  Two stochastic representations of multivariate geometric distributions are
analyzed, both are obtained by lifting the lack-of-memory (LM) property of the
univariate geometric law to the multivariate case. On the one hand, the
narrow-sense multivariate geometric law can be considered a discrete equivalent
of the well-studied Marshall-Olkin exponential law. On the other hand, the more
general wide-sense geometric law is shown to be characterized by the LM
property and can differ significantly from its continuous counterpart, e.g., by
allowing for negative pairwise correlations.
  For both families of distributions, their $d$-dimensional exchangeable
subclass is characterized analytically via $d$-log-monotone, resp.\
$d$-monotone, sequences of parameters. Using this reparameterization, the
subfamilies of distributions with conditionally i.i.d.\ components in the sense
of de Finetti's theorem are determined. For these, a third stochastic
construction based on a non-decreasing random walk is presented. The
narrow-sense family is embedded in this construction when the increments of the
involved random walk are infinitely divisible. The exchangeable narrow-sense
law is furthermore shown to exhibit the multivariate right tail increasing
(MRTI) dependence.
