Consistent Estimation in Partially Observed Random Walks

Abstract

If a discrete-time random walk, consisting of the sum of independent increments from an unknown underlying distribution, is observed at every time instant then it is clear that the underlying distribution can be consistently estimated. If, however, we are restricted to observing only a subset of the times, and if this subset is too sparse, then a central limit effect takes over and only two moments can be consistently estimated. We show that divergence of the sum of the reciprocals of the observation time-intervals is a necessary and sufficient condition to permit consistent estimation of the third moment. Corresponding conditions permit consistent estimation of moments of higher order. An explicit consistent estimator for the distribution itself is presented when all moments can be consistently estimated and the distribution is determined by its moments.