Dawson–Gärtner Theorem
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Dawson–Gärtner theorem is a result 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 ...
. Heuristically speaking, the Dawson–Gärtner theorem allows one to transport a
large deviation principle 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 qu ...
on a “smaller”
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
to a “larger” one.


Statement of the theorem

Let (''Y''''j'')''j''∈''J'' be a
projective system In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ca ...
of Hausdorff topological spaces with maps ''p''''ij'' : ''Y''''j'' → ''Y''''i''. Let ''X'' be the projective limit (also known as the inverse limit) of the system (''Y''''j'', ''p''''ij'')''i'',''j''∈''J'', i.e. :X = \varprojlim_ Y_ = \left\. 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 σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
s on ''X''. Assume that, for each ''j'' ∈ ''J'', the push-forward measures (''p''''j''∗''μ''''ε'')''ε''>0 on ''Y''''j'' satisfy the large deviation principle with good rate function ''I''''j'' : ''Y''''j'' → R ∪ . Then the family (''μ''''ε'')''ε''>0 satisfies the large deviation principle on ''X'' with good rate function ''I'' : ''X'' → R ∪  given by :I(x) = \sup_ I_(p_(x)).


References

* (See theorem 4.6.1) {{DEFAULTSORT:Dawson-Gartner theorem Asymptotic analysis Large deviations theory Theorems in probability theory