# How can I spot positive recurrence?

Can someone please explain to me the intuition behind Positive recurrence. What does it mean and why is it different to normal recurrence?

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.

See the Wikipedia article on Markov chains for more details.

In a simple symmetric 1D random walk, the probability of first return after $2n$ steps is $\dfrac{\frac{1}{2n-1}{2n \choose n}}{2^{2n}}$. Since $\sum_{n=1}^\infty \dfrac{\frac{1}{2n-1}{2n \choose n}}{2^{2n}} =1$, the probability of first return in finite time is $1$, so this is recurrent. But since $\sum_{n=1}^\infty 2n \dfrac{\frac{1}{2n-1}{2n \choose n}}{2^{2n}}$ is infinite, the expected time of the first return is infinite, so this is null-recurrent.
• @Rosie: The probability of (any, not necessarily first) return after $nd$ steps need not be $0$ for a null-recurrent state (it is not in the simple random walk) but it does need to tend towards $0$ as $n$ increases. – Henry Jun 3 '12 at 20:39