In mathematics, 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 probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...

s are "the same" from the point of view of large deviations theory.

Definition

Let

(M,d)

be a

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

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) 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. It is required to have several properties which assist in the formulation of the large devia ...

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)