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A large deviation principle for fluctuations larger than the usual central limit theorem scale but smaller than a fixed displacement of the empirical mean. For the displayed sum scaling, the natural large-deviation speed is $b_n^2$. Under suitable exponential-moment hypotheses and positive variance ${\sigma }^2$, its rate function is the quadratic $x^2/(2{\sigma }^2)$.

A large deviation principle for random elements $X^L$ in a Hausdorff space $E$, at large-deviation speed $a_L\to \infty$, has a lower semicontinuous rate function $I:E\to [0,\infty ]$. For every open set $G$ and every closed set $F$,Equivalently, for every Borel set $A$, the lower and upper bounds use its interior and closure, respectively. The conventions are $\inf \varnothing =+\infty$ and $\log 0=-\infty$. A good rate function has compact sublevel sets $\{I\le r\}$ for every finite $r$. Here take $a_L=L$; the different speeds in the moderate deviation principle are stated explicitly when used.