Rate function

In mathematics — specifically, in large deviations theory — a rate function is a function used to quantify the probabilities 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.

Rate function An extended real-valued function I : X → [0, +∞] defined on a Hausdorff topological space X 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 X. If, furthermore, they are compact, then I is said to be a good rate function.

A family of probability measures (μ_δ)_δ > 0 on X is said to satisfy the large deviation principle with rate function I : X → [0, +∞) (and rate 1 ⁄ δ) if, for every closed set F ⊆ X and every open set G ⊆ X,

If the upper bound (U) holds only for compact (instead of closed) sets F, then (μ_δ)_δ>0 is said to satisfy the weak large deviations principle (with rate 1 ⁄ δ and weak rate function I).

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 (μ_δ)_δ > 0 is said to converge weakly to μ if, for every closed set F ⊆ X and every open set G ⊆ X,

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