Can someone please explain to me the intuition behind Positive recurrence. What does it mean and why is it different to normal recurrence?
How can I spot positive recurrence?
1 Answers
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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0@Rosie: If the probability of any return after $nd$ steps tended to some positive fraction $c$ then intuitively you will return roughly a proportion $\frac{c}{d}$ of the time so the expected time between returns (i.e. for first return) is $\frac{d}{c}$. – 2012-06-04