# Exponentially equivalent measures

In mathematics, exponential equivalence of measures is how two sequences or families of probability measures are “the same” from the point of view of large deviations theory.

## Definition

Let (Md) be a metric space and consider two one-parameter families of probability measures on M, say (με)ε>0 and (νε)ε>0. These two families are said to be exponentially equivalent if there exist

• a one-parameter family of probability spaces ((Ω, ΣεPε))ε>0,
• two families of M-valued random variables (Yε)ε>0 and (Zε)ε>0,

such that

• for each ε > 0, the Pε-law (i.e. the push-forward measure) of Yε is με, and the Pε-law of Zε is νε,
• for each δ > 0, “Yε and Zε are further than δ apart” is a Σε-measurable event, i.e.
${\displaystyle {\big \{}\omega \in \Omega {\big |}d(Y_{\varepsilon }(\omega ),Z_{\varepsilon }(\omega ))>\delta {\big \}}\in \Sigma _{\varepsilon },}$
• for each δ > 0,
${\displaystyle \limsup _{\varepsilon \downarrow 0}\varepsilon \log \mathbf {P} _{\varepsilon }{\big [}d(Y_{\varepsilon },Z_{\varepsilon })>\delta {\big ]}=-\infty .}$

The two families of random variables (Yε)ε>0 and (Zε)ε>0 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 (με)ε>0 with good rate function I, and (με)ε>0 and (νε)ε>0 are exponentially equivalent, then the same large deviations principle holds for (νε)ε>0 with the same good rate function I.

## References

• Dembo, Amir; Zeitouni, Ofer (1998). Large deviations techniques and applications. Applications of Mathematics (New York) 38 (Second ed.). New York: Springer-Verlag. pp. xvi+396. ISBN 0-387-98406-2. MR 1619036. (See section 4.2.2)