About the Cram\'er Large Deviation Property for Bell Polynomials

Abstract

If $\boldsymbol{w} = (w_1,w_2,\dots)$ is a sequence in $\mathbb{N}=\{1,2,\dots\}$, the partial Bell polynomials based on $\boldsymbol{w}$ are $B_{n,k}$ for $k \in \mathbb{N}$ and $n\in\{k,k+1,\dots\}$. Let $F(z) = \sum_{n=1}^{\infty} (w_n/n!)z^n$ be the exponential generating function for $\boldsymbol{w}$, and assume the radius of convegence is positive $R>0$. Then $F(z)^k = \sum_{n=k}^{\infty} (k!/n!) z^n B_{n,k}$ for $|z|<R$. Alternatively, defining $Q_{k,n} = (k!/n!)B_{n,k}$, we have $Q_{1,n} = w_n/n!$, and $Q_{k+1,n}=\sum_{m=1}^{n-k} Q_{1,m} Q_{k,n-m}$ for $k\geq 1$. Let us say that the Cram\'er-type large deviation property holds if $$ \lim_{\substack{n \to \infty\\ k/n \to \kappa}} \frac{1}{n}\, \ln\left(Q_{k,n}\right)\, =\, \mathcal{G}(\kappa)\, ,$$ for every $\kappa \in (0,1)$, where $\mathcal{G}(\kappa)=\inf_{r \in (0,R)} (\kappa \ln(F(r))-\ln(r))$. The (Hardy-Ramanujan) Erd\"os induction argument suggests this should generally be true as long as two technical conditions are true: one an initial step, and the other a condition for small densities $\kappa$.