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Abstract

Kingman, in his seminal work [13], introduced a new type of convolution of distributions that is naturally related to spherically symmetric random walks. Motivated by this paper, Urbanik in a series of papers [17] established a theory of generalized convolutions ⋄ as certain binary commutative and associative operations that include classical and Kingman's convolutions as a special case. This theory was further developed by Bingham ([2, 3]) in the context of regularly varying functions. There is a rich class of examples of generalized convolutions that are motivated by problems in applications of probability theory. For instance, the distribution of the maximum of two independent random variables is a generalized convolution fundamentally associated with the extreme value theory, and extensively applied to model events that rarely occur, but the appearance of which causes large losses. Similarly, to the classical theory, we define infinite divisibility with respect to generalized convolution ⋄ and establish Lévy-Khintchine representation [11]. Lévy and additive stochastic processes under generalized convolutions are constructed as the Markov processes in ([5]). In this paper we survey examples of generalized convolutions and related Lévy-Khintchine representation. Results on Kendall convolution and extreme Markov chains driven by the Kendall convolution ([1, 5, 10]) using Williamson transform ([18]) are also presented.
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Infinitely divisible probability measures under
generalized convolutions
B.H. Jasiulis - Gołdyn1; M. Arendarczyk1; M. Borowiecka-Olszewska2; J.K.
Misiewicz3; E. Omey4; J. Rosiński5
1 Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384
Wrocław, Poland
2 Faculty of Mathematics, Computer Science and Econometrics, University of
Zielona Góra, ul. Prof. Z. Szafrana 4A, 65-516 Zielona Góra, Poland
3 Faculty of Mathematics and Information Science, Warsaw University of
Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland
4 KU Leuven, Warmoesberg 26, 1000 Brussels, Belgium
5 Department of Mathematics, 227 Ayres Hall, University of Tennessee,
Knoxville TN 37996, USA
Abstract
Kingman, in his seminal work [13], introduced a new type of convolution of
distributions that is naturally related to spherically symmetric random walks.
Motivated by this paper, Urbanik in a series of papers [17] established a theory
of generalized convolutions as certain binary commutative and associative
operations that include classical and Kingman’s convolutions as a special case.
This theory was further developed by Bingham ([2, 3]) in the context of
regularly varying functions. There is a rich class of examples of generalized
convolutions that are motivated by problems in applications of probability
theory. For instance, the distribution of the maximum of two independent
random variables is a generalized convolution fundamentally associated with
the extreme value theory, and extensively applied to model events that rarely
occur, but the appearance of which causes large losses. Similarly, to the
classical theory, we define infinite divisibility with respect to generalized
convolution and establish Lévy-Khintchine representation [11]. Lévy and
additive stochastic processes under generalized convolutions are constructed
as the Markov processes in ([5]). In this paper we survey examples of
generalized convolutions and related Lévy-Khintchine representation. Results
on Kendall convolution and extreme Markov chains driven by the Kendall
convolution ([1, 5, 10]) using Williamson transform ([18]) are also presented.
Keywords
infinitely divisible probability measure; generalized convolution; Kendall
random walk; LévyKchintchine representation; regular variation
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1. Introduction
Notation:
Throughout this paper, the family of all probability measures on the Borel
subsets of R+ is denoted by P+. For a probability measure λ є P+ and a є R+ the
rescaling operator is given by  if  denotes the
distribution of the random element X.
Finally a measurable function f(∙) is regularly varying at infinity with index
(notation ) if, for all , it satisfies  
(see, e.g., [4]).
2. Methodology
The main unconventional tool used here is generalized convolution ([17]),
which is a generalization of the classical convolution corresponding to the sum
of independent random elements. Generalized convolutions were explored
with the use of regular variation ([2,3]) and were applied to construct Lévy
processes and stochastic integrals ([5]). Their origin can be found in delphic
semigroups ([12]). The development of generalized convolutions was
motivated by spherically symmetric random walks (see [13]). Hence
generalized convolutions are closely related to multidimensional distributions.
Definition 1.
A generalized convolution is a binary, symmetric, associative and
commutative operation on having the following properties:
(i)  ;
(ii)  for each  and

(iii)  for all and 
(iv) if and then where denotes
weak convergence;
(v) there exists a sequence of positive numbers such that 
converges weakly to a measure (here 
denotes the generalized convolution of n identical measures).
The pair () is called a generalized convolution algebra. We define a
continuous mapping h: R+, called the homomorphism of the algebra
(), such that for  we have:
The homomorphism in ( ) plays an important role in the theory of
generalised convolutions and if it is not trivial, then it defines, for any measure
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 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 .
Example 1. The -convolution, , is defined, for  by
, where and with homomorphism 
.
Example 2. The Kendall convolution Δα is defined in the following way:
for  and , where  denotes a Pareto distribution
measure with the density  In this case
we have
where  if  and  if . 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  and properly chosen  the function
is the kernel of a Kendall type (see [14]) generalized convolution defined
for  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
c = (p-1)-1 then
and
It is natural to consider infinitely divisible measures with respect to generalized
convolutions.
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Definition 2.
A measure  is said to be infinitely divisible with respect to the
generalized convolution (- infinitely decomposable) in the algebra () if
for every  there exists a probability measure  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  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 
. For every generalized convolution on there exists a constant 
called a characteristic exponent, such that for every  there exists
a measure with the -generalized characteristic function 
 if  and  otherwise. For example 
for classical convolution and  for the case of Kendall convolution.
Moreover, the set of all - stable measures coincides with the set 

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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  and 
 is such that  whenever .
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  There exists a Markov process
 with  and transition probability:
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 
 driven by the Kendall convolution (called also Kendall random walks
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introduced in [6]) with  assuming that is finite
or the truncated moment 
  is regularly varying.
Theorem 2. Let  be a Kendall random walk with parameter 
unit step distribution  and 


(i) If  then as 
where the cdf of random variable X is given by
(ii) Suppose that H є RVθ where 0 θ < α. Then there exists an increasing
function U(x) such that U(1/(1-G(x)) x and
where has distribution, which is a convex linear combination of an
exponential and a gamma distribution
where  denotes the measure with the density

Proof.
(i) Let F denotes the cdf of the unit step . First notice that 
 and , as . Since the Williamson transform for
 is given by the following formula (see [1]):
then we obtain  as  To complete the
proof it suffices to check that the limiting measure has exactly the Williamson
transform 
(ii) The second part of theorem can be proved using limit theorem for
renewal process  constructed by the Kendall
convolution (see Theorem 6 in [10]). We use the result that
Since  is asymptotically equal to a strictly increasing
function  (see [4], Section 1.5.2, Theorem 1.5.4, p.23) and
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 in the sense of distribution. We denote the inverse of  by
 and then Clearly  implies that 

Now we have
Since  we have 

Finally we conclude that  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.
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The paper deals with renewal theory for a class of extremal Markov sequences connected with the Kendall convolution. We consider here some particular cases of the Wold processes associated with generalized convolutions. We prove an analogue of the Fredholm theorem for all regular generalized convolutions algebras. Using regularly varying functions we prove a Blackwell theorem and a limit theorem for renewal processes defined by Kendall random walks. Our results set new research hypotheses for other generalized convolution algebras to investigate renewal processes constructed by Markov processes with respect to generalized convolutions.
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The paper deals with the notions of weak stability and weak generalized convolution with respect to a generalized convolution, introduced by Kucharczak and Urbanik. We study properties of such objects and give examples of weakly stable measures with respect to the Kendall convolution. Moreover, we show that in the context of non-commutative probability, two operations: the q-convolution and the (q,1)-convolution satisfy the Urbanik’s conditions for a generalized convolution, interpreted on the set of moment sequences. The weak stability reveals the relation between two operations.