Are all transient states positive recurrent? In my Stochastic process lecture notes, I noticed that in problems related to a finite Markov chain, the mean recurrence time are calculated only for recurrent states. I calculated the mean recurrence time of all transient states and found them to be positive recurrent. 
Is mean recurrence time calculated only for recurrent states? And is it true that all transient states are positive recurrent?
 A: Transient states are those which may never be visited again. Positive recurrent states will always be revisited within some (expected) finite time. Therefore, no transient state is positive recurrent.
A: To quote from Meyn & Tweedie's Markov Chains & Stochastic Stability, on page 175, a state $\alpha$ is recurrent for a Markov chain $(\Phi_t)_t$ if $\mathbb{E_\alpha}[\eta_\alpha]=\infty$ (infinite expected number of visits) and transient if $\mathbb{E_\alpha}[\eta_\alpha]<\infty$ (finite expected number of visits), when$$\eta_\alpha=\sum_{t=1}^\infty \mathbb{I}_\alpha(\Phi_t)$$is the number of visits. This definition does not involve nullity or positivity. To further quote from Meyn & Tweedie's Markov Chains & Stochastic Stability, on page 453, a state $\alpha$ is null if$$\lim_{t \to \infty} P^t(\alpha,\alpha) = 0$$and positive if$$\lim\sup_{t \to \infty} P^t(\alpha,\alpha) > 0$$If a state $\alpha$ is transient then it is necessary null, while if a state $\alpha$ is positive, it is necessary recurrent. An important result (see Theorems 10.0.1 & 10.4.9 in Meyn & Tweedie's Markov Chains & Stochastic Stability) is that an irreducible recurrent Markov chain $(\Phi_t)_t$ admits a unique (up to multiplicative constants) invariant measure. When the measure is finite the chain is necessarily positive recurrent, while in the opposite case, it is null recurrent.
