  We show that a subclass of infinite-state probabilistic programs that can be
modeled by probabilistic one-counter automata (pOC) admits an efficient
quantitative analysis. In particular, we show that the expected termination
time can be approximated up to an arbitrarily small relative error with
polynomially many arithmetic operations, and the same holds for the probability
of all runs that satisfy a given omega-regular property. Further, our results
establish a powerful link between pOC and martingale theory, which leads to
fundamental observations about quantitative properties of runs in pOC. In
particular, we provide a "divergence gap theorem", which bounds a positive
non-termination probability in pOC away from zero.
