= Laplace principle
{disambiguate=large deviations theory}
{c}
{title2=$\lim_{\theta\to\infty}\theta^{-1}\log\int_Ae^{-\theta\phi(x)}dx=-\operatorname*{ess\,inf}_A\phi$}
{wiki}
= Laplace principle
{c}
{synonym}
The exponential integral of a fixed measurable function is controlled, on its logarithmic scale, by the function's <essential infimum>. Suppose $0<\int_Ae^{-\phi}dx<\infty$ and $\alpha=\operatorname*{ess\,inf}_A\phi$ is finite. For $\theta\geq1$, the integral is at most $e^{-(\theta-1)\alpha}\int_Ae^{-\phi}dx$. For every $\varepsilon>0$, the set $A_\varepsilon=\{x\in A:\phi(x)<\alpha+\varepsilon\}$ has positive finite <Lebesgue measure>, and the integral is at least $|A_\varepsilon|e^{-\theta(\alpha+\varepsilon)}$. Taking logarithms, dividing by $\theta$, and then letting $\varepsilon\downarrow0$ 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.
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