In this paper, we develop bounds on the individual session backlog and delay distribution under the generalized processor sharing (GPS) scheduling discipline. This work is motivated by, and is an extension of, Parekh and Gallager's deterministic study of the GPS scheduling discipline with leaky-bucket token controlled sessions , . Using the exponentially bounded burstiness (E.B.B.) process model introduced in  as a source traffic characterization, we establish results that extend the deterministic study of GPS: For a single GPS server in isolation, we present statistical bounds on the distributions of backlog and delay for each session. In the network setting, we show that networks belonging to a broad class of GPS assignments, the so-called consistent relative session treatment (CRST) GPS assignments, are stable in a stochastic sense. In particular, we establish simple bounds on the distribution of backlog and delay for each session in a rate proportional processor sharing (RPPS) GPS network with arbitrary topology.