  We discuss several constraints on the properties of gamma-ray bursts (GRB) at
cosmological distances. First we use the requirement that burst sources must be
optically thin to pair production by photon-photon collisions in order to
produce the observed nonthermal spectra. We derive probability distributions
for the minimum Lorentz expansion factor, the radiation energy, the maximum
baryonic mass, and the maximum surrounding gas density in the events, based on
254 events from the second BATSE catalog. Secondly, the time variabilities of
the bursts in the BATSE catalog are used to place an upper bound on the
curvature radius of the emitting surface in the events. This is based on the
requirement that the emitting region seen by the observer must be sufficiently
small to produce the observed variability without violating causality. Finally,
we discuss the association of cosmological GRB with galaxies. We consider eight
bright and well-localized bursts detected by the Pioneer Venus Orbiter, whose
positional error boxes contain no bright galaxies. Using the local luminosity
function of galaxies, we calculate the probability for not seeing the GRB host
galaxy. This probability tends to increase as the width of the GRB luminosity
function increases. However, the allowed width of the GRB luminosity function
is restricted by the number-count statistics of the bursts.
