Free infinite divisibility for generalized power distributions with free Poisson term

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1 Free infinite divisibility for generalized power distributions with free Poisson term Junki Morishita and Yuki Ueda Abstract arxiv: v [math.pr] 5 Jun 08 We study free infinite divisibility for the class of generalized power distributions with free Poisson term by using a complex analytic technique and a calculation for the free cumulants and Hankel determinants. In particular, our main result implies that i) if X follows the free Generalized Inverse Gaussian distribution, then X r follows the classes UI Univalent Inverse Cauchy transform) and FR Free Regular) when r, ii) if S follows the standard semicircle law and u, then S + u) r follows the classes UI and FR when r, and iii) if B p follows the beta distribution with parameters p and 3/, then a) B r p follows the classes UI and FR when r and 0 < p /, and b) Br p follows the classes UI and FR when r and p > /. Introduction In classical probability theory, many people studied the infinitel divisible distributions as laws of Lévy processes and as the general class of important distributions e.g. normal, Cauchy, gamma, t-student distributions, etc) for probability theory, mathematical statistics and finance models. A probability measure µ on R is said to be infinitely divisible if for each n N there exists a probability measure ρ n on R such that µ = ρ n n where the operation is the classical convolution. We recall that a probability measure µ on R is infinitely divisible if and only if its characteristic function ˆµ has the Lévy-Khintchine representation: ˆµu) = exp iηu au + e iut iut [,] t) ) ) dνt), u R,.) R where η R, a 0 called the gaussian component) and ν is a Lévy measure on R, that is, ν{0}) = 0 and R t )dνt) <. The triplet η,a,ν) is uniquely determined and is called the characteristic triplet. For example, the normal distribution Nm,σ ) m R, σ > 0) has the characteristic triplet m,σ,0), so that it is infinitely divisible). In general, it is very difficult to find the characteristic triplet of probability measures, in other words, to write the characteristic functions in the Lévy-Khintchine representation). Many people found and studied subclasses of probability measures as criteria for infinite divisibility. In particular, subclasses of infinitely divisible distributions were studied to understand infinitely divisible distributions which are preserved by powers, products and quotients of independent random variables. We say that a random variable X follows the class of mixture of exponential distributions ME) if X is of the form EZ where two random variables E, Z are independent, E follows an exponential distribution and Z is positive, that is, µ ME means that µ = π ρ for some probability measure ρ on the positive real line and π is an exponential distribution. By the Goldie-Steutel theorem, the class ME is a subclass of infinitely divisible distributions see [8, 5]). The class is important to understand infinitely divisible distributions which is preserved by powers and products of independent random variables, that is, if X follows the class ME then so

2 is X r when r, and if X, Y are independent and follow the class ME then so is XY. As one of the most important subclasses of infinitely divisible distributions, the class GGC generalized gamma convolution) was introduced by Thorin see [7, 8]). Bondesson see [5]) proved that the subclass HCMhyperbolically completely monotone) of the class GGC is preserved by powers if X follows the class HCM then so is X r if r ), products and quotients of independent random variables. Moreover Bondesson see [6]) proved that if X,Y follow the GGC and X,Y are independent then so is XY. However, we are not sure about whether the class is preserved by powers of random variables. In free probability theory, we have the corresponding problems for free infinitely divisible distributions which will be defined in Section.. As one of the most important subclass of free infinitely divisible distributions, we have the class of Free Regular distributions FR) and the class of Univalent Inverse Cauchy transform U I). A probability measure on the positive real line is said to be in the class FR if it is the law of a nonnegative free Lévy process. The class FR is closed with respect to the product XY X for free independent non-commutative) random variables X and Y are positive. The details for the class FR are studied by Arizmendi, Hasebe, Sakuma, and Pérez-Abreu, etc see [, 0, 4]). The class UI was introduced by Arizmendi and Hasebe see []) as a subclass of free infinitely divisible distributions. We are not sure about whether the classes U I and FR are closed with respect to powers, products and quotients of free independent random variables. Hasebe [0]) studied the powers and products for the classes U I and FR by using complex analytic techniques. In this paper, we studied the classes UI and FR, specifically, we proved that if a random variable X follows the class of the generalized power distributions with free Poisson term GPFP) X b x)x a) x N k= α k x l k a,b)x)dx,.) where 0 < a < b, N N, α = α,...,α N ) 0, ) N satisfying that the right hand side of.) becomes a probability density function, and l R N < := {l,...,l N ) R N : l < < l N }, then i) X r follows the class UI FR if r in the case when l [t,t+] N < for some t 0, and ii) X r follows the class UI FR if r in the case when l [0,] N <. By the way, due to a calculation of Hankel determinants for the free cumulants, we proved that iii) there exist five parameters a,b,n,α,l) with 0 < a < b, N N, α 0, ) N, l [t,t+] N < for some t 0 such that X GPFPa,b,N,α,l), but X does not follow the class FID, and iv) there exist five parameters a,b,n,α,l) with 0 l < < l k < l k < < l N, N a,b,α are usual parameters) for some < k N, such that X does not follow the class FID. The class of GPFP contains free Poisson distributions which have no atoms at 0, free positive stable laws with index /, the class of free Generalized Inverse Gaussian distributions fgig), shifted semicircle laws and a partial class of beta distributions. From our result, we obtain some properties for powers of random variables which follow the fgig distributions, shifted semicircle laws, and beta distributions. This paper consists of 5 sections. In section, we introduce the free infinitely divisible distributions and theclasses UI and FR. Insection 3, we prove that probability measures whose pdfsatisfieds some assumptions are in the class UI FR by using complex analytic techniques. In section 4, we prove our results i) and ii) as above by using results in section 3. In section 5, we prove our results iii) and iv) as above by using the Hankel determinants for free cumulants.

3 Preliminaries. Notations In this paper, we use the following notations. Let X be a non-commutative) random variable and µ a probability measure on R. The notation X µ means that X follows µ. The domain C + resp. C ) means the complex upper resp. lower) half plane. Let I be an interval in R. The notation I ±i0 means the set {x±i0 : x I}. The complex function z p p R) means the principal value defined on C\,0]. The complex function argz) means the argument of z defined on C\,0] taking values in π,π). Let N be a positive integer and I an interval in R. The set I N < means the cone of I N, that is, the set of all elements l,...,l N ) I N satisfying l < < l N.. Analytic tools in free probability theory Let µ be a probability measure on R. The Cauchy transform of µ is defined by G µ z) := dµx), z C\suppµ)..) z x R In particular, if X µ then we sometimes write G X as the Cauchy transform of µ. The function G µ is analytic on the complex upper half plane C +. Bercovici and Voiculescu [4, Proposition 5.4]) proved that for any γ > 0 there exist λ,m,δ > 0 such that G µ is univalent in the truncated cone Γ λ,m := {z C + : λ Rez) < Imz),Imz) > M},.) and the image G µ Γ λ,m ) contains the triangular domain Λ γ,δ := {z C : γ Rez) < Imz),Imz) > δ}..3) Therefore the right inverse function G µ exists on Λ γ,δ. We define the Voiculescu transform of µ by ) φ µ z) := G µ z z, z Λ γ,δ..4) It was proved in [4] historically, see also [, 9]) that for any probability measures µ and ν on R, there exists a unique probability measure λ on R such that φ λ z) = φ µ z)+φ ν z),.5) for all z in the intersection of the domains of the three transforms. We write λ := µ ν and call such λ the additive free convolution for short, free convolution) of µ and ν. The operation is the free analogue of the classical convolution. The free cumulant transform of µ is defined by ) C µ z) := zφ µ, z Λ γ,δ..6) z In particular, if X µ then we sometimes write C X as the free cumulant transform of µ. By definition, we have that C µ ν z) = C µ z) +C ν z) for all z in the intersection of the domains of the three transforms. The transform is the free analogue of log ˆµ. Aprobability measureµonris saidtobefree infinitely divisibleifforeach n Nthere exists a probability measure ρ n on R such that µ = ρ n n. In particular, if X µ and µ is 3

4 free infinitely divisible then we write X FID. Bercovici and Voiculescu [4, Theorem 5.0]) proved that µ is free infinitely divisible if and only if the Voiculescu transform φ µ has analytic continuation to a map from C + taking values in C R, and therefore it has the Pick-Nevanlinna representation φ µ z) = γ + R +xz z x dσx), z C+,.7) for some γ R and a nonnegative finite measure σ on R. The pair γ,σ) is uniquely determined and called the free generating pair. In this case, we can rewrite the free cumulant transform C µ as ) C µ z) = ηz +az + R tz tz [,]t) dνt), z C,.8) where η R, a 0 called the semicircular component) and ν is a nonnegative measure on R satisfying ν{0}) = 0 and R t )dνt) < see [3, Proposition 5.]). The measure ν is called the free Lévy measure of µ. The representation.8) is called the free Lévy-Khintchine representation and it is the free analogue of classical Lévy-Khintchine representation of the Fourier transform ˆµ. The triplet η, a, ν) is uniquely determined by µ and called the free characteristic triplet. Moreover the relation between the free generating pair γ,σ) and the free characteristic triplet η,a,ν) is given by a = σ{0}), dνt) = +t t R\{0} t)dσt), η =γ + t [,] t) ) +t dνt). R.9) Example.. ) The semicircle distribution Sm,σ ) is a probability measure with the following probability density function see Figure ): πσ 4σ x m) m σ,m+σ) x), m R,σ > 0..0) It is known for example, see [3]) that C Sm,σ )z) = mz +σ z, z C..) ThereforeSm,σ )isfreeinfinitelydivisibleandhasthefreecharacteristictripletm,σ,0). This distribution appears as the limit of eigenvalue distributions of Wigner matrices as the size of the random matrices goes to infinity. ) The free Poisson distributionor Marchenko-Pastur distribution) fpp, θ) is a probability measure given by θ p+) x ) x θ p ) ) max{ p,0}δ 0 + πx θ p ),θ p+) )x)dx.) for p,θ > 0 see Figure ). It has a free cumulant transform C fpp,θ) z) = pθz θz..3) Therefore fpp, θ) is free infinitely divisible. This distribution appears as the limit of eigenvalue distributions of Wishart matrices as the size of the random matrices goes to infinity. 4

5 Figure : Semicircle distribution S0, ) Figure : Free Poisson Marchenko- Pastur) distribution fp4, ).3 Subclass of free infinitely divisible distributions In section., we introduced the free infinitely divisible distributions. Free infinite divisibility for probability measures is equivalent to that its free cumulant transform has the free Lévy-Khintchine representation.8)or its Voiculescu transform has the Pick-Nevanlinna representation). However, in general, it is very difficult to find the free characteristic triplet or free generating pair) of probability measures to check free infinite divisibility for probability measures. Many people studied conditions of free infinite divisibility. Pérez-Abreu and Sakuma [4]) introduced the class of free regular probability measures in terms Bercovici-Pata bijection. Definition.. A probability measure µ on [0, ) is said to be in the class FR Free Regular) if µ is free infinitely divisible and µ t is a probability measure on [0, ) for all t > 0. In particular, if X µ and µ FR, then we write X FR. In [], the class FR was developed as a characterization of nonnegative free Lévy processes. Moreover the paper [] gave us the following properties for the class FR. Lemma.3. ) The class FR is closed with respect to weak convergence. ) Suppose that µ is free infinitely divisible distribution which has the free generating pair γ,σ). Then we have that µ FR if and only if σ,0)) = 0 and φ µ 0) 0. Arizmendi and Hasebe [, Definition 5.]) defined a special class of probability measures as a criterion for free infinite divisibility. Definition.4. A probability measure µ on R is said to be in the class UI Univalent Inverse Cauchy transform) if the right inverse function G µ, originally defined in a triangular domain Λ γ,δ, has univalent analytic continuation to C. In particular, if X µ and µ UI, then we write X UI. In [, Proposition 5. and p.763], the class U I has the following properties. Lemma.5. µ UI implies that µ is free infinitely divisible, and the class UI is closed with respect to weak convergence. Example.6. ) The semicircle distribution Sm,σ ) is in the class UI. The free Poisson distribution fpp,θ) is in the class UI FR see [0, Section.3 ),)]). ) The beta distribution β p,q is the probability measure with pdf Bp,q) xp x) q 0,) x), p,q > 0..4) The beta distribution β p,q is in the class UI if i) p,q 3/, ii) 0 < p /, q 3/ or iii) 0 < q /, p 3/ see [9, Theorem.] and [0, Theorem 3.4]). 5

6 Remark.7. Hasebe [0]) studied the free infinite divisibility for powers of random variables which follow the class U I FR by using complex analytic techniques. ) Let X be a random variable which follows the free Poisson distribution fpp) := fpp,). Then we have that X r UI FR if i) p, r,0] [, ) or ii) 0 < p <, r see [0, Theorem 3.5]). ) Let X be a random variable which follows the beta distribution β p,q. Then X r UI FR if r, q [3/,5/], 0 < p r and q 3/)r p + q r see [0, Theorem 3.4]). 3) Let X be a random variable which follows the HCM class whose pdf is of the form C x p exp log 0, ) ) Γdt) t+x ),.5) where Γ is a finite measure on 0, ), 0 < p / and 0 < p + Γ0, )) /. If W 0 is a random variable independent of X and r then WX r UI FR. In particular, if W > 0 a.s. then WX r UI FR if r see [0, Theorem 3.]). 3 UI and FR property In this section, G µ z) denotes the Cauchy transform of a complex measure µ on R defined in.). In [9, Proposition 4.], Hasebe proved that the Cauchy transform has an analytic continuation to a domain C + suppµ) D, for some subdomain D of C in the case when a pdf of µ has some analytic properties. Lemma 3.. Consider 0 < a < b. Let a,b) be an interval of R. Suppose that f is analytic in a neighborhood of a,b) {z C : argz) θ,0)} for some 0 < θ π and that f is integrable on a,b) with respect to the Lebesgue measure. We define the complex measure σdx) := fx) a,b) x)dx. Then the Cauchy transform G σ := G σ C + defined on C + has an analytic continuation to C + a,b) {z C : argz) θ,0)}, which we denote by the same symbol G σ, and G σ z) = G σ z) πifz), z {z C : argz) θ,0)}. 3.) We prove that a probability measure whose pdf satisfies the following properties is in the class UI FR and therefore the measure is free infinitely divisible. Theorem 3.. Suppose that 0 < a < b. Let µ be a probability measure on a,b) which has a pdf fx) where f is real analytic, positive and there exists θ 0,π] such that A) f analytically extends to a function also denote by f) defined in {z C \ {0} : argz) θ,0)} a,b). Moreover f extends to a continuous function on {z C\{0} : argz) [ θ,0]}; A) Refx i0)) = 0 for all 0 < x < a and x > b; A3) There exist α > 0 and 0 < l < π θ θ such that fz) = iα z+l+o)), as z 0, argz) θ,0); 3.) A4) Refue iθ )) 0 if θ = π, then Ref u)) 0) for all u > 0; A5) lim z,argz) θ,0) fz) = 0. By Lemma 3. and our assumption A), the Cauchy transform Gz) := G µ z) of µ has analytic continuation to C + a,b) {z C : argz) θ,0)}, and we denote by the same symbol G. Then A6) Gz) can extend to a continuous function on C + [a,b] {z C\{0} : argz) [ θ,0)},a] [b, )+i0) [0,a] [b, ) i0). Then µ UI FR. 6

7 Proof. We first prove that µ is in UI. We define the following 8 lines and curves see Figure 3). In the following η > 0 is supposed to be large and δ > 0 is supposed to be small. c is the real line segment from η +i0 to a+i0; c is the real line segment from a i0 to δ i0; c 3 is the clockwise circle δe iψ where ψ starts from 0 and ends with θ; c 4 is the line segment from δe iθ +0 to ηe iθ +0; c 5 is the counterclockwise circle ηe iψ where ψ starts from θ and ends with 0; c 6 is the real line segment from η i0 to b i0; c 7 is the real line segment form b+i0 to η +i0; c 8 is the counterclockwise circle ηe iψ where ψ starts from 0 and ends with π. Figure 3: The curves c k Figure 4: The curves g k Definethe curves g k := Gc k ) for all k =,...,8 see Figure4). Itis easy to prove that g is the negative real line, g 7 is the positive real line and g 8 is the small clockwise circle whose argument starts from 0 and ends with π. By our assumption A6), the curves g g and g 6 g 7 are continuous since so are c c and c 6 c 7. By our assumption A), we have that Refx i0)) = 0 for all 0 < x < a and x > b. Hence ImGx i0)) = 0 for all 0 < x < a, x > b, and therefore g and g 6 are real lines. By our assumption A4), we have ImGue iθ )) = Im Gue iθ )) πrefue iθ )) > 0, u > ) Therefore g 4 is contained in C +. Let ε > 0 be supposed to be small. Since our assumption A3) and the condition 3.) hold and Gz) = o/z) as z 0, z C, we have that Gz) = πα z+l+o)), as z 0, argz) θ,0). 3.4) By asymptotics 3.4), we can take δ 0,παε) +l ) small enough such that uniformly on ψ θ,0). Hence we have Gδe iψ )+παδe iψ ) +l) < παδ +l), 3.5) Gδe iψ ) > παδ +l) > ε, 3.6) and therefore the distance between the curve g 3 and 0 is larger than ε. Since we have that Gδe iψ ) = πα e π ψ+l))i, and the argument ψ starts from 0 and ends with θ, δ +l 7

8 the g 3 is the large counterclockwise circle whose argument starts from π and ends with π +θ+l) < π by A3)). Note that Gz) = z +o)) as z, z C. By our assumption A5), we have that fz) 0 as z, argz) θ,0), and therefore Gz) Gz) +π fz) 0, 3.7) as z, argz) θ,0). Hence there exists η > 0 large enough such that z η implies that Gz) < ε, that is, g 5 is contained in a small circle with radius ε > 0. Therefore every point of D ε := {z C : ε < z < ε } is surrounded by the closed curve g g 8 exactly once. By the argument principle, for any w D ε, there exists only one element z in the bounded domain surrounded by the closed curve c c 8, such that Gz) = w. Therefore we can define a right inverse function G on D ε. By the open mapping theorem, the function G is univalent and analytic in D ε. By analytic continuation, we can define a univalent right inverse function as the same symbol G ) in C. By the identity theorem, the right inverse function G, originally defined in some triangular domain Λ γ,δ has a univalent analytic continuation to our function G on C. Hence we can conclude that µ UI. Next, we prove that µ is in FR. Define two lines c := t + i0) t,a] and c = a t i0) t [0,a]. By the definition of G, we can conclude that for any t,a) there exists δ > 0 such that Gt + iy) C for all y 0,δ ). Since G is univalent on the domain G C ) and the assumption A) holds, we can conclude that for any t 0,a) there exists δ > 0 such that Ga t iy) C for all y 0,δ ). Note that Gc ) = [α,0) and Gc ) =,α], where α := Ga 0) < 0. Therefore we have that G,α] i0) = c, G [α,0) i0) = c. 3.8) Hence we have that Imφx + i0)) = 0 for all x < 0 see.4) for the definition of Voiculescu transform φ). By Stieltjes inversion formula, the free Lévy measure has no mass on,0). The formula 3.4) implies that φ 0) = G i0) = 0. By Lemma.3 ) and the relation.9), we conclude that µ FR. 4 Free infinite divisibility for generalized power distributions with free Poisson term In[0]orseeExample.63)), ifarandomvariablex followsthefreepoissondistribution Marchenko-Pastur distribution) fpp), then X r follows UI FR, where p > 0 and r R have some properties. We prove that X follows the generalized power distributions with free Poisson term depend on five parameters a,b,n,α,l), then X r follows UI FR when l [t,t + ] N < for some t 0 and r in particular l [0,] N <, r ). The class of these distributions is the generalization of the free Poisson distributions and the free Generalized Inverse Gaussian fgig) distributions See Example 4.4). Theorem 4.. Let A,φ) be a C -probability space. Consider 0 < a < b. Suppose that a random variable X A follows the generalized power distributions with free Poisson term for short, GPFP distributions), that is, X b x)x a) x N k= α k x l k a,b)x)dx =: f a,b,n,α,l x)dx =: GPFPa, b, N, α, l), 4.) wheren N,α = α,...,α N ) 0, ) N with b a f a,b,n,α,lx)dx = andl = l,...,l N ) R N <. If l [t,t+]n < for some t 0, then Xr UI FR if r. 8

9 Proof. Fix a number N N. Consider first that r > and l = l,...,l N ) [t,t +] N < satisfies t < l < < l N < t + for some t 0. Assume that [t] = n for some n N. Define l k := l k [t] = l k n ) 0,) for all k =,...,N. By a calculation for φfx r )) for all Borel measurable functions f on R), we have X r s B s x s )x s A s ) x N k= α k x sn + l k ) A,B)x)dx =: hx)dx, 4.) where s = r 0,), A = ar and B = b r. We define a function h k x) as the k-th term of hx). Set n 0 := sn ) +. Note that sn + l k ) = n 0 + l k s for each k =,...,N. Consider θ = π n 0. Let G = G X r be the Cauchy transform of X r. We prove that θ, h and G satisfy the assumptions from A) to A6) in Theorem 3.. Suppose that z C\{0} and argz) θ,0). By a similar proof of [0, Theorem 3.5], we have that argb s z s )z s A s )) π,π). Moreover we also have that argz sn + l k ) ) θsn + l k ),0) π,0). Thus every h k analytically extends to a function defined in {z C \ {0} : argz) θ,0)} A,B), so that the assumption A) holds. For 0 < x < A we have h k x i0) = sα k A s x s )B s x s ) i. 4.3) x +sn + l k ) For x > B we have h k x i0) = sα k x s A s )x s B s ) i. 4.4) x +sn + l k ) Therefore Reh k x i0)) = 0 for all 0 < x < A and x > B, so that the assumption A) holds. For each k =,,N, we have h k z) = isα k AB) s +o)), as z 0,argz) θ,0). 4.5) z +sn + l k ) Moreover we have that 0 < sn + l k ) < π θ θ. Therefore the assumption A3) holds. Consider first 0 < θ < π equivalently, n > ). For all k =,...,N, u > 0 we have ) argh k ue iθ sα k B )) = arg s u s e isθ )u s e isθ A s ) u +sn + l k ) e iθ+sn + l k )) π +θ+sn + l k )), π sθ ) +θ+sn + l k )) = π +θn 0 +s l k ), π sθ ) +θn 0 +s l k ) 4.6) π = +θs l k, 3 ) π θs l k ) π, 3 ) π, since we have ) arg B s u s e isθ )u s e isθ A s ) π, π sθ ). 4.7) Therefore Reh k ue iθ )) 0 for all u > 0. When θ = π equivalently, n = ), for all 9

10 k =,...,N, u < 0 we have that ) sα k B argh k u)) = arg s u i0) s )u i0) s A s ) u +sn + l k ) ) sα k B = arg s u i0) s )u i0) s A s ) u +s l ke iπ+s l k ) π +π+s l k ), s ) π +π+s l k ) π = +πs l k, 3 ) π πs l k ) π, 3 ) π, 4.8) since ) arg B s u i0) s )u i0) s A s ) π, s ) π. 4.9) Therefore Reh k u)) 0 for all u < 0. Hence we can conclude that the assumption A4) holds. Finally, we have h k z) isα k z +sn + l k ), as z,argz) θ,0). 4.0) Hence lim z,argz) θ,0) h k z) = 0, so that the assumption A5) holds. Finally, we can conclude that the function h satisfies from A) to A5) since so is every function f k, k =,...,N). By taking α = 3, x 0 = A and x 0 = B in [9, Theorem )], the Cauchy transform G satisfies the assumption A6). By Theorem 3., we have that X r UI FR. Bythew-closedness ofui FRseeLemmas.3and.5), wehave thatx r UI FR even if r and l [t,t+] N < satisfies that t l < < l N t+ for t 0. Corollary 4.. Let A,φ) be a C -probability space. If X GPFPa,b,N,α,l) and l [0,] N < then Xr UI FR if r. Proof. Suppose that r. By a calculation for φfx r )) f: a Borel measurable function on R), we have X r t AB) t B t x t )x t A t ) x N k= α k x t l k) A,B)x)dx := px)dx, 4.) where t := r 0,), A := b /t and B := a /t. We set θ = π and a function p k x) as the k-th term of px). By a similar proof of Theorem 4., we have that θ, p k, p, and G = G X r satisfy the assumptions from A) to A6) in Theorem 3.. Therefore X r UI FR if r. Finally, we have that X GPFPa,b,N,α,l) and l [0,] N < implies that Xr UI FR if r by Theorem 4. and Corollary 4., but we will find a counterexample in the case when X GPFPa,b,N,α,l) and l [t,t+] N < for some t 0, by calculating free cumulants and Hankel determinants in section 5). Remark 4.3. ConsiderarandomvariableX whichsatisfiesthatx GPFPa, b, N, α, l) where α = α,...,α N ) and l = l,...,l N ). By a similar calculation in Corollary 4., we have that X GPFP b, ) a, N, α N ab,...,α ab), ln,..., l ). 4.) 0

11 We give five examples of GPFP distributions. Example 4.4. ) We put p >, a = p ) and b = p+), N =, α = π, l = 0. Then we have that GPFP a = p ), b = p+), N =, α = ) π, l = 0 p+) x ) x p ) ) 4.3) = πx p ), p+) )x)dx = fpp). In this case, the GPFP distribution corresponds to the free Poisson distributionmarchenko- Pastur distribution). By Theorem 4. and Corollary 4., we have that X fpp) implies that X r UI FR if r, but the result in [0] is stronger than this one). ) Consider n N, b > 0. Define the following constant number: b x /n)b x) cn,b) := dx). 4.4) πx n Note that lim n cn,b) = 4 b for each b > 0. By Theorem 4. and Corollary 4., for all n N we have that ) cn,b) b S n,b GPFP, n, N =, α = b π n, l = 4.5) b x /b)n x) = cn,b) n πx /b,n) x)dx, implies that Sn,b r UI FR for all r. Let S b be a random variable such that S b 4 bx b πx /b, ) x)dx. 4.6) In particular, if b = 4, then the probability measure in 4.6), corresponds to the free positive stable law fs / with an index /. It is easy to check that φsn,b r ) φsr b ) as n for all r and b > 0. Therefore the w-closedness of UI FR implies that Sb r UI FR for all r. By a similar proof, we have that Sr b UI FR for all r. In particular, we have that S4 fp) see the diagram 4.7)). S n,4 GPFP /4, n,, cn,4) π X S n,4 GPFP /n, 4,, cn,4) π, 0 ) ) n 4/n, S 4 fs / X n S 4 fp) 4.7) ) 3) Consider λ R. We put N =, α = α π, α π and l = 0,), where 0 < a < b. are ab the solution of { a+b λ+α ab α ab = 0 +λ+ α a+b 4.8) ab α = 0. Then we have GPFP a, b, N =, α = x a)b x) = πx α ) ), l = 0,) π, α π ab α + α ) a,b) x)dx =: fgigα,α,λ). abx 4.9)

12 In this case, the GPFP distribution corresponds to the free Generalized Inverse Gaussian fgig) distribution. This distribution was studied by [] free version of the generalized inverse gaussian distributions, free selfdecomposability, free regularity and unimodality) and [6] free version of Matsumoto-Yor property and R-transform of fgig distribution). Moreover X fgigα,α,λ) implies that X r UI FR if r by Theorem 4. and Corollary 4.. 4) Let 0 < a < b. We put N = and l =. Then we have GPFP a, b, N =, α = Cπ ) b a, l = = C ) x a+b ) π a,b)x)dx, 4.0) where C > 0 is a constant such that the function 4.0) becomes a pdf. If a random variable X follows the probability measure 4.0), then Remark 4.3 yields that X GPFP b, ) a,, C ab,. 4.) π By Theorem 4., we have that X r = X ) r UI FR for r. Suppose that S S0,) and u >. Then we have that S +u GPFP a = u, b = u+, N =, α = ) π, l =. 4.) See Figure 5 of the GPFP distribution in the case when u = 3. For the above reason, we have that S +u) s UI FR for all s. By the w-closedness of UI FR, we can also conclude that S+) r UI FR for all r. However, since S+u) FID for u 0 from the paper [7], we can conclude that if X GPFPa = u, b = u+, N =, α = π, l = ) then X FID for all u > Figure 5: GPFP a =, b = 5, N =, α = π, l = ) 5) Consider n N and l < /. Define the following constant number: αn,l) := B l, 3 ) n ) x)x /n) dx. 4.3) Note that lim n αn,l) = for each l < /. For all n N and l < /, we consider that ) αn,l) B n,l GPFP,, N =, n B/ l,3/), l. 4.4) x +l

13 By Remark 4.3, we have that B n,l GPFP, n,, ) n αn, l) B/ l,3/), l. 4.5) By Theorem 4. andcorollary 4., wehave that i) if0 l < / then Bn,l r UI FRfor r, and ii) if l < 0 then the condition 4.5) implies that B r n,l = B n,r) r UI FR for r. For each l < /, we consider a random variable B l which follows the beta distribution β / l,3/. Recall that the beta distributions β p,q are defined in Example.6 ). It is easy to check that for all l < /, we have that φbn,l r ) φbr l ) as n. By the w-closedness of UI FR, we have that i) if 0 l < / then Bl r UI FR for r, and ii) if l < 0 then Bl r UI FR for r. 5 Non free infinite divisibility for GPFP distributions Let A,φ) be a C -probability space and X A a random variable. The n-th moment m n = m n X) of X is defined by the value φx n ). In particular, if X µ, we write m n µ) as the n-th moment of X or µ). The n-th free cumulant κ n = κ n X) of X is defined as the n-th coefficient of power series of the free cumulant transform C X z). If X µ, we write κ n µ) as the n-th free cumulant of X or µ). We have moment-cumulant formula as follows κ n = m V )µπ, n ), n N, 5.) π NCn) V π where NCn) is the set of all non-crossing partitions of the finite set {,...,n}, the symbol n isthenon-crossingpartitionwhichhastheblock{,...,n}, thefunctionµπ,σ) π σ in NCn)) is the Möbius function of NCn), and V is the number of elements in a block V of π for detail, see [3]). By the moment-cumulant formula, we can compute free cumulants by using the moments. For example we have that κ = m, κ = m m, κ 3 = m 3 3m m +m 3, κ 4 = m 4 4m m 3 m +0m m 5m 4, κ 5 = m 5 5m m 4 5m m 3 +5m m 3 +5m m 35m3 m +4m 5. 5.) The n-th free cumulant of fpp) is given by κ n fpp)) = p for all n N see Example. )). Then the formula 5.) yields that m fpp)) = p, m fpp)) = pp+), m 3 fpp)) = pp +3p+), m 4 fpp)) = pp 3 +6p +6p+), m 5 fpp)) = pp 4 +0p 3 +0p +0p+). 5.3) We define moments with degree of a complex number. In particular, we need to study the moments of fpp) with degree of a complex number: m s fpp)) = This is analytic as a function of s C. 3 0 x s fpp)dx), s C, p >. 5.4)

14 Lemma 5.. For s C and p >, we have that Proof. Consider s C and p >. Then we have that m s fpp)) = m s fpp)). 5.5) p ) s m s fpp)) = p+) p ) x s p ) = p ) x s p+) = p+) x ) x p ) ) dx π ) x p ) p+) p ) s x s p ) = m s fpp)) p ) s, x p+) ) dx πx p+) x ) x p ) ) dx π 5.6) where the second equality holds by changing the variables from x to x and the third equality holds by changing the variables from x to x p ). By Lemma 5.4, we have that for all p >, m fpp)) = m 0fpp)) p m fpp)) = m fpp)) p ) 3 = = p, p p ) ) We study a condition for the following measures to become a GPFP distribution. Lemma 5.. Let p > and α,α > 0. ) The following measure p+) x ) x p ) ) πx α + α x ) p ), p+) )x)dx, 5.8) is a probability measure on the positive real line if and only if α + p p ) 3 α =. ) The following measure x ) p+) p ) x πx α + α x ) ) / p+),/ p ) )x)dx, 5.9) is a probability measure on the positive real line if and only if α +α p = p. Proof. ) Let µ p,α,α be the measure 5.8) on the positive real line. It is a probability measure if and only if = p+) p ) µ p,α,α dx) = α m 0 fpp))+α m fpp)). 5.0) The calculation 5.7) implies our conclusion. 4

15 ) Assume that the measure 5.9) is a probability measure. Suppose that X is a random variable which follows the probability measure 5.9). Then we have x ) p ) p+) x ) X p α +α x) p ), p+) )x)dx. πx 5.) By our assumption the measure is also a probability measure. Equivalently, we have = p α m 0 fpp))+α m fpp))) = p α +α p). 5.) Therefore the relation α +α p = p holds. The inverse implication is clear. As one of criteria for non free infinite divisibility, we have the method of Hankel determinants of a sequence of free cumulants. First we find an example of parameters such that X GPFPa,b,N,α,l), l [t,t+] N < for some t 0 and X FID. Theorem 5.3. There exist 5 parameters a,b,n,α,l) with l [t,t+] N < for some t 0, such that X GPFPa,b,N,α,l), but X FID Proof. We consider a random variable X satisfying X GPFP a = +), b = ), N =, α = α π, α ) ), l =,) π ) ) x +) x ) α = πx x + α ) x / +),/ ) ) x)dx, 5.3) where α,α > 0 satisfy the relation α +α = 0 < α < /) by Lemma 5. ). By Remark 4.3, we have X GPFP ), +) α,, π, α ) ),,0) π x ) ) +) x ) 5.4) = α +α x) πx ), +) ) x)dx. The relation between α and α and the above formulas 5.3) yield that m X ) = α )m fp))+α m fp)) = +α ), m X ) = α )m fp))+α m 3 fp)) = 3+5α ), m 3 X ) = α )m 3 fp))+α m 4 fp)) = +3α ), m 4 X ) = α )m 4 fp))+α m 5 fp)) = 45+07α ). 5.5) The formula 5.) and the above calculations 5.5) follow that κ := κ X ) = α α ), κ 3 := κ 3 X ) = 8α 3 6α α +), κ 4 := κ 4 X ) = α )0α 3 0α 3α +), 5.6) where κ n is the n-th free cumulant of X. This implies that the nd Hankel determinant of {κ n } n : κ κ 3 κ 3 κ 4 = κ κ 4 κ 3, 5.7) is negative for all 0 < α < / see Figure 6). Therefore X FID. 5

16 Figure 6: κ κ 4 κ 3 ) as a function of α. It is negative when 0 < α < /. Next we find an example of 5 parameters a, b, N, α, l) with N and the width of l is grater than one such that GPFPa, b, N, α, l) is not free infinitely divisible. Theorem 5.4. There exist 5 parameters a,b,n,α,l) with N and l [0, ) N < satisfying 0 l < < l k < l k < < l N for some < k N such that GPFPa,b,N,α,l) is not free infinitely divisible. Proof. Consider the probability measure η = η α,α as follows η α,α : = GPFP a = ), b = +), N =, α = +) x ) x ) ) = πx α π, α ) ), l = 0,) π α + α x ) ), +) ) x)dx, 5.8) whereα,α > 0satisfy therelation α +α = by Lemma5. ). Therelation between α and α and the above formulas 5.3) and 5.7) yield that m η) = α )m fp))+α m fp)) = 3α, m η) = α )m fp))+α m 0 fp)) = 6 α, m 3 η) = α )m 3 fp))+α m fp)) = α ), m 4 η) = α )m 4 fp))+α m fp)) = 65 9α ). 5.9) The formula 5.) and the above calculations 5.9) imply that κ := κ η) = 9α +α +, κ 3 := κ 3η) = 54α 3 +9α +6α +, κ 4 := κ 4 η) = 405α 4 +90α 3 +34α +0α ) This implies that thendhankel determinant of {κ n} n, is negative if 0 < α see Figure 7). For example, the measure η 0.7,0.5 is not free infinitely divisible. Acknowledgement Both authors would like to express his hearty thanks to Professor Takahiro Hasebe who is their advisor in Hokkaido university. In particular, thanks to his important advices, the authors could advance their researches. 6

17 Figure 7: κ κ 4 κ 3 ) as a function of α. It is negative when 0 < α Figure 8: η 0.7,0.5 References [] O. Arizmendi, T. Hasebe, On a class of explicit Cauchy-Stieltjes transforms related to monotone stable and free Poisson laws, Bernoulli 9 5B), ). [] O. Arizmendi, T. Hasebe and N. Sakuma, On the law of free subordinators. ALEA Lat. Am. J. Probab. Math. Stat. 0 ), ). [3] O. E. Barndorff-Nielsen and S. Thorbjørnsen, Lévy laws in free probability, Proc. Natl. Acad. Sci. USA 99 00) [4] H. Bercovici and D. V. Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. 4 3), ). [5] L. Bondesson, Generalized gamma convolutions and related classes of distributions and densities, volume 76 of Lecture Notes in Statistics. Springer-Verlag, NewYork 99). ISBN [6] L. Bondesson, A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables, J. Theoret. Probab. 8 3), ). [7] N. Eisenbaum, Another failure in the analogy between Gaussian and semicircle laws, In Séminaire de Probabilités XLIV, volume 046 of Lecture Notes in Math. pages Springer, Heidelberg 0). [8] C. Goldie. A class of infinitely divisible random variables. Proc. Cambridge Philos. Soc. 63, ). [9] T. Hasebe, Free infinite divisibility for Beta distributions and related ones, Electron, J. Probab. 9 04), no. 8, -33. [0] T. Hasebe, Free infinite divisibility for powers of random variables, ALEA, Lat. Am. J. Probab. Math. Stat. 3, ). [] T. Hasebe, K. Szpojankowski, On the free generalized inverse gaussian distributions, to appear in Complex Analysis and Operator Theory, arxiv: [] H. Maassen, Addition of freely independent random variables, J. Funct. Anal. 06, ). [3] A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, London Math. Soc. Lecture Notes Series 335, Cambridge University Press, 006. [4] V. Pérez-Abreu and N. Sakuma, Free infinite divisibility of free multiplicative mixtures of the Wigner distribution, J. Theoret. Probab. 5 ), 00-0). [5] F. W. Steutel. Note on the infinite divisibility of exponential mixtures. Ann. Math. Statist 38, ). [6] K. Szpojankowski, On the Matsumoto-Yor property in free probability, J. Math. Anal. Appl. 445 ) 07),

18 [7] O. Thorin, On the infinite divisibility of the lognormal distribution, Scand. Actuar. J. 3), ). [8] O. Thorin, On the infinite divisibility of the Pareto distribution, Scand. Actuar. J. ), ). [9] D. V. Voiculescu, Addition of certain non-commuting random variables, J. Funct. Anal. 66, ). Junki Morishita Department of Mathematics, Hokkaido University, Kita 0, Nishi 8, Kita-Ku, Sapporo, Hokkaido, , Japan Yuki Ueda Department of Mathematics, Hokkaido University, Kita 0, Nishi 8, Kita-Ku, Sapporo, Hokkaido, , Japan 8