The statistical representation of a spray as a finite point process is investigated. One objective is to develop a better understanding of how single-point statistical information contained in descriptions such as the droplet distribution function ~ddf!, relates to the probability density functions ~pdfs! associated with the droplets themselves. Single-point statistical information contained in the droplet distribution function ~ddf! is shown to be related to a sequence of single surrogate-droplet pdfs, which are in general different from the physical single-droplet pdfs. It is shown that the ddf contains less information than the fundamental single-point statistical representation of the spray, which is also described. The analysis shows which events associated with the ensemble of spray droplets can be characterized by the ddf, and which cannot. The implications of these findings for the ddf approach to spray modeling are discussed. The results of this study also have important consequences for the initialization and evolution of direct numerical simulations ~DNS! of multiphase flows, which are usually initialized on the basis of single-point statistics such as the droplet number density in physical space. If multiphase DNS are initialized in this way, this implies that even the initial representation contains certain implicit assumptions concerning the complete ensemble of realizations, which are invalid for general multiphase flows. Also the evolution of a DNS initialized in this manner is shown to be valid only if an as yet unproven commutation hypothesis holds true. Therefore, it is questionable to what extent DNS that are initialized in this manner constitute a direct simulation of the physical droplets. Implications of these findings for large eddy simulations of multiphase flows are also discussed.