0
votes
1answer
56 views

Tips on how I would find the transition matrix for the following phenomenon?

how would I go about finding the transition matrix for the following phenomenon (which can be modeled as a Markov process)? Any hints or advice is appreciated! During a study break, a student's ...
2
votes
1answer
62 views

Monotonicity and Convexity of Stochastic Matrices

The definition of stochastic monotonicity and convexity is given by "Stochastic Orders and Their Applications" by Moshe Shaked and George Shanthikumar (1994) as: Let $P = \{p_{i,j} \}$ be a ...
0
votes
3answers
85 views

How fast does this Markov chain converge?

Observe the above a Markov chain and limiting matrix of it. Finding the limiting matrix if it exists is easy but I am curious as to how fast this given matrix converges to its limiting matrix. Is ...
0
votes
0answers
16 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 ...
4
votes
1answer
75 views

How is the Chapman-Kolmogorov Equation not a fancy name for matrix multiplication?

The Chapman-Kolmogorov Equation: $$p^{m+n}(i,j)=\sum_kp^m(i,k)p^n(k,j)$$ Matrix Multiplication (with $[A]_{i,j}=a_{i,j}$ where $A$ is a linear map "" for B) $$[AB]_{i,j}=\sum_ka_{i,k}b_{k,j}$$ In ...
2
votes
1answer
51 views

Inverse of a regular stochastic matrix

Is it true that the inverse of a regular stochastic matrix is also regular? Are there any other interesting features that the inverse may have of a regular stochastic matrix? Hope someone could answer ...
1
vote
0answers
48 views

How's the damping factor in Google PageRank algorithm calculated

I'm doing some researches about Google's PageRank algorithm for my thesis, I've found that the damping factor x (for example), where x is in : P` = x.P + (1-x)Q where P is the original ...
0
votes
0answers
15 views

radius of markov chain

For an irreducible markov chain one can show, that $\limsup \sqrt[n]{p_{ij}^{(n)}}$ is independent of the choice of the states $i$ and $j$ where $p_{ij}^{(n)}$ is the probability to get from $x$ to ...
1
vote
2answers
97 views

Expected number of steps between states in a Markov Chain

Suppose I am given a state space $S=\{0,1,2,3\}$ with transition probability matrix $\mathbf{P}= \begin{bmatrix} \frac{2}{3} & \frac{1}{3} & 0 & 0 \\[0.3em] \frac{2}{3} ...
0
votes
0answers
70 views

A matrix of size $n\times n$ with several properties like Markov matrices

Could you find a square matrix $A=[a_{ij}]$ of size $n$ such that satisfies to following properties 1) For all $1\le i\le n$, $\sum_{j=1}^n a_{ij}=0$ 2) For all $i$, $a_{ii}<0$ and for $1\le i\ne ...
2
votes
2answers
54 views

Question on MIT Markov Matrices video

Markov matrices are pretty new to me and I'm a little rusty with my linear algebra. My question stems from watching this video from YouTube on Markov matrices. For those who wish to skip the video, ...
4
votes
0answers
174 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 ...
7
votes
0answers
130 views

Uniqueness of an infinite system of linear ODEs

How to prove that $\dot{x}=ax,\space x(0)=1$ has a unique solution if $a,x$ are infinite dimensional matrices? More specifically, let $Q$ be a bounded infinitesimal generator, i.e. ...
0
votes
1answer
135 views

Steps to convert one linear algebra equation to another

I'm reading a paper that jumps from eq (1) to eq (2) without describing the steps taken. I'd like to understand how/why/what steps allow the transform. (My linear algebra is a bit rusty...) (1) ...