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.

我们考虑一个无限有向图，其顶点由整数编号\(\ldots,-2, -1,0,1,2,\ldots\strut\)，其中任何一对顶点\(j< k\)由一条边\((j,k)\)从\(j\)指向\(k\)并具有随机权重\(v_{j,k}\in [-\infty,\infty)\ )。这里，\(\{v_{j,k},\: j< k\}\)是一族独立同分布的随机变量，它们采用有限值（任何符号）或值\(-\infty \)。图中的路径是一系列连接的边\((j_0,j_1),(j_1,j_2),\ldots,(j_{m-1},j_m)\)（其中\(j_0< j_1< \ldots < j_m\))，其权重是边权重之和\(\sum\limits_{s=1}^m v_{j_{s-1},j_s}\ge -\infty\)。让\(w_{0,n}\)是从\(0\)到\(n\)的所有路径的最大权重。假设\({\boldsymbol{\rm{P}}}(v_{0,1}>0)>0\)，则\({\boldsymbol{\rm{P}}}(v_ {0,1}\in\cdot\,\,|\, v_{0,1}>0)\)是非退化的，并且\({\boldsymbol{\rm{E}}}\exp (Cv_{ 0,1})< \infty\)对于某些\(C={\rm{const}} >0\)，我们将随机序列\(w_{0,n}\)的渐近行为研究为\(n \to\infty\). 在正态和中等大偏差的域中，当随机变量\(v_{i,j}\) 的分布是算术分布时，我们获得局部极限定理，如果该分布是非晶格分布，则获得积分局部极限定理。