Page 118 - Special Topic Session (STS) - Volume 3

STS518 B.H. Jasiulis G. et al.
є a counterpart of a classical characteristic function called generalized
+
characteristic function

Each generalized convolution is uniquely determined by the probability kernel
⋄ , i.e.

for every , .
2
1
+

Example 1. The -convolution, > 0, is defined, for , , ≥ 0, by ⋄

= , where = + and with homomorphism ℎ( ) =

{− }.

Example 2. The Kendall convolution Δα is defined in the following way:

for 0 ≤ ≤ 1 and > 0 , where 2 denotes a Pareto distribution
measure with the density (dx) = 2α x −( 2α+1) (1,∞) (x)dx. In this case
2
we have

where = if > 0 and + = 0 if ≤ 0 . The corresponding
+
generalized characteristic function is the Williamson transform (for more
details on the transform see, e.g., [14, 15, 16, 18])

Example 3. For every ≥ 2 and properly chosen > 0 the function

is the kernel of a Kendall type (see [14]) generalized convolution ⋄ defined
for є [0,1] by the formula:

where λ1, λ2 are probability measures absolutely continuous with respect
to the Lebesgue measure and that does not depend on x. For example if
-1
c = (p-1) then

and

It is natural to consider infinitely divisible measures with respect to generalized
convolutions.

107 | I S I W S C 2 0 1 9