= Moderate deviation principle
{title2=$\frac{S_n-n\mu}{\sqrt n\,b_n},\quad b_n\to\infty,\quad b_n/\sqrt n\to0$}
= Moderate deviation
{synonym}
Let $X_i$ be <independent and identically distributed> with <mean> $\mu$, <variance> $\sigma^2\in(0,\infty)$, and a <moment-generating function> finite on an open <interval> about zero. Choose $b_n>0$ with $b_n\to\infty$ and $b_n/\sqrt n\to0$. The <moderate deviation principle> for
$$
Z_n=\frac{\sum_{i=1}^nX_i-n\mu}{\sqrt n\,b_n}
$$
is a <large deviation principle> of speed $b_n^2$ and <good rate function> $I(x)=x^2/(2\sigma^2)$.
To see the <Gaussian distribution>[Gaussian] quadratic rate, center one summand and expand its logarithmic <moment-generating function> at zero. For every fixed real $\theta$,
$$
\frac1{b_n^2}\log\mathbb E e^{b_n^2\theta Z_n}
=\frac n{b_n^2}\log\mathbb E\exp\left[\frac{\theta b_n}{\sqrt n}(X_1-\mu)\right]
\longrightarrow\frac{\sigma^2\theta^2}{2}.
$$
The limiting <function> is finite and differentiable everywhere, so the <Gärtner–Ellis theorem> applies; its <Legendre-Fenchel transform> is the displayed rate. This proof states an actual sufficient exponential-moment hypothesis rather than leaving the assumptions implicit.
For $b_n=n^{1/4}$, a fixed positive $Z_n=x$ corresponds to a sample-mean displacement of order $n^{-1/4}$, between the ordinary $n^{-1/2}$ central-limit scale and a fixed order-one displacement. Its tail has logarithmic <Probability> of order $-\sqrt n\,x^2/(2\sigma^2)$. The scaling is intermediate, but the rate is still quadratic under these hypotheses.
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