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
Categories: math.PR