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 ...

, Varadhan's lemma is a result from the

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 ...

named after

S. R. Srinivasa Varadhan Sathamangalam Ranga Srinivasa Varadhan, (born 2 January 1940) is an Indian American mathematician. He is known for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations. He is re ...

. The result gives information on the

asymptotic distribution In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the limiting distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing appr ...

of a statistic ''φ''(''Z''_''ε'') of a family of

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

s ''Z''_''ε'' as ''ε'' becomes small in terms of a

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 ...

for the variables.

Statement of the lemma

Let ''X'' be a

regular topological space In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' have non-overlapping open neighborhoods. Thus ''p'' and ''C'' can b ...

; let (''Z''_''ε'')_''ε''>0 be a family of random variables taking values in ''X''; let ''μ''_''ε'' be the law (

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 ...

) of ''Z''_''ε''. Suppose that (''μ''_''ε'')_''ε''>0 satisfies the

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 ...

with good rate function ''I'' : ''X'' → , +∞ Let ''ϕ'' : ''X'' → R be any

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 ...

. Suppose that at least one of the following two conditions holds true: either the tail condition :

big) = -\infty,

where 1(''E'') denotes the

indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...

of the event ''E''; or, for some ''γ'' > 1, the moment condition :

big) < \infty.

Then :

= \sup_ \big( \phi(x) - I(x) \big).

References

* {{cite book , last= Dembo , first = Amir , author2=Zeitouni, Ofer , title = Large deviations techniques and applications , series = Applications of Mathematics (New York) 38 , edition = Second , publisher = Springer-Verlag , location = New York , year = 1998 , pages = xvi+396 , isbn = 0-387-98406-2 , mr=1619036 (See theorem 4.3.1) Asymptotic analysis Lemmas Theorems in probability theory Theorems in statistics Large deviations theory

Statement of the lemma

See also

References