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 ...

, exponential equivalence of measures is how two sequences or families 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 are "the same" from the point of view of

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 ...

Definition

Let

(M,d)

be a

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

and consider two one-

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

families of probability measures on

M

, say

(\mu_\varepsilon)_

and

(\nu_\varepsilon)_

. These two families are said to be exponentially equivalent if there exist * a one-parameter family of probability spaces

(\Omega,\Sigma_\varepsilon,P_\varepsilon)_

, * two families of

M

-valued random variables

(Y_\varepsilon)_

and

(Z_\varepsilon)_

, such that * for each

\varepsilon >0

, the

P_\varepsilon

-law (i.e. the

push-forward measure In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function. Definition Given mea ...

) of

Y_\varepsilon

\mu_\varepsilon

, and the

P_\varepsilon

-law of

Z_\varepsilon

\nu_\varepsilon

, * for each

\delta >0

, "

Y_\varepsilon

and

Z_\varepsilon

are further than

\delta

apart" is a

\Sigma_\varepsilon

- measurable event, i.e. ::

\big\ \in \Sigma_,

* for each

\delta >0

, ::

\limsup_\, \varepsilon \log P_\varepsilon \big( d(Y_\varepsilon, Z_\varepsilon) > \delta \big) = - \infty.

The two families of random variables

(Y_\varepsilon)_

and

(Z_\varepsilon)_

are also said to be exponentially equivalent.

Properties

The main use of exponential equivalence is that as far as large deviations principles are concerned, exponentially equivalent families of measures are indistinguishable. More precisely, if a large deviations principle holds for

(\mu_\varepsilon)_

with good

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

, and

(\mu_\varepsilon)_

and

(\nu_\varepsilon)_

are exponentially equivalent, then the same large deviations principle holds for

(\nu_\varepsilon)_

with the same good rate function

I

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 section 4.2.2) Asymptotic analysis Probability theory Equivalence (mathematics)