Drought problems are critical aspects of water resources conservation, development, and control at present. Continuous pressure on limited water resources will make drought problems much more serious in the future. Therefore, intensive and systematic investigations on drought problems are urgent and necessary. However, the definition of drought is a controversial subject. Since Yevjevich (1967) introduced the theory of runs as an objective definition of hydrologic drought, several theoretical and experimental studies of runs related to drought problems have been done. Among properties of run-length, runsum, and run-intensity defined in Fig. 1, the run-length has been most widely investigated because the run-length properties are free of underlying distribution of input processes. However, applications of these developed properties of runs are limited to univariate or bivariate stationary series. In this paper, some statistical properties of the runs for an independent nonstationary process (periodic-stochastic process) are studied on some simple but realistic assumptions (Eq. (4) through Eq. (8)). On the assumption of Eq. (7), the properties of deficit or negative run-lengths are free of the underlying distribution as in the case of stationary process. The properties of run-sums are considered with the corresponding variables of run-length and onset time of the run-length. Expected value, variance, conditional expected values, and conditional variances are found analytically. The theoretical values are compared with those of generated data.
