Rate function

In mathematics — specifically, in large deviations theory — a rate function is a function used to quantify the probabilities of rare events. Such functions are used to formulate large deviation principles. A large deviation principle quantifies the asymptotic probability of rare events for a sequence of probabilities.
A rate function is also called a Cramér function, after the Swedish probabilist Harald Cramér.

Definitions

Rate function An extended real-valued function defined on a Hausdorff topological space is said to be a rate function if it is not identically and is lower semi-continuous i.e. all the sub-level sets
are closed in.
If, furthermore, they are compact, then is said to be a good rate function.
A family of probability measures on is said to satisfy the large deviation principle with rate function if, for every closed set and every open set,
If the upper bound holds only for compact sets, then is said to satisfy the weak large deviations principle.

Remarks

The role of the open and closed sets in the large deviation principle is similar to their role in the weak convergence of probability measures: recall that is said to converge weakly to if, for every closed set and every open set,
There is some variation in the nomenclature used in the literature: for example, den Hollander uses simply "rate function" where this article — following Dembo & Zeitouni — uses "good rate function", and "weak rate function". Rassoul-Agha & Seppäläinen uses the term "tight rate function" instead of "good rate function" due to the connection with exponential tightness of a family of measures. Regardless of the nomenclature used for rate functions, examination of whether the upper bound inequality is supposed to hold for closed or compact sets tells one whether the large deviation principle in use is strong or weak.

Properties

Uniqueness

A natural question to ask, given the somewhat abstract setting of the general framework above, is whether the rate function is unique. This turns out to be the case: given a sequence of probability measures _δ>0 on X satisfying the large deviation principle for two rate functions I and J, it follows that I = J for all x ∈ X.

Exponential tightness

It is possible to convert a weak large deviation principle into a strong one if the measures converge sufficiently quickly. If the upper bound holds for compact sets F and the sequence of measures _δ>0 is exponentially tight, then the upper bound also holds for closed sets F. In other words, exponential tightness enables one to convert a weak large deviation principle into a strong one.

Continuity

Naïvely, one might try to replace the two inequalities and by the single requirement that, for all Borel sets S ⊆ X,
The equality is far too restrictive, since many interesting examples satisfy and but not. For example, the measure μ_δ might be non-atomic for all δ, so the equality could hold for S = only if I were identically +∞, which is not permitted in the definition. However, the inequalities and do imply the equality for so-called I-continuous sets S ⊆ X, those for which
where and denote the interior and closure of S in X respectively. In many examples, many sets/events of interest are I-continuous. For example, if I is a continuous function, then all sets S such that
are I-continuous; all open sets, for example, satisfy this containment.

Transformation of large deviation principles

Given a large deviation principle on one space, it is often of interest to be able to construct a large deviation principle on another space. There are several results in this area:

the contraction principle tells one how a large deviation principle on one space "pushes forward" to a large deviation principle on another space via a continuous function;
the Dawson-Gärtner theorem tells one how a sequence of large deviation principles on a sequence of spaces passes to the projective limit.
the tilted large deviation principle gives a large deviation principle for integrals of exponential functionals.
exponentially equivalent measures have the same large deviation principles.

History and basic development

The notion of a rate function emerged in the 1930s with the Swedish mathematician Harald Cramér's study of a sequence of i.i.d. random variables _i∈. Namely, among some considerations of scaling, Cramér studied the behavior of the distribution of the average as n→∞. He found that the tails of the distribution of X_n decay exponentially as e^−nλ where the factor λ in the exponent is the Legendre–Fenchel transform of the cumulant-generating function For this reason this particular function λ is sometimes called the Cramér function. The rate function defined above in this article is a broad generalization of this notion of Cramér's, defined more abstractly on a probability space, rather than the state space of a random variable.