Exponentially equivalent measures

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

Definition

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

a one-parameter family of probability spaces ((Ω, Σ_ε, P_ε))_ε>0,
two families of M-valued random variables (Y_ε)_ε>0 and (Z_ε)_ε>0,

such that

for each ε > 0, the P_ε-law (i.e. the push-forward measure) of Y_ε is μ_ε, and the P_ε-law of Z_ε is ν_ε,
for each δ > 0, “Y_ε and Z_ε are further than δ apart” is a Σ_ε-measurable event, i.e.

$\big\{ \omega \in \Omega \big| d(Y_{\varepsilon}(\omega), Z_{\varepsilon}(\omega)) > \delta \big\} \in \Sigma_{\varepsilon},$

for each δ > 0,

$\limsup_{\varepsilon \downarrow 0} \varepsilon \log \mathbf{P}_{\varepsilon} \big[ d(Y_{\varepsilon}, Z_{\varepsilon}) > \delta \big] = - \infty.$

The two families of random variables (Y_ε)_ε>0 and (Z_ε)_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 (μ_ε)_ε>0 with good rate function I, and (μ_ε)_ε>0 and (ν_ε)_ε>0 are exponentially equivalent, then the same large deviations principle holds for (ν_ε)_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 edition ed.). New York: Springer-Verlag. pp. xvi+396. ISBN 0-387-98406-2. MR 1619036. (See section 4.2.2)