View source for Convergence, almost-certain

''almost-sure convergence, convergence with probability one''

Convergence of a sequence of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026020/c0260201.png" /> defined on a certain probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026020/c0260202.png" />, to a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026020/c0260203.png" />, defined in the following way: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026020/c0260204.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026020/c0260205.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026020/c0260206.png" />-almost certain) if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026020/c0260207.png" /></td> </tr></table>

In mathematical analysis this form of convergence is called almost-everywhere convergence. [[Convergence in probability|Convergence in probability]] follows from almost-certain convergence.



====Comments====
See also [[Convergence, types of|Convergence, types of]]; [[Weak convergence of probability measures|Weak convergence of probability measures]]; [[Distributions, convergence of|Distributions, convergence of]].