We consider a processor sharing queue with several customer classes. For an arbitrary customer of class i we show that the sojourn time distribution is regularly varying of index -\nu_i iff the service time distribution is regularly varying of index -\nu_i, and derive an explicit asymptotic formula. Furthermore, the tail of the sojourn time distribution of customer class i is shown to be unaffected by the tails of the service time distributions of other customer classes, even if some of the latter tails are heavier. This result implies that, when the sojourn time of a customer is large, this is not due to long service requirements of other customer types. In particular, short-range dependent traffic does not suffer from longe-range dependent traffic if processor sharing is used as a service discipline.