= Fixed exponential perturbation of a Gaussian empirical mean
{title2=$K(x)=\inf_{b\geq0}\{\lambda b+(x-b-\mu)^2/(2\sigma^2)\}$}
Let $A_i$ be <independent random variables> with <normal distribution> $N(\mu,\sigma^2)$, $\sigma^2>0$, and let $B$ be an independent <exponential random variable> of rate $\lambda>0$. At speed $L$, $(B+\sum_{i=1}^LA_i)/L$ has the <good rate function>
$$
K(x)=\begin{cases}(x-\mu)^2/(2\sigma^2),&x\leq\mu+\lambda\sigma^2,\\ \lambda(x-\mu)-\lambda^2\sigma^2/2,&x\geq\mu+\lambda\sigma^2.\end{cases}
$$
The <product large-deviation principle> gives the joint cost $\lambda b+(z-\mu)^2/(2\sigma^2)$ for $b\geq0$, and the <contraction principle for large deviations> gives its <infimal convolution> under $b+z=x$. The minimum occurs at $b=(x-\mu-\lambda\sigma^2)^+$. The fixed exponential perturbation changes the far upper tail even though it vanishes in probability after division by $L$; a fixed <Gaussian random variable> divided by $L$ instead has <exponential equivalence> with zero at speed $L$.
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