A general correlation inequality for level sets of sums of independent random variables using the Bernoulli part with applications to the almost sure local limit theorem

Abstract

Let $X=\{X_j , j\ge 1\}$ be a sequence of independent, square integrable variables taking values in a common lattice $\mathcal L(v_{ 0},D )= \{v_{ k}=v_{ 0}+D k , k\in \Z\}$. Let $S_n=X_1+\ldots +X_n$, $a_n= {\mathbb E\,} S_n$, and $\s_n^2={\rm Var}(S_n)\to \infty$ with $n$. Assume that for each $j$, $\t_{X_j} =\sum_{k\in \Z}{\mathbb P}\{X_j=v_k\}\wedge{\mathbb P}\{X_j=v_{k+1}\}>0$. Using the Bernoulli part, we prove a general sharp correlation inequality extending the one we obtained in the i.i.d.\,case in \cite{W3}: Let $0<\t_j\le \t_{X_j}$ and assume that $ \nu_n =\sum_{j=1}^n \t_j \, \uparrow \infty$, $n\to \infty$. Let $\k_j\in \mathcal L(jv_0,D)$, $j=1,2,\ldots$ be a sequence of integers such that \begin{equation*} {\rm(1)}\qquad\frac{\kappa_j-a_j}{\s_j}=\mathcal O(1 ), \qq\quad {\rm(2)}\qquad \s_j \,{\mathbb P}\{S_j=\kappa_j\} ={\mathcal O}(1). \end{equation*} Then there exists a constant $C $ such that for all $1\le m<n$, \begin{align*} \s_n&\s_m \, \Big|{\mathbb P}\{S_n=\k_n, S_m=\k_m\}- {\mathbb P}\{S_n=\k_n \}{\mathbb P}\{ S_m=\k_m\} \Big| \cr & \,\le \, \frac{C}{D^2}\, \max \Big(\frac{\s_n }{\sqrt{\nu_n}},\frac{\s_m }{\sqrt {\nu_m}} \Big)^3 \,\bigg\{ \nu_n^{1/2} \prod_{j=m+1}^n\vartheta_j + {\nu_n^{1/2} \over (\nu_n-\nu_m) ^{3/2}}+{ 1\over \sqrt{{\nu_n\over \nu_m}}-1} \bigg\}. \end{align*} We derive a sharp almost sure local limit theorem