Enlargement of the class of geometrically infinitely divisible random variables

Slobodanka Jankovi\'c

Abstract: The class of negative binomial infinitely divisible random variables is introduced in the following way: Random variable $Y$ is called {\it negative binomial infinitely divisible\/} if there exist i.i.d.\ random variables $X^{(1)}_p,X^{(2)}_p,\dots$, $p\in(0,1)$, independent of $Y$ and $\nu^{(r)}_p$ and such that $$ Y \mathrel{\mathop=^{\text {\rm d}}} \lim_{p\to 0} \sum^{\nu_p^{(r)}}_{j=1} X^{(j)}_p, $$ where $\nu^{(r)}_p$ has negative binomial law. \par The representation of characteristic functions from the class of negative binomial infinitely divisible random variables is given and also some related properties discussed. When $r=1$ the above class reduces to the well known class of geometrically infinitely divisible random variables.

Classification (MSC2000): 60E07

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