Page 117 - Special Topic Session (STS) - Volume 3

STS518 B.H. Jasiulis G. et al.
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 є ) if, for all > 0, 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) ⋄ = є ;
0
+
(ii) ( + (1 − ) ⋄ = ( ⋄ ) + (1 − )( ⋄ ) for each [0,1] and
1
2
1
2
, , ;
2
+
1
(iii) ( ⋄ ) = ( ) ⋄ ( ) for all ≥ 0 and , ;
2

1
2
1
2
+
1
(iv) if ⟶ and ⟶ , then ( ⋄ ) ⟶ ( ⋄ ), where ⟶ denotes

weak convergence;
⋄
(v) there exists a sequence of positive numbers such that 1

converges weakly to a measure ≠ (here = ⋄ ⋄ . .. ⋄
⋄
0
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 , є , є [0,1], 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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