2.5 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Almost surely | 2/2 | https://en.wikipedia.org/wiki/Almost_surely | reference | science, encyclopedia | 2026-05-05T07:23:15.024547+00:00 | kb-cron |
== Asymptotically almost surely == In asymptotic analysis, a property is said to hold asymptotically almost surely (a.a.s.) if over a sequence of sets, the probability converges to 1. This is equivalent to convergence in probability. For instance, in number theory, a large number is asymptotically almost surely composite, by the prime number theorem; and in random graph theory, the statement "
G
(
n
,
p
n
)
{\displaystyle G(n,p_{n})}
is connected" (where
G
(
n
,
p
)
{\displaystyle G(n,p)}
denotes the graphs on
n
{\displaystyle n}
vertices with edge probability
p
{\displaystyle p}
) is true a.a.s. when, for some
ε
>
0
{\displaystyle \varepsilon >0}
p
n
>
(
1
+
ε
)
ln
n
n
.
{\displaystyle p_{n}>{\frac {(1+\varepsilon )\ln n}{n}}.}
In number theory, this is referred to as "almost all", as in "almost all numbers are composite". Similarly, in graph theory, this is sometimes referred to as "almost surely".
== See also ==
Almost Almost everywhere, the corresponding concept in measure theory Convergence of random variables, for "almost sure convergence" With high probability Cromwell's rule, which says that probabilities should almost never be set as zero or one Degenerate distribution, for "almost surely constant" Infinite monkey theorem, a theorem using the aforementioned terms List of mathematical jargon
== Notes ==
== References == Rogers, L. C. G.; Williams, David (2000). Diffusions, Markov Processes, and Martingales. Vol. 1: Foundations. Cambridge University Press. ISBN 978-0521775946. Williams, David (1991). Probability with Martingales. Cambridge Mathematical Textbooks. Cambridge University Press. ISBN 978-0521406055.