Laplace principle (large deviations theory)

The exponential integral of a fixed measurable function is controlled, on its logarithmic scale, by the function's essential infimum. Suppose $0<{\int }_A{e}^{-\varphi }dx<\infty$ and $\alpha ={\mathrm{essinf}}_A\varphi$ is finite. For $\theta \ge 1$, the integral is at most $e^{-(\theta -1)\alpha }{\int }_A{e}^{-\varphi }dx$. For every $\epsilon >0$, the set $A_{\epsilon }=\{x\in A:\varphi (x)<\alpha +\epsilon \}$ has positive finite Lebesgue measure, and the integral is at least $|A_{\epsilon }|e^{-\theta (\alpha +\epsilon )}$. Taking logarithms, dividing by $\theta$, and then letting $\epsilon \downarrow 0$ proves the displayed limit. The argument needs no isolated or nondegenerate minimizer. The Laplace principle for probability measures treats families whose distributions also change with the scaling parameter.