In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, a pregaussian class or pregaussian set of functions is a set of functions, square integrable with respect to some probability measure

Probability measure

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...

, such that there exists a certain Gaussian process

Gaussian process

In probability theory and statistics, a Gaussian process is a stochastic process whose realisations consist of random values associated with every point in a range of times such that each such random variable has a normal distribution...

, indexed by this set, satisfying the conditions below.

Definition

For a probability space

Probability space

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

 (S, Σ, P), denote by a set of square integrable with respect to P functions , that is


Consider a set . There exists a Gaussian process

Gaussian process

In probability theory and statistics, a Gaussian process is a stochastic process whose realisations consist of random values associated with every point in a range of times such that each such random variable has a normal distribution...

 , indexed by , with mean 0 and covariance

Such a process exists because the given covariance is positive definite. This covariance defines a semi-inner product as well as a pseudometric

Pseudometric space

In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space...

 on given by

Definition A class is called pregaussian if for each the function on is bounded, -uniformly continuous, and prelinear.

Brownian bridge

The process is a generalization of the brownian bridge

Brownian bridge

A Brownian bridge is a continuous-time stochastic process B whose probability distribution is the conditional probability distribution of a Wiener process W given the condition that B = B = 0.The expected value of the bridge is zero, with variance t, implying that the most...

. Consider with P being the uniform measure

Uniform distribution (continuous)

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...

. In this case, the process indexed by the indicator functions , for is in fact the standard brownian bridge

Brownian bridge

A Brownian bridge is a continuous-time stochastic process B whose probability distribution is the conditional probability distribution of a Wiener process W given the condition that B = B = 0.The expected value of the bridge is zero, with variance t, implying that the most...

 B(x). This set of the indicator functions is pregaussian, moreover, it is the Donsker class.