equicontinuity technical

In practice, showing the equicontinuity is often not so difficult.

The main condition is the equicontinuity of the family of functions.

The notion of equicontinuity was introduced at around the same time by and .

Note that the equicontinuity is essential here.

Compactness may be shown by invoking equicontinuity.

For example, Ascoli introduced equicontinuity in 1884, a topic regarded as one of the fundamental concepts in the theory of real functions.

A typical Fredholm integral equation gives rise to a compact operator K on function spaces; the compactness property is shown by equicontinuity.

It is a version of equicontinuity used in the context of functions of random variables: that is, random functions.

The definitions of boundedness and equicontinuity can be generalized to the setting of arbitrary compact metric spaces and, more generally still, compact Hausdorff spaces.

For instance, stochastic equicontinuity, along with other conditions, can be used to show uniform weak convergence, which can be used to prove the convergence of extremum estimators.

If estimator T is defined implicitly, for example as a value that maximizes certain objective function (see extremum estimator), then a more complicated argument involving stochastic equicontinuity has to be used.

In estimation theory in statistics, stochastic equicontinuity is a property of estimators or of estimation procedures that is useful in dealing with their asymptotic behaviour as the amount of data increases.

For a fixed x X and ε, the sets N(ε, U) form an open covering of F as U varies over all open neighborhoods of x. Choosing a finite subcover then gives equicontinuity.

The equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only if it is closed, pointwise bounded and equicontinuous.