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

TL;DR

This paper investigates the asymptotic behavior of the maximum path weight in an infinite directed graph with random edge weights, deriving limit theorems under various distributional assumptions.

Contribution

It provides new local and integro-local limit theorems for the maximal path weight in a directed graph with i.i.d. random edge weights, covering both lattice and non-lattice cases.

Findings

Established normal and moderate deviation limit theorems for lattice distributions.

Derived integro-local theorems for non-lattice distributions.

Analyzed asymptotic behavior of maximal path weights under specified conditions.

Abstract

We consider the infinite directed graph with vertices the set of integers ...,-2,-1,0,1,2,... . Let v be a random variable taking either finite values or value "minus infinity". Consider random weights v(j,k), indexed by pairs (j,k) of integers with j<k, and assume that they are i.i.d. copies of v. The set of edges of the graph is the set (j,k), j<k. A path in the graph from vertex j to vertex k, j<k, is a finite sequence of edges (j(0), j(1)), (j(1), j(2)), ..., (j(m-1), j(m)) with j(0)=j and j(m)=j; the weight of this path is taken to be the sum v(j(0),j(1))+v(j(1),j(2))+...+v(j(m-1),j(m)) of the weights of its edges. Let w(0,n) be the maximal weight of all paths from 0 to n. We study the asymptotic behaviour of the sequence w(0,n), n=1, 2, ..., as n tends to infinity, under the assumptions that P(v>0)>0, the conditional distribution of v, given v>0, is not degenerate, and that E exp(Cv) is finite, for some C>0. We derive local limit theorems in the normal and moderate large deviations regimes in the case where v has an arithmetic distribution. We also derive an integro-local theorem in the case where v has a non-lattice distribution.