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Difference between revisions of "Stochastic sequence"

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A sequence of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s0903001.png" />, defined on a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s0903002.png" /> with an increasing family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s0903003.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s0903004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s0903005.png" />, on it, which is adapted: For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s0903006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s0903007.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s0903008.png" />-measurable. In writing such sequences, the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s0903009.png" /> is often used, stressing the measurability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s09030010.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s09030011.png" />. Typical examples of stochastic sequences defined on a probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s09030012.png" /> are Markov sequences, martingales, semi-martingales, and others (cf. [[Markov chain|Markov chain]]; [[Martingale|Martingale]]; [[Semi-martingale|Semi-martingale]]). In the case of continuous time (where the discrete time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s09030013.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s09030014.png" />), the corresponding aggregate of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s09030015.png" /> is called a [[Stochastic process|stochastic process]].
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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|Markov chain]]; [[Martingale|Martingale]]; [[Semi-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|stochastic process]].
  
  
  
 
====Comments====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090300/s09030016.png" /> is given, one assumes, as in the main article, adaptation of the process. Cf. also [[Stochastic process, compatible|Stochastic process, compatible]].
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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|Stochastic process, compatible]].

Latest revision as of 23:31, 25 November 2018

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.

How to Cite This Entry:
Stochastic sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_sequence&oldid=43493
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article