Varadhan's integral lemma
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Varadhan's integral lemma is the large-deviation generalisation of Laplace's method for asymptotics of integrals. Laplace's method says that integral of e^(n F(x)) dx is dominated, to exponential order, by the maximum of F; Varadhan's lemma replaces the deterministic measure dx by a family of probability measures satisfying an LDP, so that the exponential growth of a weighted exponential integral is governed by a clean variational competition between the integrand and the rate function. It is the tool that turns an LDP into asymptotics of expectations.
Suppose mu_n satisfies an LDP with good rate function I, and F is a continuous function (satisfying a tail/moment condition so the integral does not blow up — typically that lim sup (1/n) log E[e^(n gamma F)] < infinity for some gamma > 1). Then lim (1/n) log E[e^(n F(X_n))] = sup over x of (F(x) - I(x)). In words, the exponential rate of the exponentially weighted average is the best trade-off between the reward F(x) and the cost I(x) of reaching x. This is also called the Laplace principle, and it is logically equivalent to the LDP itself for good rate functions: knowing the variational limit for all continuous F is the same information as the LDP.
Varadhan's lemma is the workhorse for computing free energies, partition function asymptotics, and tilted-measure normalisations, because all of these are exactly exponential integrals against an LDP. It is also the analytic engine behind the exponential change of measure: the normalising constant of a tilted law e^(n F) dmu_n is precisely such an integral, so its exponential rate is sup(F - I). The continuity and moment condition on F cannot be dropped — a discontinuous or super-exponentially growing F can make the integral dominated by exponentially rare but unboundedly rewarded outcomes, breaking the clean sup(F - I) formula.
If S_n is the mean of n iid N(0,1) (so I(x) = x^2/2) and F(x) = a x, then E[e^(n a S_n)] = e^(n a^2/2), and indeed (1/n) log of this is a^2/2 = sup_x (a x - x^2/2), the Legendre value attained at x = a. Varadhan recovers the moment generating function as a sup(F - I) optimisation.
Varadhan: exponential integrals reduce to the variational sup(F - I).
The moment/tail condition on F (a uniform-integrability-type control of the exponential tails, e.g. the gamma > 1 condition) is essential. If F grows too fast or is unbounded above where I grows too slowly, the integral is dominated by rare events and the limit exceeds sup(F - I) or diverges.