By using the matrix formulation of the two-step approach to the distributions of runs, a recursive relation and an explicit expression are derived for the generating function of the joint distribution of rises and falls for multivariate random sequences in terms of generating functions of individual letters, from which the generating functions of the joint distribution of rises, falls, and number of runs are obtained. An explicit formula for the joint distribution of rises and falls with arbitrary specification is also obtained.

翻译：通过使用两步法处理游程分布的矩阵表述，本文基于单个字母的生成函数推导了多元随机序列中上升与下降联合分布的生成函数的递归关系及显式表达式，并由此获得上升、下降与游程数联合分布的生成函数。此外，还得到了任意指定条件下上升与下降联合分布的显式公式。