The distributions of the $m$-th longest runs of multivariate random sequences are considered. For random sequences made up of $k$ kinds of letters, the lengths of the runs are sorted in two ways to give two definitions of run length ordering. In one definition, the lengths of the runs are sorted separately for each letter type. In the second definition, the lengths of all the runs are sorted together. Exact formulas are developed for the distributions of the m-th longest runs for both definitions. The derivations are based on a two-step method that is applicable to various other runs-related distributions, such as joint distributions of several letter types and multiple run lengths of a single letter type.

翻译：本文考虑了多元随机序列中第m长游程的分布。对于由k种字母构成的随机序列，游程长度通过两种方式排序，从而给出游程长度排序的两种定义。在第一种定义中，各字母类型的游程长度分别排序；在第二种定义中，所有游程长度统一排序。针对这两种定义，本文推导了第m长游程分布的精确公式。推导基于一种两步法，该方法可适用于多种其他游程相关分布，例如多种字母类型的联合分布以及单一字母类型的多重游程长度分布。