I am studying continuous time Markov chains on a countable state space and my professor said the following: A state $i$ is positive recurrent if and only if $$\liminf_{t\to\infty} \frac{1}{t} \int_0^t P_{i,i}(s)\textit{d}s>0$$ where P is the semigroup associated to the Markov chain. I do not understand why this is true. Might someone have some insights?

Thank you in advance!


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