= Large-deviation rate of a minimum of independent copies
{title2=$J(m)=I(m)+(k-1)\inf_{x\ge m}I(x)$}
For fixed $k$ <independent random variables> whose families have the same <large deviation principle> with <good rate function> $I$, the <product large-deviation principle> gives the sum of coordinate costs. Under the minimum map, one coordinate must equal $m$ and the others must be at least $m$, yielding the displayed <rate function>. If $I$ is <convex> with unique zero $\mu$, then $J(m)=I(m)$ for $m\le\mu$ and $J(m)=kI(m)$ for $m\ge\mu$.
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