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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$\begingroup$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.
Added as an example:
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.
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