We propose to perform statistical probabilistic model checking by using perfect simulation in order to verify steady-state and unbounded until formulas over Markov chains. The model checking of probabilistic models by statistical methods has received increased attention in the last years since it provides an interesting alternative to numerical model checking which is poorly scalable with the increasing model size. In the previous statistical model checking works, the unbounded until formulas can not be efficiently verified and the steady-state formulas have not been considered, due to the burn-in time problem to detect the steady-state. Perfect simulation by coupling in the past is an extension of Markov chain Monte Carlo MCMC method that allows us to obtain the samples according to the steady-state distribution of the underlying Markov chain and thus it avoids the burn-in time problem to detect the steady-state. Therefore we propose to verify unbounded until and steady-state dependability properties for large Markov chains by combining perfect simulation and statistical hypothesis testing.