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 "nice" properties which assist in the formulation of the large deviation principle.
Formally, a extended real-valued function ''I'' : ''X'' → [0, +∞] defined on a Hausdorff topological space ''X'' is said to be a 'rate function' if it is lower semi-continuous, i.e. all the sub-level sets
:$\{\; x\; in\; X\; |\; I(x)\; leq\; c\; \}\; mbox\{\; for\; \}\; c\; geq\; 0$
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, +∞] if
:$limsup\_\{delta\; downarrow\; 0\}\; delta\; log\; mu\_\{delta\}\; (F)\; leq\; -\; inf\_\{x\; in\; F\}\; I(x)$
for every closed set ''F'' ⊆ ''X'' and
:$liminf\_\{delta\; downarrow\; 0\}\; delta\; log\; mu\_\{delta\}\; (G)\; geq\; -\; inf\_\{x\; in\; G\}\; I(x)$
for every open set ''G'' ⊆ ''X''. If the upper bound hold only for compact (not just closed) ''F'', then (''μ''_''δ'')_''δ''>0 is said to satisfy the 'weak large deviation principle' (with rate function ''I'').

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References

References

★ Large deviations techniques and applications, , Amir, Dembo, Springer-Verlag, 1998,

★ Some large deviation results for diffusion processes, , Jochen, Voß, PhD thesis, 2004, (Chapter 2)