In a context of $\omega$-regular specifications for infinite execution

sequences, the classical B\"uchi condition, or repeated liveness

condition, asks that an accepting state is visited infinitely often. In

this paper, we show that in a probabilistic context it is relevant to

strengthen this infinitely often condition. An execution path is now

accepting if the \emph{proportion} of time spent on an accepting state

does not go to zero as the length of the path goes to infinity. We

introduce associated notions of recurrence and transience for

non-homogeneous finite Markov chains and study the computational

complexity of the associated problems. As Probabilistic B\"uchi Automata

(PBA) have been an attempt to generalize B\"uchi automata to a

probabilistic context, we define a class of Constrained Probabilistic

Automata with our new accepting condition on runs. The accepted language

is defined by the requirement that the measure of the set of accepting

runs is positive (probable semantics) or equals 1 (almost-sure

semantics). In contrast to the PBA case, we prove that

the emptiness problem for the language of a constrained probabilistic

B\"uchi automaton with the probable semantics is decidable.