In this paper, we study the problem of state observation of nonlinear systems over an erasure channel. Stochastic notions of stability are adopted from ergodic theory of random dynamical systems for the analysis. We use stability with probability one and mean square exponential stability to analyze the stability property of observer error dynamics. The main results of this paper prove that there is no limitation for stabilization with probability one, however fundamental limitation arises for mean square exponential stabilization of the error dynamics. We provide necessary condition for the mean square exponential stability of the error dynamics, expressed in terms of the probability of channel erasure and the positive Lyapunov exponents of the system dynamics. Simulation results are presented to verify the main results of the paper.