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...

time will almost surely type any given text, including the complete works of William Shakespeare. In fact, the monkey would almost surely type every...

converges almost completely, or almost in probability towards X. When Xn converges almost completely towards X then it also converges almost surely to X....

{\mathcal {F}}} is called almost surely invariant if and only if its indicator function 1 S {\displaystyle 1_{S}} is almost surely equal to the indicator...

Almost periodic function - and Operators Almost all Almost surely Approximation List of mathematical jargon "Almost All Real Numbers are Transcendental -...

{ X n } {\displaystyle \{X_{n}\}} converges to X {\displaystyle X} almost surely, it means that the set of points O = { ω ∣ lim X n ( ω ) ≠ X ( ω ) }...

complement is of measure zero. In probability theory, the terms almost surely, almost certain and almost always refer to events with probability 1 not necessarily...

{\mathcal {N}}(0,u).} W {\displaystyle W} has almost surely continuous paths: W t {\displaystyle W_{t}} is almost surely continuous in t {\displaystyle t} . That...

\alpha }} has almost surely pure point spectrum and exhibits Anderson localization. (It is known that almost surely can not be replaced by surely.) That the...

flips will almost surely converge to 1⁄2 as n approaches infinity. Although the proportion of heads (and tails) approaches 1⁄2, almost surely the absolute...

]}\leq \liminf _{n\to \infty }\,\mathbb {E} [X_{n}|{\mathcal {G}}]}    almost surely. Note: Conditional expectation for non-negative random variables is...

of the following three conditions holds: (a) The stopping time τ is almost surely bounded, i.e., there exists a constant c ∈ N {\displaystyle \mathbb...

generally, "almost all" is sometimes used in the sense of "almost everywhere" in measure theory, or in the closely related sense of "almost surely" in probability...

{\displaystyle \mathbf {x} } and z {\displaystyle \mathbf {z} } , then almost surely we have m ~ M , n ( x , Θ 1 , … , Θ M ) = ∑ i = 1 n Y i K M , n ( x...

|X_{t+1}-X_{t}|\leq k} almost surely for some fixed bound k and all t. Also assume that | X 1 | ≤ k {\displaystyle |X_{1}|\leq k} almost surely. Define σ t 2 =...

such that a i ≤ X i ≤ b i {\displaystyle a_{i}\leq X_{i}\leq b_{i}} almost surely. Consider the sum of these random variables, S n = X 1 + ⋯ + X n . {\displaystyle...

{1}{t}}Y_{t}={\frac {1}{\operatorname {E} [S_{1}]}}\operatorname {E} [W_{1}]} almost surely. Renewal processes additionally have a property analogous to the central...

points have infinitely many intersections almost surely, but for dimensions higher than 5, they almost surely intersect only finitely often. The asymptotic...

"almost surely"—a central property of the Lebesgue integral. Basically, one says that an inequality like X ≥ 0 {\displaystyle X\geq 0} is true almost surely...

G(n, p) will almost surely have no connected components of size larger than O(log(n)). If np = 1, then a graph in G(n, p) will almost surely have a largest...

discovered to be exceptionally complex mathematically. The Wiener process is almost surely nowhere differentiable; thus, it requires its own rules of calculus...

the intersection of two almost sure events is almost sure. By definition, we conclude that g(Xn) converges to g(X) almost surely. Slutsky's theorem Portmanteau...

an instantiation of the St. Petersburg paradox. Since a gambler will almost surely eventually flip heads, the martingale betting strategy is certain to...

) (almost surely) {\displaystyle {\frac {R_{1}+\cdots +R_{n}}{n}}\rightarrow {\frac {\mu (X)}{\mu (A)}}\quad {\text{(almost surely)}}} (See almost surely...

{\textstyle X_{t}^{-}=-\min(X_{t},0)} . Then the sequence converges almost surely to a random variable X {\displaystyle X} with finite expectation. There...

name for a clade). Therefore, in practice the name Liliopsida will almost surely refer to the usage as in the Cronquist system. In summary the monocotyledons...

from an almost surely constant random variable, which may take other values, but only on events with probability zero. Constant and almost surely constant...

namely a tail event of independent σ-algebras, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring...

event is independent of itself if and only if it almost surely occurs or its complement almost surely occurs; this fact is useful when proving zero–one...

blessing, as it ensures that the nearest-neighbor of an instance is almost-surely its most closely related instance. From this perspective, contrast-loss...