In mathematics — specifically, in

large deviations theory In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to Laplace, the formalization started with insura ...

— the contraction principle is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

that states how a large deviation principle on one space "pushes forward" (via the

pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...

of a probability measure) to a large deviation principle on another space ''via'' a continuous function.

Statement

Let ''X'' and ''Y'' be Hausdorff

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 ...

s and let (''μ''_''ε'')_''ε''>0 be a family of

probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...

s 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 ''T'' : ''X'' → ''Y'' be a continuous function, and let ''ν''_''ε'' = ''T''_∗(''μ''_''ε'') be the push-forward measure of ''μ''_''ε'' by ''T'', i.e., for each

measurable set In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simi ...

/event ''E'' ⊆ ''Y'', ''ν''_''ε''(''E'') = ''μ''_''ε''(''T''⁻¹(''E'')). Let :

J(y) := \inf \,

with the convention that the

infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...

of ''I'' over the

empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...

∅ is +∞. Then: * ''J'' : ''Y'' → , +∞is a rate function on ''Y'', * ''J'' is a good rate function on ''Y'' if ''I'' is a good rate function on ''X'', and * (''ν''_''ε'')_''ε''>0 satisfies the large deviation principle on ''Y'' with rate function ''J''.

References

* (See chapter 4.2.1) * {{cite book , last = den Hollander , first = Frank , title = Large deviations , series =

Fields Institute The Fields Institute for Research in Mathematical Sciences, commonly known simply as the Fields Institute, is an international centre for scientific research in mathematical sciences. It is an independent non-profit with strong ties to 20 Onta ...

Monographs 14 , publisher =

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...

, location = Providence, RI , year = 2000 , pages = x+143 , isbn = 0-8218-1989-5 , mr = 1739680 Asymptotic analysis Large deviations theory Mathematical principles Probability theorems