Stability theorems and the second-order asymptotics in threshold phenomena for boundary functionals of random walks

In Section 1, we prove stability theorems for a series of boundary functionals of random walks. In Section 2, we suggest a new simpler proof of the theorem on threshold phenomena for the distribution of the maximum of the consecutive sums of random variables. In Section 3, we find the second-order asymptotics for this distribution under the assumption that the third moments of the random variables exist.