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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Given irreducible transition matrix $P$, why does the matrix $(P−I|\mathbb{1})(P−I|\mathbb{1})^T$ have full rank?

Given irreducible transition matrix $P$, why does the matrix $(P−I|\mathbb{1})(P−I|\mathbb{1})^T$ have full rank? I know this is partially due to the fact that since $P$ is irreducible, there exists ...
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1answer
33 views

About the expected transitions in Markov Chain

The problem is here: The given answer is here: K = $2+ X_1 + X_2$, where $X_1$ and $X_2$ are independent exponential random variables with parameters $2/3$ and $3/5$. $$ E[K] = 2=2+1/p_1 +1/p_2 = ...
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1answer
17 views

Finding Transition Probabilities using Metropolis Hastings

I want to find the $4$x$4$ Probability Transition Matrix under the temperature parameter T=2 of Metropolis Hastings. I know that, if x and y are neighbors, $p(x,y) =$ $$ f(x) = \left\{ \...
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0answers
30 views

Series with Markov Chains Probabilities

Notation For each $t \in \mathbb{N}$, let $h_t \in H$ be a random variable that follows a Markov chain, and $h^t \equiv \{h_0,h_1,\dots,h_t\} \in H^t$. Let $\Pi(h^{t})$ be the probability that a ...
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8 views

Ergodicity coefficient of block matrix

I have a stochastic matrix of the following form \begin{equation} X=\begin{bmatrix}A/3&B/3&C/3\\I_n&&\\&I_n&&\\\end{bmatrix}, \end{equation} where $A,B,C$ are all $n$ by $n$...
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1answer
29 views

Finding the mean given the probability

I'm doing some work on branching processes and would like to know where the process becomes extinct. If $X$ is the number of offspring of an individual, then the process goes extinct when $\mathbb{E}[...
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2answers
30 views

Show that this MC is ergodic?

Suppose I have a Markov Chain with States, $S = {1,2,3,4}$ and a PTM given by $P =$ $\begin{pmatrix} .25 & .25 & .25 & .25 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 &...
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1answer
66 views

Markov Chain Transition Matrix Question

Ok, so my question is pretty simple, the question states: A spider web is only big enough to hold 2 flies at a time. Assuming that the flies fly into the web independently: -The probability that no ...
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1answer
22 views

Strong Markov Property for Markov Chains - Statement Verification

I suspect that my handwritten lecture notes for the Strong Markov Property are wrong. I'd appreciate corrections to them. We first define the following: A random variable $\tau$ is called a ...
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0answers
9 views

Markov Chain: Aperiodicity => Primitivity

Hellooo, I would like to know how I can show that the transition Matrix $P$ of an aperiodic Markov chain is primitive. Any suggestions?
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15 views

Markov chain nulls

hope the question is ok for this forum. I am a developer and not a mathematician but realise your group is likely to know the answer for these questions. The background is that I am writing a program ...
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1answer
30 views

Limiting Distribution of a Gibbs Distribution

I know that the Gibbs distribution at a particular state, x, is given by $\frac{e^{-\beta E_i}}{\sum_j e^{-\beta E_j}}$ with $\beta = \frac{1}{T}$, but I do not understand what a limiting distribution ...
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1answer
67 views

A Question Regarding Markov Chains and Ergodicity

Suppose the Markov chain with Probability Transition Matrix, $P$ = ($p{_x}{_y}$) is ergodic and $p{_m}(x, y) > 0$ for all states $x$ and $y$. If $n ≥ m$, show that $p_n(x, y) > 0$ for all ...
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1answer
32 views

Finding Initial state vector with given values

I am not sure how to use a given values to form a initial state vector. There are 30% of customers from Company A switch to company B every month, and 35% customers from Company B switch to company A ...
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24 views

Correlation Matrix Question

Why is this not a possible correlation matrix for any three random variables X, Y, and Z? $\begin{pmatrix} 1 & -1 & -1 \\ -1 & 1 & -1\\ -1 & -1 & 1\end{pmatrix}$
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3answers
893 views

Expected time between successive visits to state $i$ in a Markov chain conditionally on no visit to state $j$

Say I have a Markov chain $\{X_n: n \geq 1\}$ with state space $E = \{1,2,3,4,5\}$ and transition matrix, $$ P = \begin{bmatrix} 0 & 1/2 & 1/2 & 0 & 0 \\\ 1/2 & 0 & 1/2 & ...
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23 views

First Time Passage References

Does someone have any references for this topic regarding distribution for first time passage? Details included in this topic. Hitting times of Markov chain/process have always finite moments? Best ...
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1answer
19 views

Simple Bayes? Probability of a state at time t in hidden markov model

Suppose we have a HMM with $2$ states -- $A$ and $B$, with $P(A) = 0.4$ and $P(B) = 0.6$. $A$ has a probability of $0.9$ of outputting "hot," and $B$ has a probability of $0.1$ of outputting "hot." ...
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1answer
130 views

What does a steady state vector tell us if the Markov chain is irregular?

When a Markov chain is regular, the finding the steady state vector (i.e. the eigenvector corresponding to the eigenvalue $1$) will tell us the long term probability of ending up in any of the states, ...
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30 views

Partition Theorem and Markov Chains

Suppose a Markov chain has $s$ states, $S = {1, 2, . . . , s}$, with PTM $P =$ ($p_{ij}$). That is, $p_{ij} = P[X_{n+1} = j | X_n = i]$. Use the Partition Theorem to verify that if $X_n ∼ ν$, then $X_{...
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14 views

Analysis of incomplete system of diferential equations

I need to find information about the kinetics of a reaction. I tried to solve this problem first generalizing the equations for the different kind of reactions yielding an equation like: $$ \dot{x} =...
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1answer
78 views

Distribution of a process dependent on a Markov chain's states

Consider a Markov chain $X_t$ with state space $\{0,1\}$, initial distribution $$ \begin{array}{l} \mathbf{P}(X_0=1)=\lambda \\ \mathbf{P}(X_0=0)=1-\lambda \end{array} $$ and transition ...
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1answer
40 views

Probability that two random walks on $\mathbb{Z}^2$ meet at the origin

Suppose $X,Y$ are symmetric, independent random walks on the lattice $\mathbb{Z}^2$. I am trying to find the probability: $$\mathbb{P}\big(X_n=Y_n=(0,0)\;\text{for some}\;n\,\big|\,X_0=Y_0=(0,N)\big)$$...
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23 views

MCMC and Metropolis-Hastings problem(s)

What does it mean for a particular state to be a "ground" state or a "stable" state? I should make clear that this is final exam review material and not homework. Also, how does one compute a Gibbs ...
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2answers
33 views

Transition Matrix and Invariant Probability

Given the transition matrix for a 2 state Markov Chain, how do I find the n-step transition matrix P^n? I also need to take n--> inf and find the invariant probability pi?
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1answer
20 views

A formula for an expected value

We have a Markov chain with $X_0 = z$, the return time $\tau_z$ of the first time at which we return to $z$, and some other state $y$. A proof I'm reading states: $$\operatorname{E}(\text{number of ...
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1answer
31 views

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
749 views

Expected state of a Markov chain

Let's start with a slightly trivial Markov chain defined as follows: the beginning state is called $1$ and the set of states is $\mathbb{N}$. At each step, when the current state is $n$, the ...
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1answer
30 views

Why do we have these probability functions for this Markov Chain?

The following shows one of the questions we were given in lectures a while back: We have been given the following solutions to this question: I'm rather confused by these. Take, for example, the ...
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What is an example of a second-order markov chain? [closed]

I'd like to see an example of a second-order markov chain. Haven't found one over google or in any of my textbooks
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14 views

Gaussian blur over (or random walk in) a surface mesh

Let $V$ be the set of mesh vertices, connected by edges $E$, forming a mesh that represents a surface embedded in $\mathbb{R}^3$. On this mesh a function $f:V\rightarrow\mathbb{R}$ is defined. For ...
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1answer
30 views

Building a hidden markov model with an absorbing state.

I'm working on trying to implement a hidden markov model to model the affect of a specific protein that can cut an RNA when the ribosome is translating the RNA slowly. Some brief background: The ...
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1answer
51 views

What is the expected number of steps in a random walk from leaf to leaf in a full binary tree?

Let $h \geq 2$ be a natural number. Consider a complete binary tree of height $h$. Say we take a random walk starting from the "leftmost" leaf. What is the expected number of steps before the "...
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1answer
40 views

Expectation in reversible Markov chain

Let $X$ be a Markov chain with transition matrix: $$\mathbf{P}=\begin{pmatrix} 0 & \frac{3}{5} & \frac{2}{5} \\ \frac{3}{4} & 0 & \frac{1}{4} \\ \frac{2}{3} & \frac{1}{3} & 0\...
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24 views

Perron-Frobenius Theorem: Markov Chain -> Matrices

I am interested in finding out a way how to transform the stochastic results of perron-frobenius for markov chains to any matrix. I am aware that perron-frobenius was originally proofed with linear ...
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0answers
19 views

How to formally justify matrix manipulation in countable-state Markov chain

I have a Markov chain with transition probabilities $t_{i,i+1} = \binom{k+i}{k}^{-1}$ and $t_{i,0} = 1-t_{i,i+1}$, i.e. we have an absorbing chain with absorption probability approaching one as $i \...
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Markov property for the gambler's ruin problem

Let $(X_n)_{n\ge 0}$ be a simple asymmetric random walk on states $0,1,\dots,M$, where $0$ and $M$ are absorbing. Initial state is $i\neq 0,M$. Let $(X_n^*)_{n\ge 0}$ be the process $(X_n)$ ...
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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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0answers
25 views

This markov-chain diagram is correct? [duplicate]

Consider two Poisson process arriving with rate $\lambda_1$ and $\lambda_2$ to a single line, and rate of handling $\mu_1$ and $\mu_2$. The time of handling is exponential with rate $\mu_i$. The ...
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1answer
144 views

Prove that this transformation of a stochastic matrix (or markov chain) is still a stochastic matrix (or markov chain)

Assume to have an $N \times N$ stochastic matrix $W$, where $\sum_j w_{ij} = 1$ and $w_{ij}$ is a generic element on row $i$ column $j$ of the matrix $W$. Moreover you have the following two $N \times ...
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1answer
59 views

Strong Markov property and time homogeneity

Let $X$ be a Markov chain with state space $\mathcal{S}$ and denote $\mathbb{N} := \{ 0, 1, \cdots\}$. We know that for any stopping time $\tau < \infty$ and any bounded measurable function $\phi : ...
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Relation between the expected number of visits to a state and reachability in a Markov chain

Let's consider a discrete time Markov chain $X_n$. Let $R_{ij} = \sum_{n=0}^\infty \mathbb{1}_{\{X_n= j | X_0 = i\}}$ be the number of visits to $j$ starting from $i$, and let $f_{ij}$ be $\text{Prob}...
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Recurrent states - proof of claim

I want to prove: If $x↔y$, then $x$ is recurrent iff $y$ is recurrent. $i\in S$ is recurrent if $P(T_i<\infty)=1$ How can I properly prove this? I don't know where to start from. Thanks
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1answer
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Recurrence of a state in a finite state space

Suppose $T_A := \inf\{ n \ge 1 : X_n \in A\}$ where $A \subset \mathcal{S}$ is finite. Assume $\mathbb{P}\{T_A < \infty \; | \; X_0 = x\}= 1$ for $\forall x \in \mathcal{S}-A$. I need to show that ...
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1answer
29 views

Recurrence of 0 in a random walk

Assume $\mathcal{S} := \{0, 1, \cdots \}$, $p(0,1)=1$ and $p(n,0)=p(n,n+1)=\frac{1}{2}$ for $n=1,2, \cdots$. Is $0$ recurrent or transient? So, basically this is an irreducible, closed but infinite ...
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What is the probability to reach every single number in an infinite random walk over $\mathbb{Z}$?

Suppose we have a Markov chain starting in $X_0 = 0$, with states $S = \mathbb{Z}$ and the transition probabilities $$P(X_{n+1} = i | X_n = j) = \begin{cases}0.9 &\text{if } i = j+1\\ ...
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1answer
56 views

Finite-state Markov chain

Suppose $X_n$ is a Markov chain with transition probability matrix $p$ where the set of possible states is $S = \{1,2, \ldots, k\}$. If we are given, $X_1$, $X_2, \ldots, X_{2000}$, can we say about ...
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1answer
52 views

How to Express the Probabilities Associated with a Third Variable in a Hidden Markov Model?

Suppose I have an observation $Y_t$ that is conditionally dependent on $X_t$. (More specifically, Y is a series of observations emitted by an underlying hidden Markov state sequence X.) I can ...
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200 views

How to Derive Gibbs Sampling Update Formula for Hidden Markov Model?

I want to understand how to derive the update formula for Gibbs sampling for Hidden Markov Model, for example, in here: $$p(z_t | \mathbf{x}, \mathbf{z}_{\setminus t}, \boldsymbol{\alpha}, > \...
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4answers
435 views

Figuring out probabilities with Hidden Markov Models

I'm really new to Math so sorry in advance if this question does not make sense. Also I cross posted this on stats.stackexchange.com also. Background: I'm trying to learn about hidden Markov models ...