Fixed exponential perturbation of a Gaussian empirical mean ($K(x)={\inf }_{b\ge 0}\{\lambda b+(x-b-\mu {)}^2/(2{\sigma }^2)\}$)

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 $L$, $(B+{\sum }_{i=1}^L{A}_i)/L$ has the good rate functionThe product large-deviation principle gives the joint cost $\lambda b+(z-\mu {)}^2/(2{\sigma }^2)$ for $b\ge 0$, 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 $L$.