Limit theorems for moment of the maximum of random walk reaching fixed level

Consider an oscillating random walk $S_n=X_1+\dots+X_n$, $n\in\mathbb{N}$, $S_0=0$, where $X_1,X_2,\dots$ are independent and identically distributed (i.i.d.) random variables. For this walk, the arcsine law is well known:
$$ {\mathbf P}\left(\frac{\tau_M}{n}\le x\right) \to \frac{2}{\pi} \arcsin\sqrt{x},\quad n\to\infty,\quad x\in [0,1], $$
where $\tau_M$ is the first moment of the maximum of random walk $S_n$. We are interested in similar results for
$$ {\mathbf P}\left(\left.\frac{\tau_M}{n}\le x\right|M_n=k\right),\quad x\in [0,1], $$
where $M_n = S_{\tau_M}$.
We obtain the asymptotic behaviour of this probability for different values of parameter $k$. We obtain the result for a random walk that belongs to the domain of attraction of a stable law in the zone of normal deviations. The case of lower deviations is considered for a random walk with finite variance. We obtain the limit theorem for a random walk with finite variance and under the right‑side Cramer condition in the zone of moderate deviations.
Our results are applied to the branching processes in a random environment. Let $\{Z_n, n\geq0\}$ be a branching process in a random environment, let $\{S_n, n\geq0\}$ be the associated random walk, $ L_n = \min\{S_0,S_1,\dots,S_n\}$. We obtain the asymptotic behaviour for the probability
$$ {\mathbf P}\left( Z_n>0 \middle| L_n=-k \right) $$
when $k$ belongs to the zone of normal or lower deviations.