Conditional limit theorems for random walks and their local times

In the classical theory of random walks, two functional limit theorems are well known: the Donsker-Prokhorov invariance principle for random walks themselves and the Borodin theorem for the local time of integer random walks. The report discusses analogs of these results obtained
1) for a random walk considered under the condition that its trajectory is positive up to time $n$,
2) for a random walk stopped at the moment $T$ of the first attaining of the non-positive semi-axis and considered either under the condition that $T>n$, or provided that it attains a certain high level of the order of root of $n$.
It is well known that conditional limit theorems for a random walk itself are in demand in the theory of branching processes in a random environment. It turns out that conditional limit theorems for the local time of a random walk find application in the classical theory of Galton-Watson branching processes. In particular, the connection of these theorems with the most important functional limit theorems for Galton-Watson branching processes is established.