Search results
Results from the WOW.Com Content Network
A measurable subset of a standard probability space is a standard probability space. It is assumed that the set is not a null set, and is endowed with the conditional measure. See (Rokhlin 1952, Sect. 2.3 (p. 14)) and (Haezendonck 1973, Proposition 5). Every probability measure on a standard Borel space turns it into a standard probability space.
An event space, which is a set of events, , an event being a set of outcomes in the sample space. A probability function, , which assigns, to each event in the event space, a probability, which is a number between 0 and 1 (inclusive).
It is easy to show that, when t ≤ s, the X t,p norm is finite whenever the X s,p norm is. Therefore, the spaces X s,p and X t,p are nested: ,,. This is consistent with the usual nesting of smoothness classes of functions f: D → R: for example, the Sobolev space H 2 (D) is a subspace of H 1 (D) and in turn of the Lebesgue space L 2 (D) = H 0 (D); the Hölder space C 1 (D) of continuously ...
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also ...
In probability theory particularly in the Malliavin calculus, a Gaussian probability space is a probability space together with a Hilbert space of mean zero, real-valued Gaussian random variables. Important examples include the classical or abstract Wiener space with some suitable collection of Gaussian random variables.
In the mathematical theory of probability, the Ionescu-Tulcea theorem, sometimes called the Ionesco Tulcea extension theorem, deals with the existence of probability measures for probabilistic events consisting of a countably infinite number of individual probabilistic events.
A probability metric D between two random variables X and Y may be defined, for example, as (,) = | | (,) where F(x, y) denotes the joint probability density function of the random variables X and Y.
In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probability space.