# Solved – Essential transient state in a Markov chain

Can a finite state Markov chain have essential transient state?

I have found out an example for an infinite state one and I have the intuition (I may be wrong) that for a finite state space .. This isn't possible… But I am not being able to prove it..

Contents

According to Wikipedia,

• A state \$i\$ is accessible from a state \$j\$ (written \$jto i\$) if a system started in state \$j\$ has a non-zero probability of transitioning into state \$i\$ at some point.

• A state \$i\$ is essential if for all \$j\$ such that \$i to j\$ it is also true that \$j to i\$.

• A state \$i\$ is said to be transient if, given that we start in state \$i\$, there is a non-zero probability that we will never return to \$i\$.

A Markov chain with an essential transient state can be constructed from three states \$i,j,k\$ for which \$ito j\$, \$jto i\$, \$jto k\$, and \$k\$ never returns to \$i\$. The transition \$jto k\$ guarantees \$i\$ is transient. The transition matrix is

\$\$pmatrix{0 & 1 & 0\ 1-rho & 0 & rho \ 0 & 0 & 1}\$\$

for some number \$rho\$ with \$0lt rho lt 1\$. It is the chance of never returning to \$i\$ when starting at \$i\$.

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