Positive stable densities and the bell-shape

We show that positive stable densities are bell-shaped, that is their n-th derivatives vanish exactly n times on (0,+oo) and have an alternating sign sequence. This confirms the graphic predictions of Holt and Crow (1973) in the positive case.


Introduction
Consider a smooth probability density defined on some open interval I ⊂ R and such that all its derivatives vanish at both ends of I. Rolle's theorem shows that the n−th derivative vanishes at least n times on I. The density is said to be bell-shaped if the n−th derivative vanishes exactly n times on I for every n ≥ 1. For n = 1 this amounts to strict unimodality. For n = 2 this means, as for the familiar bell curve, that there is one inflection point on each side of the mode and that the second derivative is successively positive, negative, and positive. Rolle's theorem also entails that the zeroes of the successive derivatives of a bell-shaped density strictly interlace, and hence that each derivative has an alternating sign sequence, starting positive. The sign sequence of the third derivative can be observed on a curve in comparing the aberrancy and the slope, as shown by Transon's formula [10], but contrary to the second derivative it requires an educated eye to guess what this sequence must be at first glance. The fourth derivative of a density has also a kinematic interpretation in terms of penosculating conics -see [10] and the references therein for a complete account, which however cannot be understood through the sole sequence of its signs.
For a given density whose derivatives vanish at both ends of its definition interval, the usual way to show the bell-shape is to factorize its n-th derivative by a positive function and a polynomial function of degree n. This entails that this n-th derivative vanishes at most, and hence exactly, n times. Basic examples -whose details can be checked by the reader and serve as an exercise in calculus for sophomores -are the Gaussian, Gumbel and centered Student densities on R, and the inverse Gamma densities on (0, +∞). When the density is not explicit, such factorizations are not appropriate anymore. Another criterion for the bell-shape is the ETP character of the associated additive convolution kernel -see Chapter 6.11.C in [7] -but the latter is rather difficult to check for non-explicit densities, and also quite stringent since it is e.g. fulfilled neither by the Student nor by the inverse Gamma densities. In this paper we are interested in the positive α−stable random variables X α and their densities f α , 0 < α < 1. The latter are characterized by their Laplace transform and we choose the normalisation It is known -see e.g. the expansions (14.31) and (14.35) in [8] -that f α is real-analytic on (0, +∞) and that all its derivatives vanish at zero and at infinity. By Proposition 7.1.3 in [7] and (14.31) in [8], the kernel f α (x − y) is not TP 2 and hence not ETP. The function f 1/2 is explicit thanks to the identity in law where here and throughout Γ a denotes the standard Gamma random variable of parameter a > 0, and is hence bell-shaped. The functions f 1/3 and f 2/3 can be written down in terms of a modified Bessel resp. a confluent hypergeometric function -see e.g. (2.8.31) and (2.8.33) in [17] -but it does not seem to the author that the associated second order ODE's can provide any substantial information on their bell-shape.
The strict unimodality of f α was first proved by Ibragimov and Chernin -see the proof of the Theorem in [6] -and then extended to all real stable densities by Sato and Yamazato -see Theorem 1.4 in [9]. Since the latter are real-analytic, this actually follows from Yamazato's original theorem [15] on self-decomposable distributions. See also [12] for a short proof. The graphic simulations made by Holt and Crow [5], partly reproduced in [17], display bell-shaped curves for all real stable densities, at least at the level of the second derivative. A consequence of Theorem 5.1.(ii) in [9] is that f α has only one inflection point on the left side of its mode, and more generally that each n-th derivative of f α vanishes only once before the first zero of the (n − 1)-th derivative. The bell-shape property was claimed in [3] for all stable densities but a serious mistake on the TP character of the kernel f α (xy −1 ) -see Remark (f) in [12] for an explanation -invalidates this result, also at the unimodal level, except for a particular class of stable densities which will be discussed in Section 3.3 and does not include the positive case. Apart from [5,12,3] the bell-shape of stable densities seems to have escaped investigation, even in the most visual case n = 2. In this note, we show the full property in the positive case.
Theorem. The densities f α are bell-shaped.
Our main argument comes from Schoenberg's variation-diminishing property and the total positivity of certain infinitely convoluted exponential kernels. The little known fact that X α admits infinite exponential sums as additive factors had been noticed in [16]. It is used here in conjunction with a precise analysis of the other factor, which contains a certain exponential mixture with no atom at zero and can be chosen in order to have a weak bell-shape property of arbitrary order.
In the case when 1/α is an integer, another proof of the bell-shape is obtained quickly by a multiplicative factorization with inverse Gamma laws -see Section 3.1 below, but the argument does not extend to the other cases. With this method, the bell-shape of the density of log X 1/n on R can also be established. In Section 3.2, we raise two natural conjectures on the bell-shape of positive self-decomposable densities.

Proof of the theorem
We first establish an additive factorization of X α via standard exponential variables. Recall that a positive random variable X ∼ µ is said to be an exponential mixture (ME) if its law has the form with c ∈ [0, 1] and f a completely monotone function on (0, +∞). When c = 0 we will use the notation X ∈ ME * , and in this case Bernstein's theorem shows that the density f writes over (0, +∞), where µ is some probability measure on (0, +∞). In other terms X where here and throughout Exp (λ) stands for the exponential law with parameter λ, and X µ ∼ µ.
The following lemma is mostly due to Yamazato -see [16] pp. 601-602 -but we rephrase it and give a proof for the reader's convenience.
Proof. We first write with the notation c α = sin(πα)/π. This yields the further decomposition where [.] stands for the integer part and is a completely monotone function. By Theorem 51.10 in [8] there exists a random variable Y α belonging to the Bondesson class such that Steutel's criterion -see Theorem 51.12 in [8] -shows that Y α is actually ME. We next prove that Y α has no atom at zero, which will entail Y α ∈ ME * . A change of variable and an integration by parts yield for every x > 0, and it is a bit tedious but elementary to see that so that Y α has infinite Lévy measure and, by Theorem 27.4 in [8], no atom at zero. On the other hand, setting κ α,n = (nπ/ sin(πα)) 1/α for every n ≥ 1, one has where the first equality comes from Fubini's theorem and the second from Frullani's. Putting everything together completes the proof.
This shows that X α and Y α have the same L p −integrability index which is α, excluded. The latter can also be seen from Theorem 25.3 in [8] and the fact that (b) Since the infinite exponential sum has a PF 2 viz. log-concave density -see the proof of the theorem below for a stronger property -and since the density of f α is not PF 2 , one can interpret Y α as the factor which breaks down the PF 2 property for f α . Recall indeed that f α is log-concave until its second inflection point -see Theorem 1.3 (vii) in [9]. On the other hand Y α has a log-convex density by Hölder's inequality. It is plausible that log f α has only one inflection point, which can be readily checked for α = 1/2.
We now introduce our notations concerning the sequence of signs of a real function defined on (0, +∞). We set Observe that by the intermediate value theorem, the zero set of f ∼ ε 1 . . . ε n corresponds to a subsequence of period 2 in {ε 1 , . . . , ε n }. In particular, a function of type ε 1 · · · ε n vanishes exactly (n/2 − 1) times on (0, +∞) if n is even and either (n − 1)/2 or (n − 3)/2 times on (0, +∞) if n is odd. For example, the n-th derivative of a completely monotone function is of type ± n 0 for every n ≥ 0. The density function f α is of type 0+0 and, by the strict unimodality of X α , its derivative f ′ α is of type 0+0−0. Introducing the sequences {a n } = 0± 0 0± 1 0 · · · ± n 0 and {b n } = ± 0 0± 1 0 · · · ± n 0 for every n ≥ 1, we can now state the central definition of this paper.
Definition. For every n ≥ 0, a smooth function f : (0, +∞) → R is said to be weakly bell-shaped The bell-shape property of a function f means that f (i) ∼ {a i } ∀ i ≥ 0, so that weakly bell-shaped functions are never bell-shaped. Notice also that WBS p ∩ WBS q = ∅ if p = q. The following result provides our key-argument.
Proof. Set f 0 for the density of X and recall from (1) that it is completely monotone. In particular f 0 is real-analytic on (0, +∞). We denote by f n the density of X + Exp(λ 1 ) + · · · + Exp(λ n ) for every n ≥ 1. The latter are connected to one another by the formula x 0 e λny f n−1 (y)dy, and hence solutions on (0, +∞) to the linear ODE's An induction hinging on (4) and (5) show that all densities f n are real-analytic on (0, +∞), that all derivatives f n (0+) > 0 for all n ≥ 0, so that 0+ is an isolated zero of f (i) n for all n ≥ i + 1 ≥ 1 (notice that this cannot be proved directly by analyticity because 0+ is also a singular zero). The weak bell-shape property will now be established through a more elaborate induction on n.
We next show the induction step and suppose that f n is WBS n−1 for some n ≥ 1. We have already seen that f n+1 ∼ {a 0 } and we will first prove by induction on i that f We begin with f ′ n+1 for the sake of clarity. Since f n+1 ∼ {a 0 }, we know that f ′ n+1 vanishes at least once on (0, +∞). Set where the strict inequality comes from the fact that 0+ is an isolated zero of f ′ n+1 . Differentiating (5) as above entails where the inequality comes from the initial profile 0+0 of f ′ n+1 . Suppose that f ′′ n+1 (x 1 ) = 0. Then by analyticity x 1 is an isolated zero of f ′′ n+1 , so that By (5), this entails f ′ n (x) < 0 as x ↑ x 1 . But the induction hypothesis yields f ′ n ∼ {b 1 } if n = 1 and f ′ n ∼ {a 1 } if n > 1, so that one cannot have f ′ n (x 1 ) = 0, a contradiction. Hence f ′′ n+1 (x 1 ) < 0 and if there exists x 2 > x 1 such that f ′ n+1 (x 2 ) = 0, then again the induction hypothesis entails f ′ n (x 2 ) < 0 so that f ′′ n+1 (x 2 ) < 0, a contradiction. This shows that f ′ n+1 vanishes only once on (0, +∞) and, all in all, that f ′ n+1 ∼ {a 1 }. We now suppose f vanishes at least i + 1 times on (0, +∞), and also from the above that f (i+1) n+1 has initial profile 0+0. Set {x j , j ≥ 1} for the ordered sequence of its zeroes on (0, +∞). Again, one gets from (5) for every j ≥ 1. For clarity we consider two separate cases.
(i) If i + 1 = n, the induction hypothesis entails f (i+1) n ∼ {b n } = ± 0 0 · · · ± n 0. Set I k n ↔ {± k 0} for the k-th half-closed interval corresponding to each part ± k 0 of the graph of f The last equality in (6) with j = 1 and the initial profile 0+0 of f (6) with j = 1, 2 and the induction hypothesis show that x 2 ∈ I k 2 n with k 2 even. If f (i+1) n (x 1 ) = 0, then x 2 ∈ I k 2 n with k 2 > k 1 . Iterating the procedure shows that there are at most, and hence exactly, n = i + 1 zeroes and that x k ∈ I k n for every k = 1 . . . n. All in all this shows that f We finally show that (−1) n+1+i f (i) n+1 ∼ {b n+1 } for every i ≥ n + 1. We only sketch the proof since the arguments are the same. Consider first the case i = n + 1. Since f n+1 vanishes at least n + 1 times on (0, +∞) and set {x k , k ≥ 1} for the ordered sequence of its zeroes. Recall f (n+1) n+1 (0+) > 0 and, from the induction hypothesis, that f (n+1) n ∼ ± 1 0 · · · ± n+1 0.
Set again I k n ↔ {± k 0} for the k-th half-closed interval corresponding to each part ± k 0 of the graph of f (n+1) n , k = 1 . . . n + 1. The same reasoning as above shows that necessarily x k ∈ I k n for all k and hence that f n ∼ ± 1 0 · · · ± n+1 0.

Remarks 2. (a)
The complete monotonicity of f 0 is crucial in the above argument, initializing all induction steps. It seems difficult to extend the proposition to broader classes than ME.
(b) When X itself has an exponential law and all parameters are different, the density of X + Exp(λ 1 ) + · · · + Exp(λ n ) is a linear combination of n + 1 distinct exponentials functions. Since the latter define a Chebyshev system -see [7] p. 24, the property WBS n is much easier to prove.
We leave the details to the reader.
(c) The method of examining the sign of the (n+1)-th derivative at the zero of the n-th derivative is used in [6] for n = 1 in order to show the strict unimodality of f α , after some explicit contour integration. The involved computations to extend this argument directly on the further derivatives of f α seem however extremely heavy, and the author does not believe that they could yield the full bell-shape. See [4] for an application of the contour integration of [6] to the asymptotic behaviour of these further derivatives.
End of the proof. Fix n ≥ 1 and set f α,n resp. g α,n for the density of with the notations of the Lemma, so that one has f α (x) = ∞ 0 f α,n (x − y)g α,n (y)dy.
By the Proposition, the function f α,n has a C n+1 extension on R and one obtains α,n (y)g α,n (x − y)dy.

Remarks and open questions
3.1. The case when 1/α is an integer. In this situation the bell-shape follows easily from the independent factorization which was pointed out in [13] and also basically shown in [7] pp. 121-122. We discard the explicit case n = 2 and set g n resp. h n for the density of n −n Γ −1 1/n resp. Γ −1 2/n × . . . × Γ −1 (n−1)/n . The multiplicative convolution yields dy y and one can clearly differentiate under the integral. This entails for every x > 0, i ≥ 1. The kernel h n (xy −1 ) is the composition of (n − 2) kernels that are STP ∞ on (0, +∞) × (0, +∞) by Theorem 2.1 p. 18 in [7], and is hence itself STP ∞ on (0, +∞) × (0, +∞) by the Binet-Cauchy formula. With the above notation for S − and S + , Theorem 3.1 (b) p. 21 in [7] and the aforementioned bell-shape property of g n yield for every i ≥ 1 so that f 1/n is bell-shaped, too. The above factorization (7) shows also easily that log X 1/n has a bell-shaped density on R, a property which we believe to be true for all α ∈ (0, 1).
When α is rational, another multiplicative factorization of X α in terms of inverse Beta and inverse Gamma random variables was obtained in Lemma 2 of [11]. But the latter seems useless for the bell-shape of X α or log X α , because inverse or log Beta laws have neither bell-shaped densities nor TP ∞ convolution kernels.

3.2.
Positive self-decomposable densities. A positive random variable X is self-decomposable when its Laplace transform reads for some γ 0 ≥ 0 and k : (0, +∞) → R + non-increasing. For example, the positive stable law is self-decomposable with γ 0 = 0 and k(x) = α/Γ(1 − α)x α . It is known and easy to see -see Theorem 27.7 in [8] -that X has a density f when k = 0. The unimodality of f was shown by Wolfe [14].
In the following we will suppose that γ 0 = 0 w.l.o.g. and set f k for the positive self-decomposable density associated with the non-degenerate spectral function k.
When k(0+) = +∞, Theorem 28.4 (ii) in [8] and Lemma 2.5 in [9] show that f k is smooth and that all its derivatives vanish at zero and at infinity, and Theorem 1.4 in [9] entails that f k is strictly unimodal. When k(0+) < +∞, it is also known -see Remark 28.6 in [8] -that f k is not smooth at zero and hence not bell-shaped. In view of the main result of the present paper, one might raise the Conjecture 1. The density f k is bell-shaped if and only if k(0+) = +∞.
Apart from the positive stable, examples of densities verifying the above conjecture are the inverse Gamma, or more generally the densities of Γ −a t with t > 0, a ≥ 1. Indeed, the latter are self-decomposable by the HCM criterion [1], and bell-shaped by a direct computation. Other selfdecomposable examples are infinite exponential sums, whose bell-shape property can be proved exactly in the same manner as for f α . As pointed out in the introduction, when k(0+) = +∞ Theorem 5.1.(ii) in [9] already shows that f with the same method seems however not quite obvious, because of the memory involved in this integro-differential equation.
Let us now consider the case where k(0+) is finite. When k(0+) < 1, Theorem 1.3 in [9] shows that f k is decreasing. When k(0+) = 1, the situation is complicated because of the three different types I 2 , I 3 and I 4 of [9]. In particular f k may have a non-trivial modal interval -see Lemma 53.2 in [8]. When k(0+) > n + 1 for some n ≥ 0, then Theorem 28.4 (i) in [8] and Lemma 2.5 in [9] show that f k is C n with derivatives vanishing at zero and at infinity, and again Theorem 1.4 in [9] entails that f k is strictly unimodal. The following is hence quite natural.