I take the limit as n goes to ...
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28 views

Limiting Distribution of a Markov Chain

I'm having trouble understanding how to find a limiting distribution. If I have a Markov Chain whose transition probability matrix is: $$ \mathbf{P} = \matrix{~ & 0 & 1 & 2 & 3 & ...
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1answer
29 views

Electrical components failing in a Poisson process

A machine has infinitely many identical components. They fail according to a Poisson process with rate $\lambda = 4$/hour. A repairman arrives at time $t$ and instantly repairs all of the broken ...
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1answer
36 views

Prove that the absolute value of the difference of two invariant distributions on a Markov chain is invariant

If we have $a(x)$, $b(x)$ which are invariant distributions on a Markov chain $X_n$ with state space $S$, how can I prove that $|a(x)-b(x)|$ is also invariant? I know that I must show that: ...
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28 views

Multiclass Markov process

There are two car M/M/1 queues Q1 and Q2. Arrival rate of Red car and Green car in Q1 is $\lambda_{1R}$ and $\lambda_{1G}$ respectively. Similarly arrival rate of red car and green car in Q2 is ...
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16 views

Converse of Perron Frobenius Theorem: Necessary and Sufficient Conditions for positivity (or non negativity)

As I understand, Perron Frobenius theorem asserts only in one direction, i.e. if Matrix A is positive then there is a perron eigenvalue, eigenvector etc. What I wanted to know is what are the ...
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29 views

Measurability of a stopping time in a Markov chain

Suppose you have a finite-state continuous-time inhomogeneous Markov chain with transition rate $Q(t)$. Further, let us suppose that $Q(t)$ is a piecewise continuous function of $t$. Two questions: ...
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52 views

Markov Property Confusion

I feel like I'm being very dense/employing some sort of circular reasoning, but I'm having trouble understanding the Markov Property. According to Durrett (ISBN-10:1461436141), $X_n$ is a Markov chain ...
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1answer
21 views

Strategy for Unbalnaced Gamber Ruin

A gambler plays the following game: A fair coin is tossed until getting three times continuously head. When that happens the Gambler gets 20$\$$. Each round costs the gambler 1$ (even if he won the ...
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19 views

Practical Way to Detect a Markov Chain is Regular Given the Transition Matrix

I understand that a Markov Chain is reducible if, given its transition matrix $P$, there exists $n$ such that every element of $P^n$ is greater than 0. However, I am wondering that if there is an ...
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59 views

Eigenvalue range of $P+P^T$ (P is a transition matrix)

$P$ is a transition matrix of dimension $N\times N$. I know $\lambda_1=1$ and $|\lambda_i|<1, 2\leq i \leq N$. I want to know the eigenvalue range of $P+P^T$. Because $P$ is not symmetric, so I ...
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2answers
13 views

why are the recurrent classes closed?

i've recently started studying about markov chain, we call a communication class a recurrent one in a markov chain if by starting from that class we infinitely return to it with probability 1,with ...
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1answer
20 views

Definition of Stationary Distributions of a Markov Chain

I'm having a lot of trouble understanding the definition of the stationary distribution of a Markov Chain from Hoel, Port, Stone's Introduction to Stochastic Processes. They define the stationary ...
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Estimating Markov transition matrix for regularization

Suppose that I have a sequence of discrete distributions: $$ p_j = (p_{1j},...,p_{Cj}), \: j=1...D,\\ p_{ij}>0 \:\: \forall i,j,\: \sum_{k=1}^Cp_{kj}=1\:\:\forall j. $$ I suppose that these ...
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Showing which classes are recurrent and which are transient

If I have a Markov chain on states {0,1,2,3,4,5} $$ \mathbf{a} = \matrix{~ & 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & 1/3 & 0 & 2/3 & 0 & 0 & 0 \\ ...
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1answer
41 views

Markov chain for two players with two coins [closed]

Two players A and B toss two fair coins independently. Whoever gets the smaller number of heads will pay that many dollars to the other player. For example, if player A tosses two coins and gets 2 ...
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24 views

Show that this Markov chain is recurrent or transient

Consider the Markov chain $(X_n)_{n\geq 0}$ with state space $E=\left\{1,2,3,4,5\right\}$ and transition matrix $$ T=\frac{1}{2}\begin{pmatrix}0 & 1 & 1 & 0 & 0\\0 & 0 ...
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1answer
34 views

Markov chain - Can anyone explain me why this is the solution?

Customers arrive according to a Poisson process at a rate of four customers per hour. A customer who finds four other customers in already waiting gives up and leaves. Some clients in the 3rd ...
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30 views

diffusion- stuck

In a round room of radius R, a large number of coins N of diameter d are randomly dispersed upon the floor. A ladybird starts from the centre of the room, crawling at speed v. Suppose that every time ...
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Show that the process created from taking kth steps of a markov chain is markov.

Suppose $(X_n)_{n\geq0}$ is a Markov chain with transition probability matrix $P$ and initial distribution $\lambda$. Show that the process $Y_n = (X_{kn})_{n\geq0}$ with $k$ fixed is Markov with ...
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51 views

How to solve $h(i) = \frac{i^2}{(n-i)^2+i^2}h(i-1) + \frac{(n-i)^2}{(n-i)^2+i^2}h(i+1)$

$h(i) = $P(reach n eventually| the initial state = i) $h(0) = 0$ $h(n) = 1$ 0 and n are stopping time. For $ 0 < i < n$, $$h(i) = \frac{i^2}{(n-i)^2+i^2}h(i-1) + ...
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27 views

Show $\sum^n_{j=0} P_{ij} = \sum^n_{j=0} P(X_1 =j | X_0 = i) = 1 $

Let $P$ be the one step transition matrix of a Markov chain with states {$0,1,...,n$}. Show $\sum^n_{j=0} P_{ij} = \sum^n_{j=0} P(X_1 =j | X_0 = i) = 1 $ I understand that this is the row sum, but ...
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18 views

Changing the index of the sums when changing the sums - why this way?

Let $(X_n)_{n\in\mathbb{N}_0}$ be a Markov chain with state space $E$. For $i,j\in E$ set $$ h_i(j):=\mathbb{P}_i(H(j)<\infty):=\mathbb{P}(H(j)<\infty|X_o=i), $$ where $H(j)\colon ...
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1answer
19 views

Adding distances/weights to absorbing markov chain

in presence of an absorbing state, I want to calculate mean/expected 'distance' from any state to that absorbing state. What I mean by distance is that I want to give different lengths from one ...
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17 views

expected value - last $k$ flips of coin are same

we flip a normal coin $n$ times. We mark $k=0.5log(n)$ and we mark the $i$'th value in $Xi$. $Y$ will be the number of times where the last $k$ flips were the same. What is $E[Y]$? I think this has ...
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25 views

Markov Chain - last $0.5log(n)$ Tosses of Coin

We toss a coin $n$ times and we mark $k=0.5log(n)$. $Y$ is the number of times where the last $k$ tosses were the same. What is $E(Y)$? I'm pretty sure I need to use Markov Chain but I'm not sure ...
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68 views

Expected value of money left from a coin flipping game

Say we were to play a game. We started off with \$100 and kept flipping a fair coin. If it turned out heads, we won \$1, else our money got inverted. For example, if on the first flip we got heads, ...
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50 views

Gibbs sampling from a 2D Gaussian

Hi I have the to do the next problem and I am kind of lost, if someone could give a litte hint of where to start I would really appreciate it. Thanks in advance! Suppose $x$~$ N(\mu;\sigma)$ where ...
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27 views

How can i interpret this absorbing markov chain to solve a probability question?

I try to solve a simple question; if I toss a coin and repeat it until a tails come up, what is the mean number of steps? (I want to solve another question but it is just a complicated version of ...
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Survival function of birth-death process

There is a linear birth-death process with $N$ states + an absorbing state $0$, with $$\Pr[X_{t+1}=0|X_{t}=0]=1, \\ \Pr[X_{t+1}=i+1|X_{t}=i]=\Pr[X_{t+1}=i-1|X_{t}=i]=q_i, i\in [1..N-1],$$ and ...
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46 views

Can you help me with this Markov Chain question?

The Problem: Prove that if the number of States in a Markov Chain is M, and that state j can be reached from state i, then it can be reached in M steps or less. The work: I assumed by contradiction ...
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No null recurrent state in finite state space from definition.

Let $\{X_n\}$ be a markov chain on finite state space $I$, with stationary transition probabilities. Let us denote $f^n(i,i):=P(X_n=i,X_{n-1}\neq i,\ldots X_1\neq i\mid X_0=i)$. We say $i$ is ...
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1answer
26 views

Forward Algorithm Hidden Markov Model matrix help [Discrete]!

So this may seem like a bioinformatics question but it is the math part that is giving me trouble. I'm using a Python package called YAHMM to model DNA sequences. I created a model with two states ...