2
votes
0answers
60 views

Intuition behind criterion for an irreducible Markov chain to be transient

I have been looking over my notes for Markov chains, and I have come across the following: Theorem: An irreducible Markov chain is transient iff for some state $i$ there exists a nonzero vector $y$ ...
4
votes
2answers
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 ...
0
votes
1answer
254 views

What is the meaning of detailed balance in Markov Chains?

I know what it means formally to say that a stochastic matrix and a measure are in detailed balance. ($ \lambda_iP_{ij} = \lambda_jP_{ji} \; \forall (i,j)$) but I'm not really sure how to interpret it ...
1
vote
2answers
222 views

Average run lengths for large numbers of trials: Intuition and proof

This article states that the formula for the average run lengths for large numbers of trials is:$$\frac{1}{1-Pr(event\ in\ one\ trial)}.$$ My questions What is the intuition behind this formula? Do ...