Verifying that a stochastic system is in a certain state when it has reached equilibrium has important applications. For instance, the probabilistic verification of the long-run behavior of a safety-critical system enables assessors to check whether it accepts a human \emph{abort}-command at any time with a probability that is sufficiently high. The stochastic system is represented as probabilistic model, a long-run property is asserted and a probabilistic verifier checks the model against the property.

However, existing probabilistic verifiers do not account for the imprecision of the probabilistic parameters in the model. Due to uncertainty, the probability of any state transition may be subject to small perturbations which can have direct consequences for the veracity of the verification result. In reality, the safety-critical system may accept the \emph{abort}-command with an insufficient probability.

In this paper, we introduce the first probabilistic verification technique that accounts for uncertainty on the verification of long-run properties of a stochastic system. We present a mathematical framework for the asymptotic analysis of the stationary distribution of a discrete-time Markov chain, making \emph{no} assumptions about the distribution of the perturbations. Concretely, our novel technique computes upper and lower bounds on the long-run probability, given a certain degree of uncertainty about the stochastic system.