# What is the difference between positive recurrent and null recurrent?

## What is the difference between positive recurrent and null recurrent?

If a state is recurrent, we say that the state is positive recurrent if the expected amount of time between recurrences (when the chain is in that state) is finite. A state that is recurrent but not positive recurrent is called null recurrent. A positive recurrent, aperiodic state is called ergodic.

How do you know if you have a positive recurrence?

1 Answer. Show activity on this post. If the probability of return or recurrence is 1 then the process or state is recurrent. If the expected recurrence time is finite then this is called positive-recurrent; if the expected recurrence time is infinite then this is called null-recurrent.

What does it mean for a Markov chain to be recurrent?

A recurrent state has the property that a Markov chain starting at this state returns to this state infinitely often, with probability 1. A transient state has the property that a Markov chain starting at this state returns to this state only finitely often, with probability 1.

### How do you prove a Markov chain is positive recurrent?

An irreducible Markov chain on Ω with the transition matrix P is positive recurrent if and only if there exists a stationary distribution π on Ω such that π = πP.

How do you know if a state is recurrent or transient?

In general, a state is said to be recurrent if, any time that we leave that state, we will return to that state in the future with probability one. On the other hand, if the probability of returning is less than one, the state is called transient.

How do you prove a state is recurrent?

We say that a state i is recurrent if Pi(Xn = i for infinitely many n) = 1. Pi(Xn = i for infinitely many n) = 0. Thus a recurrent state is one to which you keep coming back and a transient state is one which you eventually leave for ever.

## Is absorbing state positive recurrent?

You are correct: an absorbing state must be recurrent. To be precise with definitions: given a state space X and a Markov chain with transition matrix P defined on X. A state x∈X is absorbing if Pxx=1; neccessarily this implies that Pxy=0,y≠x.

How do you prove recurrent?

What is a recurrent state? 