1. Introduction
In recent decades, a number of articles have studied limit theorems with respect to the free convolution of probability measures. The concept of freeness is the key notion. It may be viewed (for non-commutative random variables) as a type of independence. Like in classical probability, in which the notion of independence leads to classical convolution, the notion of freeness gives rise to a binary operation for real measures: the free convolution. A lot of classic results for the addition and multiplication of independent random variables possess analog properties for this new theory, for example, the central limit theorem, the Lévy–Khintchine formula, the law of large numbers and others. For an introduction to these subjects, we refer to [
1]. In [
2], the authors provide the distributional model behavior of the sum of free random variables distributed in an identical manner. They explicitly describe the relation between the limiting laws for classical and free additive convolutions. On the other hand, for measures with bounded support, Tucci [
3] proved the limiting distribution for the free multiplicative law of large numbers. This result was extended in [
4] to measures with unbounded support. Continuing the study of limiting distributions in non-commutative probability, we provide in this article some new limiting laws related to the free multiplicative law of large numbers and involving free and Boolean additive convolutions. For the clarity of the presentation of our results, we need to first recall some concepts of importance in non-commutative probability.
Denote by (respectively, ) the set of probability measures on (respectively, ).
The Cauchy–Stieltjes transform
of
is defined, for
, as
where
denotes the support of the measure
.
The free additive convolution of
and
, denoted by
, is defined by
where the free cumulant transform,
, of
is given by
See [
5] for more details about the free cumulant transform.
A measure
is ⊞-infinitely divisible if for each
, there exists
such that
Denote by
the
t-fold free additive convolution of
with itself. This operation is well defined for all
, (see [
6]) and
A measure is ⊞-infinitely divisible if is well defined for all .
The Boolean additive convolution is another interesting convolution in the theory of non-commutative probability, see [
7]. For
,
, the Boolean additive convolution
is the probability measure defined by
where
denotes the Boolean cumulant transform of the measure
.
A measure
is ⊎-infinitely divisible if for each
, there exists
such that
Note that all measures
are ⊎-infinitely divisible, see [
7], Theorem 3.6.
We come now to the concept of free multiplicative convolution. For
,
, the
-transform is given by
The multiplication of
-transforms is also an
-transform. For
,
, the multiplicative free convolution
is defined by
. Multiplicative-free convolution powers
are well defined at least for all
(see [
8], Theorem 2.17) by
.
Now, we present the notion of the free multiplicative law of large numbers. More precisely, we have the following.
Theorem 1 ([
4], Theorem 2).
Let and let be the map . Set . If we consider and thus converges weakly to a probability measure denoted on . If σ is a Dirac measure on , then . Otherwise, is the unique probability measure on characterized by for all and . The support of is the closure of the interval where . In [
9] (Theorem 3.1), an interesting description is given for the free multiplicative law of large numbers
in terms of the pseudo-variance function
of the Cauchy–Stieltjes kernel (CSK) family generated by
(see the next section for CSK families and the corresponding pseudo-variance functions). A number of explicit examples are given for
, see [
9].
In this paper, we are interested in explicitly giving the limiting laws
,
and
(for
and
) by means of
. Here,
denotes the dilation of measure
by a number
and
. Some calculations of
,
and
are provided for measures
of importance in non-commutative probability.
Section 2 will describe some basic concepts about CSK families and their pseudo-variance functions.
Section 3 is devoted to the main result of this article and concludes with some examples.
2. Cauchy–Stieltjes Kernel Families
We introduce some preliminaries about CSK families and their corresponding pseudo-variance functions, see [
10] for more details.
Let
be a probability measure which is non-degenerate with support bounded from above. The transform
converges to ∀
with
. For
, let
The one-sided CSK family generated by
is the set
Denote by
the mean of
. From [
11], we know that the function
is strictly increasing on
. Denote by
the image of
by
. It is called the (one-sided) mean domain of
. This gives a re-parametrization (by the mean) of
. Consider
, the inverse of
, and for
, write
. Then, we obtain
Consider
(Here,
is interpreted as
∞.) It was proven in [
11] that
If the support of is bounded from below, one may similarly introduce the one-sided CSK family. Denote this family by . We have , where is either or with . The interval is the mean domain for with . If is compactly supported, then and the two-sided CSK family is .
The variance function is (see [
12])
If
does not have the first moment, all the laws that belong to the CSK family generated by
have infinite variance. The concept of the pseudo-variance function
was introduced in [
11]. It is defined by
If
is finite, then
exists and (see [
11])
Throughout the following two remarks, we recall some facts that will be used in the proof of the main result of the paper given by Theorem 2.
Remark 1. - (i)
Consider , where and , and let be the image of μ by φ. Then, for all x close enough to , we have (see [11]) - (ii)
According to [11] (Proposition 3.10), for all such that is defined and for all x close enough to , - (iii)
We know from [13] that for all and for all x close enough to , Furthermore, for all and for all x close enough to , we have
Remark 2. For such that is defined, there exists an injective analytic map , called subordination, such that , for . Furthermore,and , whereFor more details about the subordination function, see [14] (Theorem 2.5). 4. Conclusions
In classical probability, the law of large numbers for classical multiplicative convolution is deduced from the law for the classical additive convolution. This is not the case in non-commutative probability. The free additive law was demonstrated in [
16] for probability measures with bounded support and extended in [
17] to all probability measures with a first moment. The free multiplicative law was demonstrated in [
3] for measures with bounded support and extended in [
4] to measures with unbounded support. Contrary to the case of classical multiplicative convolution, the limiting measure for the free multiplicative law of large numbers is not a Dirac measure, except in the case where the original measure is a Dirac measure. In [
9], an interesting description is given for the free multiplicative law of large numbers by means of the pseudo-variance function. In this paper, we have explicitly demonstrated the law of large numbers of three types of non-commutative probability measures involving the free and the Boolean additive convolutions. The results are explained using several probability measures and may be useful for researchers in the field of non-commutative probability.