Stochastic sequence

A sequence of random variables , defined on a measure space with an increasing family of -algebras , , on it, which is adapted: For every , is -measurable. In writing such sequences, the notation is often used, stressing the measurability of relative to . Typical examples of stochastic sequences defined on a probability space are Markov sequences, martingales, semi-martingales, and others (cf. Markov chain; Martingale; Semi-martingale). In the case of continuous time (where the discrete time is replaced by ), the corresponding aggregate of objects is called a stochastic process.

Comments

The expression "stochastic sequence" is rarely used in the West; one usually says "stochastic process" and adds "with discrete time" if necessary. Strictly speaking, it is just a sequence of random variables, but often, when a filtration is given, one assumes, as in the main article, adaptation of the process. Cf. also Stochastic process, compatible.