= Slower-speed degeneration of a large deviation principle
{title2=$a_n/b_n\to\infty:\quad I_b(x)=0\text{ at }\mu,\ I_b(x)=\infty\text{ elsewhere}$}
Suppose a <large deviation principle> at speed $a_n$ has a <good rate function> with unique zero mu. At any slower speed $b_n\to\infty$ with $a_n/b_n\to\infty$, its rate is zero at mu and infinity elsewhere. On a <closed set> excluding mu, the positive original rate gives decay faster than every exponential at speed $b_n$. Neighborhoods of mu have probability tending to one, giving the slower-speed open-set lower bound. The new rate is good because its finite <sublevel set> is the singleton mu.
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