Stochastic sequence

A sequence of random variables $X=(X_n)_{n\geq1}$, defined on a measure space $(\Omega,\mathcal F)$ with an increasing family of $\sigma$-algebras $(\mathcal F_n)_{n\geq1}$, $\mathcal F_n\subseteq\mathcal F$, on it, which is adapted: For every $n\geq1$, $X_n$ is $\mathcal F_n$-measurable. In writing such sequences, the notation $X=(X_n,\mathcal F_n)_{n\geq1}$ is often used, stressing the measurability of $X_n$ relative to $\mathcal F_n$. Typical examples of stochastic sequences defined on a probability space $(\Omega,\mathcal F,\mathrm P)$ are Markov sequences, martingales, semi-martingales, and others (cf. Markov chain; Martingale; Semi-martingale). In the case of continuous time (where the discrete time $n\geq1$ is replaced by $t\geq0$), the corresponding aggregate of objects $X=(X_t,\mathcal F_t)_{t\geq0}$ 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 $\mathcal F=(\mathcal F_n)_{n\geq1}$ is given, one assumes, as in the main article, adaptation of the process. Cf. also Stochastic process, compatible.