A state "i" is said to be ergodic if it is aperiodic and positive recurrent. In other words, a state "i" is ergodic if it is recurrent, has a period of "1", and has finite mean recurrence time. If all states in an irreducible Markov chain are ergodic, then the chain is said to be ergodic. It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state. More generally, a Markov chain is ergodic if there is a number "N" such that any state can be reached from any other state in any number of steps less or equal to a number "N". In case of a fully connected transition matrix, where all transitions have a non-zero probability, this condition is fulfilled with "N" = 1. A Markov chain with more than one state and just one out-going transition per state is either not irreducible or not aperiodic, hence cannot be ergodic. If the Markov chain is a time-homogeneous Markov chain, so that the process is described by a single, time-independent matrix formula_89, then the vector formula_90 is called a stationary distribution (or invariant measure) if formula_91 it satisfies  An irreducible chain has a positive stationary distribution (a stationary distribution such that formula_95) if and only if all of its states are positive recurrent. In that case, is unique and is related to the expected return time:  where formula_97 is the normalizing constant. Further, if the positive recurrent chain is both irreducible and aperiodic, it is said to have a "limiting" distribution; for any "i" and "j",  If a chain has more than one closed communicating class, its stationary distributions will not be unique (consider any closed communicating class formula_100 in the chain; each one will have its own unique stationary distribution formula_101. Extending these distributions to the overall chain, setting all values to zero outside the communication class, yields that the set of invariant measures of the original chain is the set of all convex combinations of the formula_101's). However, if a state "j" is aperiodic, then formula_103 and for any other state "i", let "fij" be the probability that the chain ever visits state "j" if it starts at "i", formula_104  If a state "i" is periodic with period "k" > 1 then the limit formula_105 does not exist, although the limit formula_106 does exist for every integer "r".