In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions — 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. Some basic ideas of the theory can be tracked back to Laplace and Cramér, although a clear unified formal definition was introduced in 1966 by Varadhan. Large deviations theory formalizes the heuristic — a rate function is a function used to quantify the probabilities Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of of rare events. It is required to have several "nice" properties which assist in the formulation of the large deviation principle. In some sense, the large deviation principle is an analogue of weak convergence of probability measures In mathematics, more specifically measure theory, there are various notions of the convergence of measures. Broadly speaking, there are two kinds of convergence, strong convergence and weak convergence, but one which takes account of how well the rare events behave.

Definitions

An extended real-valued In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . These new elements are not real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended function I : X → [0, +∞] defined on a Hausdorff In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently used and discussed. It implies the topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology X is said to be a rate function if it is not identically +∞ and is lower semi-continuous In mathematical analysis, semi-continuity is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (lower) semi-continuous at a point x0 if, roughly speaking, the function values for arguments near x0 are either close to f(x0) or less than (greater than) f(x0), i.e. all the sub-level sets

are closed In topology and related branches of mathematics, a closed set is a set whose complement is open. In a metric space, such as in real analysis, a closed set can be defined as a set which contains all its limit points in X. If, furthermore, they are compact In mathematics, more specifically general topology and metric topology, a compact space is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space. Thus a closed and bounded subset of a Euclidean space, then I is said to be a good rate function.

A family of probability measures In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area and volume. A particularly important example is the Lebesgue measure on a Euclidean (μ_δ)_δ>0 on X is said to satisfy the large deviation principle with rate function I : X → [0, +∞] (and rate 1 ⁄ δ) if, for every closed set F ⊆ X and every open set In mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space. In particular, although one cannot obtain concrete values for the distance between two points in a topological space, one may G ⊆ X,

If the upper bound (U) holds only for compact (instead of closed) sets F, then (μ_δ)_δ>0 is said to satisfy the weak large deviation principle (with rate 1 ⁄ δ and weak rate function I).

Remarks

The role of the open and closed sets in the large deviation principle is similar to their role in the weak convergence of probability measures: recall that (μ_δ)_δ>0 is said to converge weakly to μ if, for every closed set F ⊆ X and every open set In mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space. In particular, although one cannot obtain concrete values for the distance between two points in a topological space, one may G ⊆ X,

It should be noted that there is some variation in the nomenclature used in the literature: for example, den Hollander (2000) uses simply "rate function" where this article — following Dembo & Zeitouni (1998) — uses "good rate function", and "weak rate function" where this article uses "rate function". Fortunately, regardless of the nomenclature used for rate functions, examination of whether the upper bound inequality (U) is supposed to hold for closed or compact sets tells one whether the large deviation principle in use is strong or weak.

Properties

Uniqueness

A natural question to ask, given the somewhat abstract setting of the general framework above, is whether the rate function is unique. This turns out to be the case: given a sequence of probability measures (μ_δ)_δ>0 on X satisfying the large deviation principle for two rate functions I and J, it follows that I(x) = J(x) for all x ∈ X.

Exponential tightness

It is possible to convert a weak large deviation principle into a strong one if the measures converge sufficiently quickly. If the upper bound holds for compact sets F and the sequence of measures (μ_δ)_δ>0 is exponentially tight Let be a topological space, and let Σ be a σ-algebra on X that contains the topology T. (Thus, every open subset of X is a measurable set and Σ is at least as fine as the Borel σ-algebra on X.) Let M be a collection of measures defined on Σ. The collection M is called tight (or sometimes uniformly tight) if, for any ε > 0, there is a, then the upper bound also holds for closed sets F. In other words, exponential tightness enables one to convert a weak large deviation principle into a strong one.

Continuity

Naïvely, one might try to replace the two inequalities (U) and (L) by the single requirement that, for all Borel sets S ⊆ X,

Unfortunately, the equality (E) is far too restrictive, since many interesting examples satisfy (U) and (L) but not (E). For example, the measure μ_δ might be non-atomic In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no "smaller" set of positive measure. A measure which has no atoms is called non-atomic for all δ, so the equality (E) could hold for S = {x} only if I were identically +∞, which is not permitted in the definition. However, the inequalities (U) and (L) do imply the equality (E) for so-called I-continuous sets S ⊆ X, those for which

where and denote the interior In mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S". A point that is in the interior of S is an interior point of S and closure of S in X respectively. In many examples, many sets/events of interest are I-continuous. For example, if I is a continuous function In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous. An intuitive though imprecise idea of continuity is given by the common, then all sets S such that

are I-continuous; all open sets, for example, satisfy this containment.

Transformation of large deviation principles

Given a large deviation principle on one space, it is often of interest to be able to construct a large deviation principle on another space. There are several results in this area:

• the contraction principle tells one how a large deviation principle on one space "pushes forward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different, but closely related things" to a large deviation principle on another space via a continuous function In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous. An intuitive though imprecise idea of continuity is given by the common;

• the Dawson-Gärtner theorem In mathematics, the Dawson–Gärtner theorem is a result in large deviations theory. Heuristically speaking, the Dawson–Gärtner theorem allows one to transport a large deviation principle on a “smaller” topological space to a “larger” one tells one how a sequence of large deviation principles on a sequence of spaces passes to the projective limit.

• the tilted large deviation principle gives a large deviation principle for integrals of exponential functionals.

• exponentially equivalent measures have the same large deviation principles.

References

• Dembo, Amir; Zeitouni, Ofer (1998). Large deviations techniques and applications. Applications of Mathematics (New York) 38 (Second edition ed.). New York: Springer-Verlag. xvi+396. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 0-387-98406-2. MR Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations) of many articles in mathematics, statistics and theoretical computer science 1619036
• den Hollander, Frank (2000). Large deviations. Fields Institute The Fields Institute is an international centre for research in mathematical sciences at the University of Toronto. The institute is named for University of Toronto mathematician John Charles Fields, founder of the Fields Medal. It was established in 1992, and was briefly based at the University of Waterloo before relocating to Toronto in 1995 Monographs 14. Providence, RI: American Mathematical Society The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians. p. x+143. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 0-8218-1989-5. MR Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations) of many articles in mathematics, statistics and theoretical computer science 1739680

Categories: Asymptotic analysis | Probability theory Categories: Actuarial science | Probability | Measure theory | Decision theory | Mathematics of computing

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• Rate Function
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