Local limit theorems for random walks conditioned to stay positive

Let $S_{0}=0$, $\{S_{n},\,n\geq 1\}$ be a random walk generated by a sequence of i.i.d. random variables $X_{1},X_{2},\dots$, and let $\tau^-=\min\{n\geq1:S_{n}\leq 0\}$ and $\tau^+=\min\{n\geq 1:S_{n}>0\}$. Assuming that the distribution of $X_1$ belongs to the domain of attraction of an $\alpha$-stable law we study the asymptotic behavior, as $n\to\infty$, of the local probabilities $\mathbf{P}(\tau^{\pm}=n)$ and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities $\mathbf{P}(S_{n}\in \lbrack x,x+\Delta )|\tau ^{-}>n)$ with fixed $\Delta$ and $x=x(n)\in(0,\infty)$. Applications of these theorems to critical branching processes in random environment will be mentioned.