Bryc's converse lemma

Bryc's converse lemma is the partial inverse of Varadhan's integral lemma: where Varadhan turns an LDP into Laplace-type asymptotics of exponential integrals, Bryc goes the other way and produces an LDP out of the existence of those asymptotics. It answers a practical question — if you can compute the limiting logarithm of all suitably weighted exponential expectations, do you automatically have a large deviation principle? — and the answer, under an exponential tightness condition, is yes.

The statement: suppose (mu_n) is exponentially tight and that for every bounded continuous function F the limit Lambda(F) = lim (1/n) log E[e^(n F(X_n))] exists. Then (mu_n) satisfies the LDP with the good rate function recovered by the inverse Legendre-type transform I(x) = sup over bounded continuous F of (F(x) - Lambda(F)). In other words, the family of Laplace functionals Lambda(F), one number per test function, contains exactly the information of the LDP, and the rate function is read off as the conjugate of the Laplace functional over the test functions. The exponential tightness hypothesis is what supplies the compactness that upgrades pointwise functional limits into a full principle with a good rate function.

Bryc's lemma matters because in practice the limiting Laplace functional is often the computable object — it is the limiting free energy in statistical mechanics, the limiting log-Laplace transform in queueing and risk theory, the scaled cumulant in Gartner-Ellis. Establishing that these limits exist for all bounded continuous F, plus exponential tightness, is frequently easier than verifying the open-set lower bound directly, so Bryc supplies the lower bound for free. The bounded-continuous restriction on F (versus Varadhan's continuous F with moment conditions) is what keeps the converse clean; the two lemmas together say the LDP and the Laplace principle are two faces of the same fact.