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

Multivariate geometric distributions, (logarithmically) monotone sequences, and infinitely divisible laws (with erratum by Natalia Shenkman)