Abstract: Ločve in [5] introduced the classes associated with number as the classes of probability measures satisfying the condition (1). Many authors
investigated those classes ([2], [5]-[9], [20], [21]). In this paper we consider certain
subclasses of the classes We prove that they coincide with the classes
of distributions of series of some random variables and with the classes of limit distributions
of some normed sums. We give a characterization of certain classes associated
with
Urbanik in [18] introduced the concept of the decomposability semigroup associated with
probability measure as the set of all numbers such that ([11]-[14]). The
class of selfdecomposable distributions coincides with the class of probability
measures such that The class of multiply
selfdecomposable distributions may be described as the class of probability measures
such that for every or in terms of multiply
decomposability semigroups it is equivalent to the inclusion
where is the multiply decomposability semigroup defined by the formula
([3], [4], [10], [15]-[17], [19]).
associated with number 
as the classes of probability measures satisfying the condition (1). Many authors
investigated those classes ([2], [5]-[9], [20], [21]). In this paper we consider certain
subclasses 
of the classes 
We prove that they coincide with the classes
of distributions of series of some random variables and with the classes of limit distributions
of some normed sums. We give a characterization of certain classes 
associated
with 
as the set of all numbers 
such that 
([11]-[14]). The
class 
of selfdecomposable distributions coincides with the class of probability
measures 
such that ![D(P ) > [0,1].](files/19.1/HTML/19.1.13.abs13x.png)
The class 
of multiply
selfdecomposable distributions may be described as the class of probability measures 
such that 
for every ![c ,...,c (- [0,1],
1 m](files/19.1/HTML/19.1.13.abs18x.png)
or in terms of multiply
decomposability semigroups it is equivalent to the inclusion ![D (P ) > [0,1]m,
m](files/19.1/HTML/19.1.13.abs19x.png)
where 
is the multiply decomposability semigroup defined by the formula
([3], [4], [10], [15]-[17], [19]).