About the Cramér Large Deviation Property for Bell Polynomials

Abstract

If $\mathbf{w}=(w_1,w_2,\dots )$ is a sequence in $\mathbb{N}=\{1,2,\dots \}$, the partial Bell polynomials based on $\mathbf{w}$ 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 $\mathbf{w}$, 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 $k\ge 1$. Let us say that the Cramér-type large deviation property holds if $$\underset{\begin{array}{c}n\to \infty \\ k/n\to \kappa \end{array}}{\lim }\frac{1}{n}\ln \left(Q_{k,n}\right)=\mathcal{G}(\kappa ),$$ for every $\kappa \in (0,1)$, where $\mathcal{G}(\kappa )={\inf }_{r\in (0,R)}(\kappa \ln (F(r))-\ln (r))$. The (Hardy-Ramanujan) Erdös 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 $\kappa$.

16 pages, 1 figure. Added appendix on Cramér's theorem