Limit Theorems on the Distributions of Maxima of Sums of Bounded Lattice Random Variables. I

The bounded latticed indentically disturbed random variables $\xi_1,\xi_2,\dots$ are considered. The local (Chapter 2) and integral (Chapter 3) theorems for the first passage time $\eta_x$, over the barrier $x>0$ in the random wanderings along the straight line with the quantity of jump $\xi _k $ are studied. The formulas for $\mathbf P(\eta_x=n)$ and $\mathbf P(\eta_x>n)$ in obvious form are obtained for the full “spectrum” of values $x$, beginning with $x=o(n)$ until $x$, equivalent to the product maximum jump $\xi_k$ by $n$. The theorems for $\mathbf P(\eta_x>n)$ simultaneously are integral theorems for maximum of sums $\sum_{k=1}^\nu\xi_k,\nu=1, \dots ,n$. The formulas for first moments $\eta_x$ and the distribution of the quantity of the first excess over the barrier x are also obtained.
Some results were published in [9] without proofs.