Page 119 - Special Topic Session (STS) - Volume 3

STS518 B.H. Jasiulis G. et al.
Definition 2.
A measure є is said to be infinitely divisible with respect to the
+
generalized convolution ⋄ (-⋄ infinitely decomposable) in the algebra ( ,⋄) if
+
for every ⋄ є there exists a probability measure є such that

+
= ⋄ .

One of the most important examples of ⋄-infinitely divisible distribution is ⋄-
compound Poisson measure ⋄ () defined in [5, 9, 10, 11]. In next
example the Poisson probability measure in the Kendall generalized
convolution algebra is presented.

Example 4.

It is worth to notice that Poisson measures with respect to generalized
convolutions are not strictly discrete and have usually a continuous part. In [1]
we proved that the last measure at the above convex linear combination is
stable in the Kendall convolution algebra in the sense of Definition 3.

Definition 3.
Let є . We say that λ is stable in the generalized convolution algebra
+
( ,⋄), if for all a, b ≥ 0 there exists c ≥ 0 such that
+

Similarly to the classical theory stable distributions in the generalized
convolutions sense are ⋄ - infinitely divisible. The generalized characteristic

function of ⋄ - infinitely divisible distribution is exponent of for some >
0. For every generalized convolution ⋄ on there exists a constant (⋄),
+
called a characteristic exponent, such that for every є (0, (⋄)] there exists
a measure є with the ⋄-generalized characteristic function (t) =
+

p
exp{−t } if < ∞ and (t) = [0,1] (t) otherwise. For example (⋄) = 2

for classical convolution and ( ) = for the case of Kendall convolution.

Moreover, the set of all ⋄ - stable measures coincides with the set { : >

0, є (0, (⋄)]}.

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