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Let be the unique normal martingale such that
=0 and d[]=(1−−
) d
+d and let
:=
+ for all ≥0; the semimartingale arises in quantum probability, where it is the monotone-independent analogue of the Poisson process. The trajectories of are examined and various probabilistic properties are derived; in particular, the level set {≥0:
=1} is shown to be non-empty, compact, perfect and of zero Lebesgue measure....
The coefficients of the moments of the monotone Poisson law are shown to be a type of Stirling number of the first kind; certain combinatorial identities relating to these numbers are proved and a new derivation of the Cauchy transform of this law is given. An investigation is begun into the classical Azéma-type martingale which corresponds to the compensated monotone Poisson process; it is shown to have the chaotic-representation property and its sample paths are described.
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