Consider a nonlinear dynamic system where one wishes to estimate a state vector using noisy measurements, Many algorithms have been proposed to address this problem, among them the extended Kalman filter (and its variants) and constant-gain stochastic approximation. To quantify the efficacy of these algorithms, it is necessary to describe the distribution of the state estimation error. Typically, performance has been measured by the estimation error covariance alone, which does not provide enough information to probabilistically quantify the estimation accuracy. By casting the estimation error in an autoregressive-type form, this paper addresses the broader question of the distribution of the error for a general class of recursive algorithms. We illustrate the distributional results in an epidemiological problem of disease monitoring.