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 I : X → [ 0 , + ∞ ] {\displaystyle I:X\to [0,+\infty ]} defined on a Hausdorff topological space X {\displaystyle X} is said to be a rate function if it is not identically + ∞ {\displaystyle +\infty } and is lower semi-continuous i.e. all the sub-level sets
{ x ∈ X ∣ I ( x ) ≤ c } for c ≥ 0 {\displaystyle \{x\in X\mid I(x)\leq c\}{\mbox{ for }}c\geq 0}
are closed in X {\displaystyle X} . If, furthermore, they are compact, then I {\displaystyle I} is said to be a good rate function. A family of probability measures ( μ δ ) δ > 0 {\displaystyle (\mu _{\delta })_{\delta >0}} on X {\displaystyle X} is said to satisfy the large deviation principle with rate function I : X → [ 0 , + ∞ ) {\displaystyle I:X\to [0,+\infty )} (and rate 1 / δ {\displaystyle 1/\delta } ) if, for every closed set F ⊆ X {\displaystyle F\subseteq X} and every open set G ⊆ X {\displaystyle G\subseteq X} ,
lim sup δ ↓ 0 δ log μ δ ( F ) ≤ − inf x ∈ F I ( x ) , (U) {\displaystyle \limsup _{\delta \downarrow 0}\delta \log \mu _{\delta }(F)\leq -\inf _{x\in F}I(x),\quad {\mbox{(U)}}}
lim inf δ ↓ 0 δ log μ δ ( G ) ≥ − inf x ∈ G I ( x ) . (L) {\displaystyle \liminf _{\delta \downarrow 0}\delta \log \mu _{\delta }(G)\geq -\inf _{x\in G}I(x).\quad {\mbox{(L)}}}
If the upper bound (U) holds only for compact (instead of closed) sets F {\displaystyle F} , then ( μ δ ) δ > 0 {\displaystyle (\mu _{\delta })_{\delta >0}} is said to satisfy the weak large deviations principle (with rate 1 / δ {\displaystyle 1/\delta } and weak rate function I {\displaystyle I} ).
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 ( μ δ ) δ > 0 {\displaystyle (\mu _{\delta })_{\delta >0}} is said to converge weakly to μ {\displaystyle \mu } if, for every closed set F ⊆ X {\displaystyle F\subseteq X} and every open set G ⊆ X {\displaystyle G\subseteq X} ,
lim sup δ ↓ 0 μ δ ( F ) ≤ μ ( F ) , {\displaystyle \limsup _{\delta \downarrow 0}\mu _{\delta }(F)\leq \mu (F),}
lim inf δ ↓ 0 μ δ ( G ) ≥ μ ( G ) . {\displaystyle \liminf _{\delta \downarrow 0}\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.
Properties
… excerpt ends here. Continue reading the full article.
