Poisson moderate deviation principle ($\frac{S_L-L\lambda }{L^{(1+\beta )/2}},a_L=L^{\beta },I(x)=x^2/(2\lambda )$)

For a Poisson random variable $S_L$ of mean $L\lambda >0$ and $0<\beta <1$, the scaled cumulant-generating function is $\lambda L^{1-\beta }(e^{\theta L^{(\beta -1)/2}}-1-\theta L^{(\beta -1)/2})$. Its Taylor expansion tends to $\lambda {\theta }^2/2$ for every real $\theta$. The Gärtner–Ellis theorem applies because the limit is finite and differentiable everywhere. Its Legendre-Fenchel transform is the displayed quadratic good rate function.