Almost surely

In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure).^[1] In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between almost surely and surely (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set,^[2] because an infinite set can have non-empty subsets of probability 0.

Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, the continuity of the paths of Brownian motion, and the infinite monkey theorem. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of almost surely: an event that happens with probability zero happens almost never.^[3]

Let ${\displaystyle (\Omega,{\mathcal {F}},P)}$ be a probability space. An event${\displaystyle E\in {\mathcal {F}}}$ happens almost surely if ${\displaystyle P(E)=1}$. Equivalently, ${\displaystyle E}$ happens almost surely if the probability of ${\displaystyle E}$ not occurring is zero: ${\displaystyle P(E^{C})=0}$. More generally, any set ${\displaystyle E\subseteq \Omega }$ (not necessarily in ${\displaystyle {\mathcal {F}}}$) happens almost surely if ${\displaystyle E^{C}}$ is contained in a null set: a subset ${\displaystyle N}$ in ${\displaystyle {\mathcal {F}}}$ such that ${\displaystyle P(N)=0}$.^[4] The notion of almost sureness depends on the probability measure ${\displaystyle P}$. If it is necessary to emphasize this dependence, it is customary to say that the event ${\displaystyle E}$ occurs P-almost surely, or almost surely ${\displaystyle \left(\!P\right)}$.

In general, an event can happen "almost surely", even if the probability space in question includes outcomes which do not belong to the event—as the following examples illustrate.

En análisis asintótico , se dice que una propiedad se cumple asintóticamente casi seguramente (aas) si sobre una secuencia de conjuntos, la probabilidad converge a 1. Esto es equivalente a la convergencia en probabilidad . Por ejemplo, en teoría de números, un número grande es asintóticamente casi seguramente compuesto , por el teorema de los números primos ; y en teoría de grafos aleatorios , la afirmación "${\displaystyle G(n,p_{n})}$está conectado " (donde${\displaystyle G(n,p)}$denota los gráficos en${\displaystyle n}$vértices con probabilidad de arista${\displaystyle p}$) es verdadero aas cuando, para algunos${\displaystyle \varepsilon >0}$

${\displaystyle p_{n}>{\frac {(1+\varepsilon )\ln n}{n}}.}$   ^{[ 6 ]}

En teoría de números , esto se denomina " casi todos ", como en "casi todos los números son compuestos". De manera similar, en teoría de grafos, a veces se denomina "casi con seguridad". ^{[ 7 ]}

• Casi
• Casi en todas partes , el concepto correspondiente en la teoría de la medida.
• Convergencia de variables aleatorias , para "convergencia casi segura"
• Con alta probabilidad
• La regla de Cromwell , que establece que las probabilidades casi nunca deben fijarse en cero o uno.
• Distribución degenerada , para "casi seguramente constante"
• Teorema del mono infinito , un teorema que utiliza los términos antes mencionados.
• Lista de jerga matemática

• Rogers, LCG; Williams, David (2000). Difusiones, procesos de Markov y martingalas . Vol.  1: Fundamentos. Cambridge University Press. ISBN 978-0521775946.
• Williams, David (1991). Probabilidad con martingalas . Cambridge Mathematical Textbooks. Cambridge University Press. ISBN 978-0521406055.