We establish tightness of graph-based stochastic processes in the space $D[0+\epsilon,1-\epsilon]$ with $\epsilon >0$ that allows for discontinuities of the first kind. The graph-based stochastic processes are based on statistics constructed from similarity graphs. In this setting, the classic characterization of tightness is intractable, making it difficult to obtain convergence of the limiting distributions for graph-based stochastic processes. We take an alternative approach and study the behavior of the higher moments of the graph-based test statistics. We show that, under mild conditions of the graph, tightness of the stochastic process can be established by obtaining upper bounds on the graph-based statistics' higher moments. Explicit analytical expressions for these moments are provided. The results are applicable to generic graphs, including dense graphs where the number of edges can be of higher order than the number of observations.

翻译：我们在空间$D[0+\epsilon,1-\epsilon]$（其中$\epsilon>0$）中建立了允许第一类间断点的图基随机过程的紧性。图基随机过程基于由相似图构建的统计量。在此设定下，紧性的经典刻画难以处理，这使得获取图基随机过程极限分布的收敛性变得困难。我们采用了一种替代方法，研究图基检验统计量高阶矩的行为。我们证明，在图的温和条件下，通过获得图基统计量高阶矩的上界，可以建立随机过程的紧性。这些矩的显式解析表达式已被给出。该结果适用于一般图，包括边数可能远大于观测数的高密度图。