Almost sure convergence: Difference between revisions

Revision as of 06:53, 14 July 2008

Definition

In this section, a formal definition of almost sure convergence will be given for complex vector-valued random variables, but it should be noted that a more general definition can also be given for random variables that take on values on more abstract topological spaces. To this end, let $(\Omega ,{\mathcal {F}},P)$ be a probability space (in particular, $(\Omega ,{\mathcal {F}}$ ) is a measurable space). A ( $\mathbb {C} ^{n}$ -valued) random variable is defined to be any measurable function $X:(\Omega ,{\mathcal {F}})\rightarrow (\mathbb {C} ^{n},{\mathcal {B}}(\mathbb {C} ^{n}))$ , where ${\mathcal {B}}(\mathbb {C} ^{n})$ is the sigma algebra of Borel sets of $\mathbb {C} ^{n}$ . A formal definition of almost sure convergence can be stated as follows:

A sequence $X_{1},X_{2},\ldots ,X_{n},\ldots$ of random variables is said to converge almost surely to a random variable $Y$ if $\mathop {\lim } _{k\rightarrow \infty }X_{k}(\omega )=Y(\omega )$ for all $\omega \in \Lambda$ , where $\Lambda \subset \Omega$ is some measurable set satisfying $P(\Lambda )=1$ . An equivalent definition is that the sequence $X_{1},X_{2},\ldots ,X_{n},\ldots$ converges almost surely to $Y$ if $\mathop {\lim } _{k\rightarrow \infty }X_{k}(\omega )=Y(\omega )$ for all $\omega \in \Omega \backslash \Lambda '$ , where $\Lambda '$ is some measurable set with $P(\Lambda ')=0$ . This convergence is often expressed as:

$\mathop {\lim } _{k\rightarrow \infty }X_{k}=Y\,\,P{\rm {-a.s}},$

or

$\mathop {\lim } _{k\rightarrow \infty }X_{k}=Y\,\,{\rm {a.s}}$ .

Important cases of almost sure convergence

If we flip a coin n times and record the percentage of times it comes up heads, the result will almost surely approach 50% as $\scriptstyle n\rightarrow \infty$ .

This is an example of the strong law of large numbers.

@@ Line 1: / Line 1: @@
 {{subpages}}
-'''Almost sure convergence''' is one of the four main modes of [[stochastic convergence]]. It may be viewed as a notion of convergece for random variables that is similar to, but not the same as, the notion of [[pointwise convergence]] for real functions.
+'''Almost sure convergence''' is one of the four main modes of [[stochastic convergence]]. It may be viewed as a notion of convergence for random variables that is similar to, but not the same as, the notion of [[pointwise convergence]] for real functions.
 <!--==Examples==-->
 <!--===Basic example===-->
 <!--===Intermediate example===-->
+==Definition==
+In this section, a formal definition of almost sure convergence will be given for complex vector-valued random variables, but it should be noted that a more general definition can also be given for random variables that take on values on more abstract [[topological space|topological spaces]]. To this end, let <math>(\Omega,\mathcal{F},P)</math> be a [[measure space|probability space]] (in particular, <math>(\Omega,\mathcal{F}</math>) is a [[measurable space]]). A (<math>\mathbb{C}^n</math>-valued) '''random variable''' is defined to be any [[measurable function]] <math>X:(\Omega,\mathcal{F})\rightarrow (\mathbb{C}^n,\mathcal{B}(\mathbb{C}^n))</math>, where <math>\mathcal{B}(\mathbb{C}^n)</math> is the [[sigma algebra]] of [[Borel set|Borel sets]] of <math>\mathbb{C}^n</math>. A formal definition of almost sure convergence can be stated as follows:
+A sequence <math>X_1,X_2,\ldots,X_n,\ldots</math> of random variables is said to '''converge almost surely''' to a random variable <math>Y</math> if <math>\mathop{\lim}_{k \rightarrow \infty}X_k(\omega)=Y(\omega)</math> for all <math>\omega \in \Lambda</math>, where <math>\Lambda \subset \Omega</math> is some measurable set satisfying <math>P(\Lambda)=1</math>. An equivalent definition is that the sequence <math>X_1,X_2,\ldots,X_n,\ldots</math>  converges almost surely to <math>Y</math> if <math>\mathop{\lim}_{k \rightarrow \infty}X_k(\omega)=Y(\omega)</math> for all <math>\omega \in \Omega \backslash \Lambda'</math>, where <math>\Lambda'</math> is some measurable set with <math>P(\Lambda')=0</math>. This convergence is often expressed as:
-==Definition==
+<math>\mathop{\lim}_{k \rightarrow \infty} X_k = Y \,\,P{\rm -a.s},</math>
-In this section, a formal definition of almost sure convergence will be given for complex vector-valued random variables, but it should be noted that a more general definition can also be given for random variables that take on values on more abstract sets. To this end, let <math>(\Omega,\mathcal{F},P)</math> be a [[measure space|probability space]] (in particular, <math>(\Omega,\mathcal{F}</math>) is a [[measurable space]]). A (<math>\mathbb{C}^n</math>-valued) '''random variable''' is defined to be any [[measurable function]] <math>X:(\Omega,\mathcal{F})\rightarrow (\mathbb{C}^n,\mathcal{B}(\mathbb{C}^n))</math>, where <math>\mathcal{B}(\mathbb{C}^n)</math> is the [[Borel set]] of <math>\mathbb{C}^n</math>. A formal definition of almost sure convergence can be stated as follows:
-A sequence <math>X_1,X_2,\ldots,X_n,\ldots</math> of random variables is said to '''converge almost surely''' to a random variable <math>Y</math> if <math>\mathop{\lim}_{n \rightarrow \infty}X_k(\omega)=Y(\omega)</math> for all <math>\omega \in \Lambda</math>, where <math>\Lambda \subset \Omega</math> is some set satisfying <math>P(\Lambda)=1</math>. An equivalent definition is that the sequence <math>X_1,X_2,\ldots,X_n,\ldots</math>  converge almost surely to <math>Y</math> if <math>\mathop{\lim}_{n \rightarrow \infty}X_k(\omega)=Y(\omega)</math> for all <math>\omega \in \Omega \backslash \Lambda'</math>, where <math>\Lambda'</math> is some set with <math>P(\Lambda')=0</math>. This convergence is often expressed as:
+or
-<math>\mathop{\lim}_{k \rightarrow \infty} X_n = Y\,\,P-a.s </math> or <math>\mathop{\lim}_{k \rightarrow \infty} X_n = Y a.s</math>.
+<math>\mathop{\lim}_{k \rightarrow \infty} X_k = Y\,\,{\rm a.s}</math>.
 ==Important cases of almost sure convergence==
@@ Line 25: / Line 22: @@
 This is an example of the [[strong law of large numbers]].
-==References==
-#D. Williams, ''Probability with Martingales'', Cambridge : Cambridge University Press, 1991.
-#P. Billingsley, ''Probability and Measure'' (2 ed.), ser. Wiley Series in Probability and Mathematical Statistics. Wiley, 1986.
-#E. Wong and B. Hajek, ''Stochastic Processes in Engineering Systems'', Springer-Verlag, New York, 1985.
-==See also==
-*[[Stochastic convergence]]
-*[[Convergence in distribution]]
-*[[Convergence in probability]]
-*[[Convergence in rth order mean]]
-==Related topics==
-*[[Stochastic variable]]
-*[[Stochastic process]]
-*[[Stochastic diffential equation]]
-<!--==External links==-->

Almost sure convergence: Difference between revisions

Revision as of 06:53, 14 July 2008

Definition

Important cases of almost sure convergence

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