The distribution of the first ladder moment and height and fluctuations of random walk

Let $\xi_1,\xi_2,\dots$ be independent identically destributed random variables, and let $S_0=0$, $S_n=\sum_{i=1}^n\xi_i$, $T=\min\{n\colon S_n>0\}$, $H=S_T$.
In the paper, the distributions of $T$ and $H$ are investigated.