Local Theorems and Moments for Maxima of Sums of Bounded Lattice Components

Let $\xi_1,\xi_2,\dots$ – independent lattice random variables; $|\xi_k|<M$, $\bar s_n=\max_{1\leq\nu\leq n}(0,\xi _1+,\xi _2+\cdots+\xi _\nu)$. The formulas for $\mathbf P({s_n=x})$ and for the first moments $\bar s_n$ are obtained in the note.