How Sampling Reveals a Process

Abstract

A series of observations $\{\xi_1, \xi_2, \xi_3,\ldots\}$ is presented to us and at each time $n$, when we have observed the first $n$ of them, we are called upon to give our guess for what stochastic process produced the data. A universal scheme is given which, for any Bernoulli process (not necessarily independent), gives a sequence of processes that converges in a strong sense (the $\bar{d}$-metric) to the real process. In addition to this main result, many others are given which put it into proper perspective. In particular it is shown that in a certain sense the class of Bernoulli processes is the largest one for which such a universal scheme is possible.