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.
Rate function An extended real-valued function
I:X\to[0,+infty]
X
+infty
\{x\inX\midI(x)\leqc\}forc\geq0
are closed in
X
I
A family of probability measures
(\mu\delta)\delta
X
I:X\to[0,+infty)
1/\delta
F\subseteqX
G\subseteqX
\limsup\delta\deltalog\mu\delta(F)\leq-infxI(x), (U)
\liminf\delta\deltalog\mu\delta(G)\geq-infxI(x). (L)
If the upper bound (U) holds only for compact (instead of closed) sets
F
(\mu\delta)\delta
1/\delta
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
(\mu\delta)\delta
\mu
F\subseteqX
G\subseteqX
\limsup\delta\mu\delta(F)\leq\mu(F),
\liminf\delta\mu\delta(G)\geq\mu(G).
There is some variation in the nomenclature used in the literature: for example, den Hollander (2000) uses simply "rate function" where this article - following Dembo & Zeitouni (1998) - uses "good rate function", and "weak rate function". Rassoul-Agha & Seppäläinen (2015) 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 (U) is supposed to hold for closed or compact sets tells one whether the large deviation principle in use is strong or weak.
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 (μδ)δ>0 on X satisfying the large deviation principle for two rate functions I and J, it follows that I(x) = J(x) for all x ∈ X.
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 (μδ)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.
Naïvely, one might try to replace the two inequalities (U) and (L) by the single requirement that, for all Borel sets S ⊆ X,
\lim\delta\deltalog\mu\delta(S)=-infxI(x). (E)
The equality (E) is far too restrictive, since many interesting examples satisfy (U) and (L) but not (E). For example, the measure μδ might be non-atomic for all δ, so the equality (E) could hold for S = only if I were identically +∞, which is not permitted in the definition. However, the inequalities (U) and (L) do imply the equality (E) for so-called I-continuous sets S ⊆ X, those for which
I(\stackrel{\circ}{S})=I(\bar{S}),
where
\stackrel{\circ}{S}
\bar{S}
S\subseteq\bar{\stackrel{\circ}{S}}
are I-continuous; all open sets, for example, satisfy this containment.
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 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 (Zi)i∈
N
\PsiZ(t)=log\operatornameEetZ.