Limit Theorems for the Maximal Path Weight in a Directed Graph on the Line with Random Weights of Edges

Abstract

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.