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?

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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