Content uploaded by Barbara HELENA Jasiulis-Gołdyn

Author content

All content in this area was uploaded by Barbara HELENA Jasiulis-Gołdyn on Feb 19, 2022

Content may be subject to copyright.

STS518 B.H. Jasiulis G. et al.

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

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

STS518 B.H. Jasiulis G. et al.

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

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

STS518 B.H. Jasiulis G. et al.

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

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.

STS518 B.H. Jasiulis G. et al.

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

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

STS518 B.H. Jasiulis G. et al.

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

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

STS518 B.H. Jasiulis G. et al.

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

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

STS518 B.H. Jasiulis G. et al.

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

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.

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

STS518 B.H. Jasiulis G. et al.

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

2. Bingham N. H. (1971). Factorization theory and domains of attraction for

generalized convolution algebra. Proc. London Math. Sci., Infinite

Dimensional Analysis.

Quantum Probability and Related Topics

23(4), 16-

30.

3. Bingham N. H. (1984). On a theorem of Kłosowska about generalized

convolutions.

Coll. Math.

48(1), 117-125.

4. Bingham N. H., Goldie C. M. & Teugels J. L. (1987).

Regular variation.

Cambridge University Press, Cambridge.

5. Borowiecka-Olszewska M., Jasiulis-Gołdyn B.H., Misiewicz J.K. &

Rosiński J. (2015). Lévy processes and stochastic integral in the sense of

generalized convolution.

Bernoulli

21(4), 2513-2551.

6. Jasiulis-Gołdyn B.H. (2016) Kendall random walks.

Probab. Math. Stat.

36(1), 165-185.

7. Jasiulis B.H. (2010). Limit property for regular and weak generalized

convolution.

J. Theoret. Probab

. 23(1), 315-327,

8. Jasiulis-Gołdyn B.H., Kula A. (2012). The Urbanik generalized

convolutions in the noncommutative probability and a forgotten

method of constructing generalized convolution.

Proc. Math. Sci.

122(3),

437-458.

9. Jasiulis-Gołdyn B.H., Misiewicz J.K. (2015). Classical definitions of the

Poisson process do not coincide in the case of weak generalized

convolution.

Lith. Math. J.

55(4), 518-542.

10. Jasiulis-Gołdyn B.H., Misiewicz J.K., Naskręt K. & Omey E.A.M. (2018).

Renewal theory for extremal Markov sequences of the Kendall type,

submitted, arXiv:https://arxiv.org/pdf/1803.11090.pdf

11. Jasiulis-Gołdyn B.H., Misiewicz J.K. (2015). Weak Lévy-Khintchine

representation for weak infinite divisibility.

Theor. Probab. Appl.

60(1),

45-61.

12. Kendall D. G. (1968). Delphic semi-groups, infinitely divisible

regenerative phenomena, and the arithmetic of p-functions.

Z.

Wahrscheinlichkeitstheorie und Verw. Gebiete

9(3), 163-195.

13. Kingman J. G. C. (1963). Random Walks with Spherical Symmetry.

Acta

Math.

109(1), 11-53.

14. Misiewicz J. K. (2018). Generalized convolutions and Levi-Civita

functional equation.

Aequationes Math.

92(5), 911-933.

15. A.J. McNeil, J. Nešlehová. (2009). Multivariate Archimedean copulas, d-

monotone functions and l1-norm symmetric distributions.

The Annals of

Statistics

37 (5B), 3059-3097.

16. McNeil A.J., Nešlehová J. (2010). From Archimedean to Liouville Copulas.

J. Multivariate Analysis

101(8), 1771-1790.

STS518 B.H. Jasiulis G. et al.

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

17. Urbanik K., Generalized convolutions I-V,

Studia Math.

, 23(1964), 217-

245, 45(1973), 57-70, 80(1984), 167-189, 83(1986), 57-95, 91(1988), 153-

178.

18. Williamson R. E. (1956). Multiply monotone functions and their Laplace

transforms.

Duke Math. J

., 23, 189-207.