  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.
