Page 120 - Special Topic Session (STS) - Volume 3

STS518 B.H. Jasiulis G. et al.
3. Results
Let λ be an infinitely divisible measure with respect to generalized convolution
⋄. In [16] Urbanik found an analogue to the Lévy-Khintchine formula for the
generalized characteristic function

where m is a finite Borel measure on [0, ∞), ω(x) = 1 − h( {,} ) and >
0 is such that ℎ( ) < 1 whenever 0 < ≤ .

The extension of this result for the case of generalized convolutions on R
connected with weakly stable measures (see, e.g., [11]) one can found in [7]. In
[8] some connections with non-commutative probability theory are studied.
Moreover some examples of measure m being an analog of the Lévy measures
one can found in [5,11].
Using the Kołmogorov theorem we prove the existence of Lévy processes
with respect to generalized convolution (see [5]) and show that they are
Markov processes with the transition probabilities given by distributions that
are infinitely divisible with respect to generalized convolution.

Theorem 1. Let 0 < < < , > 0. There exists a Markov process
{ : > 0} with (X ) = λ є and transition probability:
+
1

Proof. We show that the probability kernels (, ) satisfy the Chapman-
,
Kołmogorov equations, i.e.

Indeed, we have:

which ends the proof.□
All these results are applied to the Kendall convolution case. We consider
infinitely divisible distributions with respect to the Kendall convolution since
except the classical and stable case this seems to be most applicable for
modeling real processes.

In particular in [1] and [10] we prove a limit theorem for Markov chains { ∶

n є N} driven by the Kendall convolution (called also Kendall random walks

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