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