= Poisson moderate deviation principle
{c}
{title2=$\frac{S_L-L\lambda}{L^{(1+\beta)/2}},\quad a_L=L^\beta,\quad I(x)=x^2/(2\lambda)$}
= Poisson moderate deviation
{c}
{synonym}
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>.
Back to article page
