Limit theorems for the maximal path weight in a directed graph on the line with random weights of edges

We consider an infinite directed graph with vertices numbered by integers $\ldots,-2, -1,0,1,2,\ldots\strut$, where any pair of vertices $j<k$ is connected by an edge $(j,k)$ that is directed from $j$ to $k$ and has a random weight $v_{j,k}\in [-\infty,\infty)$. Here, $\{v_{j,k}, j< k\}$ is a family of independent and identically distributed random variables that take either finite values (of any sign) or the value $-\infty$. A path in the graph is a sequence of connected edges $(j_0,j_1),(j_1,j_2),\ldots,(j_{m-1},j_m)$ (where $j_0< j_1< \ldots < j_m$), and its weight is the sum $\sum\limits_{s=1}^m v_{j_{s-1},j_s}\ge -\infty$ of the weights of the edges. Let $w_{0,n}$ be the maximal weight of all paths from $0$ to $n$. Assuming that ${\boldsymbol{\rm{P}}}(v_{0,1}>0)>0$ , that the conditional distribution of ${\boldsymbol{\rm{P}}}(v_{0,1}\in\cdot | v_{0,1}>0)$ is nondegenerate, and that ${\boldsymbol{\rm{E}}}\exp (Cv_{0,1})< \infty$ for some $C={\rm{const}} >0$ , we study the asymptotic behavior of random sequence $w_{0,n}$ as $n\to\infty$. In the domain of the normal and moderately large deviations we obtain a local limit theorem when the distribution of random variables $v_{i,j}$ is arithmetic and an integro-local limit theorem if this distribution is non-lattice.