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STS518 B.H. Jasiulis G. et al.
()/() → in the sense of distribution. We denote the inverse of () by
() and then U є 1/( − ) . Clearly ()/() → implies that (())/
→ .
Now we have

Since V(U(n)t)/n ⁓ V(U(n)) − /n → t α – θ , we have P( ≤ U(n)t ) →

P( Z ≥ t α – θ ) = P( Z −1/( α − θ ) ≤ t).
Finally we conclude that /U(n) → Z −1/( α − θ ) in the sense of distribution.

The construction shows that (()) ⁓ (()) ⁓ , which ends the proof. □

It is worth to notice that exponential transform of random variable X fit to
maximal daily concentration of nitrogen dioxide for data from USA and Poland
in some cases. Above stable distributions are infinitely divisible in the Kendall
generalized convolution algebra.
Since random walks with respect to the generalized convolutions form a class
of extremal Markov chains (see [1, 5, 10]), studying them in the appropriate
algebras will be a meaningful contribution to extreme value theory.
More about regular variation context for extremal Markov chains driven by the
Kendall convolution one can find in [1, 10].

4. Discussion and Conclusion:
Even though the family of generalized convolutions is pretty rich by now
we are still interested in constructing new examples and finding new methods
of constructing them on the base of these which we already know.
The next open problem lie on proposing a statistical methods to recognize
which stochastic processes are the Lévy processes with respect to some
generalized convolution. Could we recognize this generalized convolution on
the base of some empirical data?

Acknowledgements. This paper is a part of project "First order Kendall
maximal autoregressive processes and their applications", which is carried out
within the POWROTY/REINTEGRATION programme of the Foundation for
Polish Science co-financed by the European Union under the European
Regional Development Fund.

References
1. Arendarczyk M., Jasiulis-Gołdyn B.H. & Omey E.A.M. (2019). Asymptotic
properties of Kendall random walks, submitted, arXiv:
https://arxiv.org/pdf/1901.05698.pdf

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