For a broad class of one-sample rank order statistics, almost sure (a.s.) convergence and exponential bounds for the probability of large deviations, when the basic random variables are not necessarily identically distributed, are established here. In this context, extending a result of Brillinger (1962) to the case of non-iidrv (independent and identically distributed random variables), a result on the a.s. convergence of sample means for a double sequence of random variables is derived. These results are of importance for the study of the properties of sequential tests and estimates based on rank order statistics.