In
mathematics — specifically, in
large deviations theory
In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to Laplace, the formalization started with insura ...
— a rate function is a function used to quantify the
probabilities
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
of rare events. It is required to have several properties which assist in the formulation of the large deviation principle. In some sense, the large deviation principle is an analogue of
weak convergence of probability measures, but one which takes account of how well the rare events behave.
A rate function is also called a Cramér function, after the Swedish probabilist
Harald Cramér
Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of stati ...
.
Definitions
Rate function An
extended real-valued function ''I'' : ''X'' →
, +∞defined on a
Hausdorff topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' is said to be a rate function if it is not identically +∞ and is
lower semi-continuous
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, r ...
, i.e. all the sub-level sets
:
are
closed in ''X''.
If, furthermore, they are