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In mathematics — specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new
large deviation principle 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 deviat ...
from an old one by "tilting", i.e.
integration Integration may refer to: Biology * Multisensory integration * Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technolo ...
against an exponential
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional s ...
. It can be seen as an alternative formulation of
Varadhan's lemma In mathematics, Varadhan's lemma is a result from large deviations theory named after S. R. Srinivasa Varadhan. The result gives information on the asymptotic distribution of a statistic ''φ''(''Z'ε'') of a family of random variables ''Z'� ...
.


Statement of the theorem

Let ''X'' be a Polish space (i.e., a separable, completely metrizable
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
), and let (''μ''''ε'')''ε''>0 be a family of probability measures on ''X'' that satisfies the large deviation principle with
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 devia ...
''I'' : ''X'' →  , +∞ Let ''F'' : ''X'' → R be a continuous function that is
bounded Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...
from above. For each Borel set ''S'' ⊆ ''X'', let :J_ (S) = \int_ e^ \, \mathrm \mu_ (x) and define a new family of probability measures (''ν''''ε'')''ε''>0 on ''X'' by :\nu_ (S) = \frac. Then (''ν''''ε'')''ε''>0 satisfies the large deviation principle on ''X'' with rate function ''I''''F'' : ''X'' →  , +∞given by :I^ (x) = \sup_ \big F(y) - I(y) \big- \big F(x) - I(x) \big


References

* {{MathSciNet, id=1739680 Asymptotic analysis Mathematical principles Probability theorems Large deviations theory