Almost sure convergence: Difference between revisions
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'''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. | '''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. | ||
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==Definition== | ==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: | 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: | ||
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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: | 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: | ||
<math>\mathop{\lim}_{k \rightarrow \infty} X_k = Y\,\,P-a.s </math> or <math>\mathop{\lim}_{k \rightarrow \infty} X_k = Y a.s</math>. | <math>\mathop{\lim}_{k \rightarrow \infty} X_k = Y \,\,P{\rm -a.s},</math> | ||
or | |||
<math>\mathop{\lim}_{k \rightarrow \infty} X_k = Y\,\,{\rm a.s}</math>. | |||
==Important cases of almost sure convergence== | ==Important cases of almost sure convergence== | ||
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</math>. | </math>. | ||
This is an example of the [[strong law of large numbers]]. | This is an example of the [[strong law of large numbers]].[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 06:00, 9 July 2024
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.
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 be a probability space (in particular, ) is a measurable space). A (-valued) random variable is defined to be any measurable function , where is the sigma algebra of Borel sets of . A formal definition of almost sure convergence can be stated as follows:
A sequence of random variables is said to converge almost surely to a random variable if for all , where is some measurable set satisfying . An equivalent definition is that the sequence converges almost surely to if for all , where is some measurable set with . This convergence is often expressed as:
or
.
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 .
This is an example of the strong law of large numbers.