Infinitely often and almost always: Difference between revisions

Revision as of 21:17, 31 July 2019

Let $A_{1}, A_{2}, A_{3}, \dots$ be a sequence of events in some sample space $Ω$ .

perspective	infinitely often	almost always
unions and intersections	$ω \in ⋂_{N = 1}^{\infty} ⋃_{n = N}^{\infty} A_{n}$	$ω \in ⋃_{N = 1}^{\infty} ⋂_{n = N}^{\infty} A_{n}$
first-order quantifiers	$\forall N \geq 1 \exists n \geq N : ω \in A_{n}$	$\exists N \leq 1 \forall n \geq N : ω \in A_{n}$
verbal expression	$ω \in A_{n}$ for infinitely many $n \geq 1$	$ω \in A_{n}$ for almost all $n \geq 1$ , i.e. $ω \in A_{n}$ for all but finitely many $n \geq 1$ , i.e. $ω \notin A_{n}$ for finitely many $n \geq 1$
lim sup/lim inf	$ω \in {lim sup}_{n \to \infty} A_{n}$	$ω \in {lim inf}_{n \to \infty} A_{n}$
limit of sup/inf	$ω \in lim_{N \to \infty} {sup}_{n \geq N} A_{n}$	$ω \in lim_{N \to \infty} {inf}_{n \geq N} A_{n}$

@@ Line 15: / Line 15: @@
 | limit of sup/inf || <math>\omega \in \lim_{N\to\infty} \sup_{n\geq N} A_n</math> || <math>\omega \in \lim_{N\to\infty} \inf_{n\geq N} A_n</math>
 |}
+[[Category:Probability]]