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 ...
— 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 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. Such functions are used to formulate large deviation principles. A large deviation principle qu ...
from an old one by
exponential tilting
Exponential Tilting (ET), Exponential Twisting, or Exponential Change of Measure (ECM) is a distribution shifting technique used in many parts of mathematics.
The different exponential tiltings of a random variable X is known as the natural expone ...
, 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 technology, ...
against an
exponential
Exponential may refer to any of several mathematical topics related to exponentiation, including:
* Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
* Ex ...
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.
Statement of the theorem
Let ''X'' be a
Polish space
In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that h ...
(i.e., a
separable,
completely metrizable In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (''X'', ''T'') for which there exists at least one metric ''d'' on ''X'' such that (''X'', ''d'') is a complete metric space and ''d'' in ...
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 ...
), 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. Such functions are used to formulate large deviation principles. A large deviation principle qu ...
''I'' : ''X'' →
, +∞ Let ''F'' : ''X'' → R be a
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
that is
bounded from above. For each Borel set ''S'' ⊆ ''X'', let
:
and define a new family of probability measures (''ν''
''ε'')
''ε''>0 on ''X'' by
:
Then (''ν''
''ε'')
''ε''>0 satisfies the large deviation principle on ''X'' with rate function ''I''
''F'' : ''X'' →
, +∞given by
:
References
* {{MathSciNet, id=1739680
Asymptotic analysis
Mathematical principles
Theorems in probability theory
Large deviations theory