= Large-deviation speed
{title2=$a_N$}
A <large deviation principle> with speed $a_N\to\infty$ and <rate function> $I$ means that every open set $G$ and closed set $F$ satisfy
$$
-\inf_G I\leq\liminf_N a_N^{-1}\log\mathbb P(X_N\in G),\qquad
\limsup_N a_N^{-1}\log\mathbb P(X_N\in F)\leq-\inf_F I.
$$
Multiplying the <large-deviation speed> by $c>0$ divides the <rate function> by $c$. The speed must match the rare-event scale: a fixed <exponential random variable> divided by $N^p$ has speed $N^p$, whereas a sum of $N$ <independent random variables> with a <normal distribution>, divided by $N^p$, has speed $N^{2p-1}$ when $p>1$.
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