On a method of introducing free-infinitely divisible probability measures

Random integral mappings $I^{h,r}_{(a,b]}$ give isomorphisms between the sub-semigroups of the classical $(ID, \ast)$ and the free-infinite divisible $(ID,\boxplus)$ probability measures. This allows us to introduce new examples of such measures and their corresponding characteristic functionals.

Probability measures µ in (⋆) are called s-selfdecomosable ( "s", because of the shrinking operators U r .) They were introduced for measures on Hilbert spaces in Jurek (1977) and in Jurek (1981). Later on studied in Jurek (1984) and (1985). More recently, in Bradley and Jurek (2014), the Gaussian limit in (⋆) was proved in a case when the independence was replaced by some strong mixing conditions. On other hand, in Arzimendi and Hasebe (2014), Section 5, measures in class U have studied via the unimodality property of their Lévy spectral measures.
Replacing in (⋆) the U ′ r s by the linear dilations T r (x); = r x we get the Lévy class L of so called selfdecomposable distributions. In particular, we obtain stable distributions, when X ′ i s are also identically distributed. For purposes in this paper we need the descriptions of the classes L and U in terms the random integral representations like the ones in (4); cf. Jurek and Vervaat (1983) and Theorem 2.1 in Jurek (1984), respectively. 0. The isomorphism. Traditionally, let φ µ denotes the Fourier transform (the characteristic function) of a probability measure µ and let V ν denotes the Voiculescu transform of a probability measure ν (the definition is given in the subsection 1.2. below). Then for a class C of classical * -infinitely divisible probability measures we define its free ⊞ -infinitely divisible counter part C as follows: log φ µ (−v)e −tv dv, t > 0; for some µ ∈ C} (1) Conversely, for a class C of ⊞ -infinitely divisible measures we define its classical * -infinitely divisible counterpart C as follows: log φ µ (−v)e −tv dv = V ν (t), t > 0; for some ν ∈ C} (2) It is notably that above, and later on, we consider V ν (and Cauchy transforms) only on the imaginary axis. Still, it is sufficient to perform the explicit inverse procedures; cf. Section 1.3. below.
We illustrate the relation between classes C and C (in fact, an isomorphism) via examples and will prove among others that : for some constants a ∈ R, σ 2 ≥ 0 (variance) and a Lévy spectral measure M. The parameters a, σ 2 and the measure M correspond to µ = [a, σ 2 , M] in (1) and (2); for other details cf. Proposition 1.
The method (idea) of inserting the same characteristics ( parameters) into different integral kernels can be traced to Jurek and Vervaat (1983), p. 254, where it was used to describe the class L of selfdecomposable distributions. Namely, [a, R, M] L (the triple in the Lévy-Khintchine formula) was identified with [a, R, M]. For other such examples cf. Jurek (2011), an invited talk at 10th Vilnius Conference. Similarly, Bercovici and Pata (1999) introduced a bijection between the semigropus of classical and free infinite divisible probability measures. In this paper the identification is done on the level of Lévy exponents log φ µ (cummulants) and Voiculescu transforms V ν .
In this paper we show the explicit relation between V w (z) = 1/z and φ N (0,1) (t) = exp(−t 2 /2), where w is the Wigner semicircle distribution and N(0, 1) is the standard Gaussian measure.
1. The classical *and the ⊞-free infinite divisibility.
1.1. A probability measure µ is * -infinitely divisible (ID, * ) if for each natural n ≥ 2 there exists probability measure µ n such that µ * n n = µ. Equivalently, its characteristic function φ µ (Fourier transform) admits the following form (Lévy-Khinchine formula) and the triplet a ∈ R, σ 2 ≥ 0 (covariance) and a positive Borel measure M are uniquely determined by µ; in short we write µ = [a, σ 2 , M]. A sigma-finite measure M in (3) is finite on all open complements of zero and integrates |x| 2 in every finite neighborhood of zero. It is called the Lévy spectral measure of µ.
In recent year has been considerable interest in studying random integral representations of infinitely divisible probability measures or their Lévy measures M; cf. Jurek (2012) and references therein. Namely, for a continuous h and a monotone right continuous r,on an interval (a, b], one defines where L(X) denotes the probability distribution of X and Y ν is a cadlag Lévy process such that L(Y ν (1)) = ν.
In terms of characteristic functions (4) means that where the minus sign is for decreasing r and plus for increasing r; cf. Jurek and Vervaat (1983), Lemma 1.1 or Jurek (2007) (in the proof of Theorem 1) or Jurek (2012). Moreover, (I h,r (a,b] (ν)) denotes here the characteristic function of the probability measure I h,r (a,b] (ν). In (5), we may write For the purposes below we consider the following specific random integral mapping: from Jurek (2007), formula (17). There, it was done for any real separable Hilbert space. (In Barndorff-Nielsen and Thorbjornsen (2006), and in other works, the mapping (6) was denoted by the letter Υ and (originaly) was defined on the family of Lévy measures on a real line). The mapping K is an isomorphism between convolution semigroup ID and E (range of the mapping K). Moreover, if µ = [a, σ 2 , M] then from (3) we get cf. Jurek (2006), Corollary 5. For a general theory of the calculus on random integral mappings of the form (2) cf. Jurek (2012).
1.2. D. Voiculescu and others studying so called free-probability introduced new binary operations on probability measures and termed them accordingly free-convolutions; cf. Bercovici-Voiculescu (1993) and references therein. To recall the definition ⊞ convolution we need some auxiliary notions.
For a measure ν, its Cauchy transform is given as follows Furthermore, having G ν we define F ν (z) := 1/G ν (z) and then the Voiculescu transform as where a such region (called Stolz angle) exists and the inverse function is well defined on it; cf. Bercovici and Voiculescu (1993), Proposition 5.4 and Corollary 5.5.
The fundamental fact is that, for two measures ν 1 and ν 2 one has for a uniquely determined probability measure, denoted as ν 1 ⊞ ν 2 . This property allowed to introduce the notion of ⊞ free-infinite divisibility. For this new ⊞ infinite divisibility we have the following analog of the Lévy-Khintchine formula (3): for some uniquely determined real constant b and a finite Borel measure ρ.
Remark 1. Note that V ν : C + → C − is an analytic function and for the mappings Bercovici-Voiculescu (1993) or Barndorff-Nielsen (2006), Lemma 4.20. This is in contrast to characteristic functions of measures where we have φ Tcµ (t) = φ µ (ct), for all t ∈ R.
1.3. For some analogies and comparison below, let us recall from Jurek (2006) that the restricted versions of G ν and V ν are just those functions considered only on the imaginary axis. Then we have that where e · η means the probability distribution of a product of stochastically independent rv's: the standard exponential e and the variable η with probability distribution ν. The identity (11) means that we can retrieve a measure ν from the characteristic function φ e·η ; cf. Jurek (2006), the proof of Theorem 1, on p.189 and Examples on pp. 195-198. This is in sharp contrast with the classical Stieltjes inversion formula where one needs to know G ν in strips of complex plane; see it, for instance, in Nica and Speicher (2006), p. 31.
In fact, we have even more straightforward relation. Namely, cf. Jankowski and Jurek (2012), Proposition 1. Thus, restricted Cauchy transforms are just Laplace transforms of characteristic functions.
In the spirit of (11) and (12), instead of (10), let us introduce the restricted Voiculescu transform as From the inversion formula in Theorem 1 in Jankowski and Jurek (2012) and from (13), we have that (14) (See there also the comment (a paradigm) in the first paragraph in the introduction on p. 298.) In order to have (13) in a form more explicitly related to (7), let us define the new triple: shift a, the variance σ 2 and the Lévy spectral measure M, as follows: Then from (13), with some calculations, we get For computational details cf. Barndorff-Nielsen and Thorbjornsen (2006), Proposition 4.16, p. 105 1.4. The functions G ν (z) and V ν (z) are analytic in some complex domains and thus are uniquely determined by their values on imaginary axis (more generally, on subsets with limiting points in their domains).
[Note that for p(z) := iℑz and q(z) := z we have that p(it) = q(it) although they are different. Of course, p is not an analytic function! ]

1.5.
Because of (6), (7) and (16), here is the explicit relation (an isomorphism) between the free-⊞ and the classical - * infinite divisibility: Theorem 2. A probability measure ν is ⊞-infinitely divisible if and only if there exist a unique * -infinitely divisible probability measure µ such that Equivalently, we have that for ⊞-infinitely divisible ν its Voiculescu transform V ν is of the form for an uniquely determined * -infinitely divisible measure µ.
This is a rephrased version of Corollary 6 in Jurek (2007). Statements (17) and (18)  Also note that in both cases (17) and (18)  2. Examples of explicit relations between free ⊞ -and classical * -infinite divisible probability measures.
In the first three subsections, for a given class C classical * -infinitely divisible measures we identify its counterpart C of free ⊞ -infinitely divisible companions.

For the free ⊞ analog of s-selfdecomposable distributions we have
Proposition 1. A probability distribution u is free ⊞ s-selfdecomposable, in symbols, u ∈ (U, ⊞), if and only it there exist a unique µ = [a, σ 2 , M] ∈ (ID, * ) such that its Voilculescu transforms have representations s ds, x > 0, is the incomplete Euler gamma function.
Proof. Recall that λ is classical * -s-selfdecomposable, i.e., λ ∈ (U, * ) if and only if λ = I s, s (0,1] (µ) for some µ ∈ ID; cf. Jurek (1984), Theorem 2.1 for different characterizations of this class or Jurek (1985). Thus for w = 0, using (3), we get [The formula (20)  Let us define the (decreasing) time change r u (v) for v > 0 as follows Then taking into account (5) and putting r u into (21) to get and this completes the proof of the part (a).
For part (b), using (17), (3) and the first line in (20), after interchanging the order of integration, we get for t < 0, Substituting −1/t for t in (22) we arrive at which gives (b). Part (c) is an analytic extension of (b) and this completes a proof of Proposition 1.
Or equivalently, using the second line in (25) and (3) we get

Now substituting for s
and hence the equality (b). Part (c) is an analytic continuation of (b) and this concludes a proof of Proposition 2.
To see (b) note that therefore for e(δ 1 ) we conclude which proves that (unexpected ?) relation (27) between Poisson (discrete) and exponential (continuous) distributions.

2.3.
For free ⊞-stable distributions we have: Proposition 3. A measure ν is non-Gaussian free-⊞ stable if and only if for t > 0 its Voiculescu transform V ν should be such that it is EITHER where a ∈ R, C > 0, 0 < p < 1 or 1 < p < 2 and |β| ≤ 1, OR p = 1 and where 1 − γ = ∞ 0 w log w e −w dw (Euler constant γ ∼ 0.577). Proof. For classical * -stable measures from Meerscheart and Scheffler (2001), Theorem 7.3.5, p. 265 we have that µ is non-Gaussian * −stable iff and only if there exist C > 0, a ∈ R, 0 < p < 1, 1 < p < 2, −1 ≤ β ≤ 1 such that for each t ∈ R and for p = 1 we have where β := 2θ − 1 is the skewness parameter; 0 ≤ θ ≤ 1 is the probability of the positive tail of Lévy measure M of µ, that is, for r > 0, we have M(x > r) = θCr −p . And 1 − θ is the probability of the negative tail of M.
In order to get (28) one needs insert (30) into first equality in (18) and perform some easy calculations. Similarly, putting (31) into (18) and using the identity log i = iπ/2 one gets equality (29), which completes a proof of Proposition 3.
Remark 5. (i) In some papers and books often there is a small but essential error. Namely, in (30), there is β instead of (−β); cf. P. Hall (1981).
(ii) Note that the expressions in square brackets in (28) and (29) are identical, up to the sign, with those in Proposition 5.12 in Bercovici-Pata (1999). Also compare Biane's formulas for free-stable distributions in the Appendix there.
for some finite Borel measure m on the real line. Moreover, ν is free-infinitely divisible if and only if for some b, c ∈ R and finite Borel measure m on R.
Proof. Since log φ e(m) (t) = R (e itx − 1)m(dx), t ∈ R, therefore by (17) which completes a proof of (32). The remaining part is a consequence of above and Theorem 1.
[Also see pp. 203-206 in Nica-Speicher (2006) for the discussion of free compound Poisson distributions] 2.5. In this subsection, for a given three examples of ⊞-infinitely divisible measures we identify their classical * -infinitely divisible companions.
Example 1. The probability measure w such that V w (z) = 1 z , z = 0, is called free-Gaussian measure. Why such a term?
Note that from Theorem 2, we get (it)V w ( 1 it ) = −t 2 . On the other hand, taking standard normal distribution N(0, 1) for the measure µ we get So, it is right to call w an analogue of free Gaussian distribution. More importantly, w is a weak limit of free-analog of CLT and w is the standard Wigner's semicircle law with the density 1 2π √ 4 − x 2 1 [−2,2] (x), mean value zero and variance 1. [Using the inversion formula in (14) for V m we get b = 0 and ρ = δ 0 in (10).] Example 2. The probability measure c with V c (z) = −i is called free-Cauchy distribution. Why such a term?
From Theorem 2, (it)V c ( 1 it ) = t. On the other hand, taking the standard Cauchy distribution (with the probability density 1 π 1 1+x 2 ) for the measure µ we get ∞ 0 log φ µ (ts)e −s ds = −|t| ∞ 0 se −s ds = t = (it)V c ( 1 it ), for t < 0, so it justifies the term free -Cauchy measure. In fact, we have that Remark 6. The measure c is the standard Cauchy distribution. To see that we use the inversion procedure from (14). Thus b = 0, c(R) = 1 and ∞ 0 φ c (r)e −wr dr = 1 w + 1 ; i. e., φ c (r) = e −r , for r > 0.
Consequently, φ c (r) = e −|r| , for r ∈ R and hence c is the standard Cauchy probability measure.
Example 3. The probability measure m such that V m (z) = z z−1 is called free-Poisson distribution. Why?