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

Let $X=\{X_j,j\ge 1\}$ be a sequence of independent, square integrable variables taking values in a common lattice $\mathcal{L}(v_0,D)=\{v_k=v_0+Dk,k\in \mathbb{Z}\}$. Let $S_n=X_1+\dots +X_n$, $a_n=\mathbb{E}{S}_n$, and ${\text{s}}_n^2=\mathrm{V}\mathrm{a}\mathrm{r}(S_n)\to \mathrm{\infty }$ with $n$. Assume that for each $j$, ${\text{t}}_{X_j}={\sum }_{k\in \mathbb{Z}}\mathbb{P}\{X_j=v_k\}\wedge \mathbb{P}\{X_j=v_{k+1}\}>0$. 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 $0<{\text{t}}_j\le {\text{t}}_{X_j}$ and assume that ${\nu }_n={\sum }_{j=1}^n{\text{t}}_j\uparrow \mathrm{\infty }$, $n\to \mathrm{\infty }$. Let ${\text{k}}_j\in \mathcal{L}(j{v}_0,D)$, $j=1,2,\dots$ be a sequence of integers such that \begin{equation*} {\rm(1)}\qquad\frac{κ_j-a_j}{\s_j}=\mathcal O(1 ), \qq\quad {\rm(2)}\qquad \s_j ,{\mathbb P}{S_j=κ_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{ν_n}},\frac{\s_m }{\sqrt {ν_m}} \Big)^3 ,\bigg{ ν_n^{1/2} \prod_{j=m+1}^n\vartheta_j + {ν_n^{1/2} \over (ν_n-ν_m) ^{3/2}}+{ 1\over \sqrt{ν_n\over ν_m}-1} \bigg}. \end{align*} We derive a sharp almost sure local limit theorem