On infinitely divisible solutions to indeterminate moment problems

Christian Berg

January 8, 2001

Abstract

For a convolution semigroup of measures with moments of any order we prove that the n’th moment is a polynomial of degree at most n in the time parameter. Special focus is on the case where the measures are indeterminate, in particular the log-normal and the q-Laguerre case.

1991 Mathematics Subject Classification: primary 44A60, secondary 60E07, 33D45. Keywords: Moment problem, infinitely divisible probability, q-Laguerre weight.

1 Introduction and results

This paper treats only the second part of the lecture given at the “International Workshop on Special Functions, Asymptotics, Harmonic Analysis and Mathematical Physics”, Hong Kong June 21-25, 1999.^{1}

In the first part of the lecture we considered a fixed indeterminate moment sequence (s_{n}) and the set V of solutions μ to the moment equations

∫ x^{n} dμ(x) = s_{n}, n = 0, 1, ….  (1.1)

The n’th moment of a solution μ will be denoted s_{n}(μ).

Let (μ_{t})_{t∈I} be a one-parameter family of solutions in V. The basic assumptions are that I is an interval and the mapping t ↦ μ_{t} of I into V is continuous, when V is equipped with the weak topology. It is well-known that V is a compact convex set in this topology, so for any probability measure τ on I the vector integral κ = ∫ μ_{t} dτ(t) belongs to V. In the lecture we showed how this simple result can be used to calculate solutions to the following moment problems: log-normal, generalized Stieltjes-Wigert or q-Laguerre and discrete q-Hermite II. The results have appeared in [7]. See also [6] for some calculation of moments and [1] for the general theory.

1Proceedings of the International Workshop “Special Functions ”, Hong Kong June 21-25, 1999.

Ed. C. Dunkl, M. Ismail, R. Wong. World Scientific, Singapore 2000, pp. 31-41.

In the second part of the lecture we considered probabilities with moments of any order and which are infinitely divisible. A probability μ with these properties can be imbedded in a convolution semigroup (μ_{t})_{t>0} such that μ_{1} = μ. For the precise conditions assumed for a convolution semigroup see section 2 below. We shall prove that all the measures in the convolution semigroup have moments of any order, and the n’th moment s_{n}(μ_{t}) of μ_{t} is a polynomial in t of degree at most n without constant term if n ≥ 1. The coefficients of this polynomial can be expressed in terms of the values at zero of the derivatives of the negative definite function of the convolution semigroup. These derivatives are closely related to the moments of the Lévy measure for the convolution semigroup. For convolution semigroups supported by the half-line and not equal to the trivial semigroup (ε_{bt})_{t>0} of uniform motion the n’th moment is a polynomial of degree n with positive coefficients. Here and in the following ε_{a} denotes the unit mass at the point a.

The set V of solutions to an indeterminate moment problem can contain no, one or infinitely many infinitely divisible distributions, other possibilities do not happen. The occurence of the two last cases depends on the determinacy/indeterminacy of the Lévy measure ([16]).

The log-normal distribution with parameter σ > 0 has density on ]0, ∞[ given by

d_{σ}(x) = (2πσ^{2})^{− \frac{1}{2}} x^{−1} exp (−\frac{(log x)^{2}}{2σ^{2}} ) .  (1.2)

It is indeterminate and infinitely divisible. As far as the author knows the convolution roots of this distribution are not explicitely known. We show that they are also indeterminate probabilities as a consequence of the fact that there are more than one infinitely divisible distribution in the set V. This is because the densities

ω_{c}(x) = \frac{x^{c−1}}{M_{c}(q, −q^{\frac{1}{2} −c}x, −q^{\frac{1}{2} +c}/x; q)_{∞}}  (1.3)

are infinitely divisible, as we shall see below, and they have the same moments

s_{n} = q^{− \frac{1}{2} n^{2}}

as (1.2) when q = exp(−σ^{2}). The latter follows from the Askey-Roy beta-integral, [4], [3]. See also [7]. For the notation above concerning q-special functions we refer to [15], [17]. The parameter c is real, but since ω_{c+1}(x) = ω_{c}(x) we may restrict c to the interval [0, 1[. The constant M_{c} normalizing ω_{c} to a probability can be expressed using the Γ-function and Jackson’s Γ_{q}-function and is given in [7]. The density (1.3) for c = \frac{1}{2} was given by Askey in [2].

Another important class of indeterminate and infinitely divisible distributions is connected to the q-Laguerre moment problem with moments

s_{n}(α; q) = q^{−αn−\binom{n+1}{2}}(q^{α+1}; q)_{n},  (1.4)

where 0 < q < 1, α > −1.

We shall see below that the density

k_{α} \frac{x^{α}}{(−x; q)_{∞}} ,  (1.5)

which has the moments (1.4), is infinitely divisible. The measure (1.5) corresponds to c = α + 1 in the one-parameter family of densities

\frac{q^{c(α+ \frac{1}{2} )}}{M_{c}} x^{c−1} \frac{(q^{α+1}, −q/x; q)_{∞}}{(q, −q^{α+1−c}x, −q^{c−α}/x; q)_{∞}}  (1.6)

all having the q-Laguerre moments, cf. [7]. This family is also periodic in c with period 1. We have only been able to prove that the special case (1.5) is infinitely divisible.

It is well-known that the log-normal case is a limiting case of q-Laguerre for α tending to infinity, cf. [2].

At the end of the paper we give evidence to the following conjecture about convolution semigroups (μ_{t})_{t>0} with moments of any order: Either all the measures μ_{t} are determinate or all the measures are indeterminate.

2 Convolution semigroups of measures with moments of any order

In 1977 O. Thorin, cf. [19], established that the log-normal distribution (1.2) is infinitely divisible. This is probably the first known explicit example of an indeterminate and infinitely divisible probability measure. The existence of such measures was noticed in Heyde [16]. We recall that a probability measure μ on ℝ is called infinitely divisible, if μ has n’th probability roots under convolution for any n, i.e. for any n ∈ ℕ there shall exist a probability σ such that σ^{∗n} = μ. It is well-known that such an n’th root is uniquely determined and that μ has arbitrary positive convolution roots. This means that there exists a (uniquely determined) family (μ_{t})_{t>0} of probabilities with μ = μ_{1} such that (μ_{t})_{t>0} is a convolution semigroup, i.e. has the properties

μ_{t} ∗ μ_{s} = μ_{t+s},  (2.1)

t ↦ μ_{t} is weakly continuous for t > 0,  (2.2)

lim_{t→0} μ_{t} = ε_{0} weakly.  (2.3)

We recall the Lévy-Khinchine formula for the Fourier transform of a convolution semigroup, cf. Lukacs [18],

μ̂_{t}(x) = ∫_{−∞}^{∞} e^{−ixy}dμ_{t}(y) = e^{−tψ(x)}, t > 0, x ∈ ℝ,  (2.4)

ψ(x) = αx^{2} + iβx + ∫_{−∞}^{∞}(1 − e^{−ixy} − \frac{ixy}{1 + y^{2}} )dλ(y), x ∈ ℝ,  (2.5)

where α ≥ 0, β ∈ ℝ, and λ is a non-negative measure on ℝ \ {0} satisfying

∫_{−∞}^{∞} \frac{y^{2}}{1 + y^{2}} dλ(y) < ∞.  (2.6)

Condition (2.6) implies that λ(ℝ\]−1, 1[) < ∞. The function ψ and the measure λ are called the negative definite function and the Lévy measure for the convolution semigroup.

Conversely, given α ≥ 0, β ∈ ℝ and a non-negative measure λ on ℝ\{0} satisfying (2.6), there exists a uniquely determined convolution semigroup (μ_{t})_{t>0} such that (2.4) and (2.5) hold. Any non-negative measure λ on ℝ \ {0} satisfying (2.6) will be called a Lévy measure.

Concerning the existence of moments of all orders for the measures in a convolution semigroup we prove:

Theorem 2.1 For a convolution semigroup (μ_{t})_{t>0} as above the following conditions are equivalent:

(i) μ_{t_{0}} has moments of all orders for some t_{0} > 0,

(ii) μ_{t} has moments of all orders for all t > 0,

(iii) ψ ∈ C^{∞}(ℝ),

(iv) ∫_{−∞}^{∞} y^{2n} dλ(y) < ∞ for n = 1, 2, . . ..

In case (i)-(iv) hold, then s_{n}(μ_{t}) is a polynomial of degree ≤ n in t without constant term (for n ≥ 1), the coefficients of which are uniquely determined by the sequence (ψ^{(n)}(0)).

Proof: We shall use the fact that a probability has moments of every order if and only if its Fourier transform is a C^{∞}-function, cf. e.g. [18]. From this the equivalence of (i)-(iii) is straight-forward.

Suppose next that (i)-(iii) hold. We establish by induction that s_{n}(μ_{t}) for n ≥ 1 is a polynomial of degree ≤ n in t without constant term. Differentiating (2.4) at x = 0 we clearly get

s_{1}(μ_{t}) = −itψ′(0),

so the result holds for n = 1. Differentiating (2.4) n + 1 times we find

(−i)^{n+1}s_{n+1}(μ_{t}) = \frac{d^{n+1}}{dx^{n+1}} \{e^{−tψ(x)}\}_{x=0} = \frac{d^{n}}{dx^{n}} \{−tψ′(x)e^{−tψ(x)}\}_{x=0}
= −t ∑_{k=0}^{n} \binom{n}{k}\frac{d^{k}}{dx^{k}} \{e^{−tψ(x)}\}_{x=0}ψ^{(n−k+1)}(0)

= −t ∑_{k=0}^{n} \binom{n}{k}(−i)^{k}s_{k}(μ_{t})ψ^{(n−k+1)}(0),

which is a polynomial as required, if we assume the result up to order n.

Since t^{−1}μ_{t}| (ℝ \ {0}) converges vaguely on ℝ \ {0} to the Lévy measure λ for t → 0, cf. [9], we get for n ≥ 1

∫_{−∞}^{∞} y^{2n} dλ(y) ≤ lim inf_{t→0} t^{−1} ∫_{−∞}^{∞} y^{2n} dμ_{t}(y) = \frac{d}{dt}\{s_{2n}(μ_{t})\}_{t=0} < ∞,

which shows that (iv) holds.

Suppose finally that (iv) holds. The assumption ∫ y^{2} dλ(y) < ∞ implies that ψ is C^{2} with derivatives

ψ′(x) = ∫_{−∞}^{∞} iy(e^{−ixy} − \frac{1}{1 + y^{2}} ) dλ(y) + 2αx + iβ

ψ′′(x) = ∫_{−∞}^{∞} y^{2}e^{−ixy} dλ(y) + 2α,

and since the finite measure y^{2}λ has moments of all even orders and hence of all orders, we see that ψ is C^{∞} and

ψ′(0) = i ∫_{−∞}^{∞} \frac{y^{3}}{1 + y^{2}} dλ(y) + iβ,

ψ′′(0) = ∫_{−∞}^{∞} y^{2} dλ(y) + 2α,

ψ^{(n)}(0) = (−i)^{n−2} ∫_{−∞}^{∞} y^{n} dλ(y), n ≥ 3.

□

Remark 2.2 (a) Defining

σ_{n} = −i^{n+1}ψ^{(n+1)}(0), n ≥ 0,  (2.7)

we see from above that σ_{n} is real and

σ_{n} = s_{n−1}(2αε_{0} + y^{2}λ), n ≥ 1, σ_{0} = ∫_{−∞}^{∞} \frac{y^{3}}{1 + y^{2}} dλ(y) + β.  (2.8)

The first moments are

s_{1}(μ_{t}) = σ_{0}t
s_{2}(μ_{t}) = σ_{1}t + σ_{0}^{2}t^{2}
s_{3}(μ_{t}) = σ_{2}t + 3σ_{0}σ_{1}t^{2} + σ_{0}^{3}t^{3}.

Writing

s_{n}(μ_{t}) = ∑_{k=0}^{n} c_{n,k}t^{k}, n ≥ 0,  (2.9)

we have c_{0,0} = 1, c_{n,0} = 0 for n ≥ 1 and it is easy to see that

c_{n,n} = σ_{0}^{n}, c_{n,1} = σ_{n−1}.  (2.10)

In general the coefficients c_{n,k} satisfy the recurrence equation

c_{n+1,l+1} = ∑_{k=l}^{n} c_{k,l}\binom{n}{k}σ_{n−k}, n, l ≥ 0.  (2.11)

(b) The assumption (iv) for n = 1 is of course stronger than (2.6). On the other hand, if (2.6) holds, condition (iv) can be replaced by

(iv′) ∫_{|y|≥1} y^{2n} dλ(y) < ∞, n ≥ 1.

Example 2.3 The degree of s_{n}(μ_{t}) can be strictly less than n as the first of the following examples show.

(a) The Gaussian semigroup

μ_{t} = \frac{1}{\sqrt{4πt}} exp(−\frac{x^{2}}{4t})dx,

s_{2n}(μ_{t}) = \frac{(2n)!}{n!}t^{n}, s_{2n+1}(μ_{t}) = 0.

(b) The drift semigroup μ_{t} = ε_{βt}, s_{n}(μ_{t}) = β^{n}t^{n}, β ∈ ℝ.

(c) The Γ-semigroup

μ_{t} = \frac{1}{Γ(t)}x^{t−1}e^{−x}1_{]0,∞[}(x)dx,

s_{n}(μ_{t}) = \frac{Γ(n + t)}{Γ(t)} = (t)_{n}.

(d) The Poisson semigroup

μ_{t} = e^{−t} ∑_{k=0}^{∞} \frac{t^{k}}{k!}ε_{k},

s_{n}(μ_{t}) = e^{−t} ∑_{k=0}^{∞} \frac{k^{n}t^{k}}{k!} = e^{−t}L^{n}(e^{t}),

where L is the differential operator Lf(x) = xf′(x).

For symmetric semigroups like (a) it is clear that s_{2n+1}(μ_{t}) = 0, and it can be proved by induction that s_{2n}(μ_{t}) is of degree n unless μ_{t} is reduced to ε_{0}.

For convolution semigroups (μ_{t})_{t>0} supported by the half-line [0, ∞[ we have also more information as indicated by the examples (b)-(d).

Proposition 2.4 Let (μ_{t})_{t>0} be a convolution semigroup of measures supported by [0, ∞[ and with moments of all orders. Then s_{n}(μ_{t}) is a polynomial (2.9) of degree n with c_{n,k} > 0 for k = 1, …, n, unless μ_{t} = ε_{bt} for some b ≥ 0.

Proof: The result can be deduced from (2.8) and (2.11), but it is perhaps more instructive to make use of the Laplace transformation. The Laplace transform of an arbitrary convolution semigroup (μ_{t})_{t>0} on [0, ∞[ has the form

Lμ_{t}(s) = ∫_{0}^{∞} e^{−sy} dμ_{t}(y) = e^{−tf(s)}, t > 0,  (2.12)

where the Bernstein function f associated with (μ_{t})_{t>0} is given as

f(s) = bs + ∫_{0}^{∞}(1 − e^{−sy}) dλ(y).  (2.13)

Here b ≥ 0 and the Lévy measure λ is concentrated on ]0, ∞[ and satisfies the integrability condition

∫_{0}^{∞} \frac{y}{1 + y} dλ(y) < ∞,

cf. [9]. Note that the negative definite function is ψ(x) = f(ix). In the present case, where all the measures have moments of any order, the Lévy measure satisfies

∫_{0}^{∞} y^{n} dλ(y) < ∞, n ≥ 1.

Therefore f ∈ C^{∞}([0, ∞[) and

σ_{n} = (−1)^{n}f^{(n+1)}(0) = s_{n}(bε_{0} + ydλ(y)) < ∞

for n ≥ 0. It follows that σ_{n} > 0 for all n ≥ 0, unless λ = 0 in which case μ_{t} = ε_{bt}. The assertion now follows from the recursion formula (2.11). □

To discuss the indeterminacy of the measures in a convolution semigroup with moments of any order, we shall say that a Lévy measure λ on ℝ\{0} is indeterminate, if there exists a Lévy measure λ̃ on ℝ \ {0} with λ ≠ λ̃ and such that

∫_{−∞}^{∞} y^{n} dλ(y) = ∫_{−∞}^{∞} y^{n} dλ̃(y) for n ≥ 2.  (2.14)

Theorem 2.5 (Heyde [16]). Let (μ_{t})_{t>0} be a convolution semigroup with moments of any order and indeterminate Lévy measure. Then the semigroup is indeterminate in the sense that all the measures μ_{t} are indeterminate.

Proof: Let ψ and λ denote the negative definite function and the Lévy measure for (μ_{t})_{t>0} and let λ̃ ≠ λ satisfy (2.14). Let ψ̃ be defined by (2.5) with λ replaced by λ̃ and β replaced by β̃, the latter defined so that ψ′(0) = ψ̃′(0). Then ψ ≠ ψ̃ and ψ^{(n)}(0) = ψ̃^{(n)}(0) for n ≥ 1. Letting (μ̃_{t})_{t>0} be the convolution semigroup associated with ψ̃, it follows from (2.8) and (2.11) that s_{n}(μ_{t}) = s_{n}(μ̃_{t}) but μ_{t} ≠ μ̃_{t} for t > 0. □

Remark 2.6 (a) The theorem shows that there exist as many indeterminate convolution semigroups as indeterminate measures. In fact, if ν is any indeterminate measure with moments of all orders and with no mass at zero, then λ = y^{−2}ν is an indeterminate Lévy measure.

(b) The converse of Theorem 2.5 is not true, although it is claimed in [16, Theorem 2]. Lemma 2.2 in [5] can be used to give a counterexample. In fact, the Lemma asserts the existence of a determinate probability λ such that y^{2}λ is determinate and λ^{∗3} is indeterminate. Since the measure λ is constructed to have no mass at 0, it is a determinate Lévy measure. The convolution semigroup

μ_{t} = e^{−t} ∑_{n=0}^{∞} \frac{t^{n}}{n!}λ^{∗n}, t > 0

is indeterminate since each μ_{t} majorizes a multiple of λ^{∗3}.

(c) Let (μ_{t})_{t>0} be an indeterminate convolution semigroup with a determinate Lévy measure, and let V_{t} be the set of measures with the same moments as μ_{t}. Then for each t the set V_{t} contains only one infinitely divisible measure.

(d) In [16] Heyde gives an example of an indeterminate measure μ such that V contains no infinitely divisible measures.

Theorem 2.7 The densities (1.2) and (1.3) with log-normal moments s_{n} = q^{−n^{2}/2} are all infinitely divisible. In particular all positive convolution roots of these measures are infinitely divisible.

Proof: The infinite divisibility of (1.2) was proved first by Thorin in [19]. He introduced the subclass of infinitely divisible distributions called generalized gamma convolutions and proved that the log-normal distribution belongs to this subclass T . Bondesson found in [10] a subclass B ⊂ T given as all the probability densities on ]0, ∞[ which are pointwise limits of densities of the form

Cx^{β−1} ∏_{i=1}^{N}(1 + c_{i}x)^{−γ_{i}}, β, c_{i}, γ_{i} > 0.  (2.15)

In later papers [11],[12] Bondesson found an extremely elegant characterization of B as the probability densities f on ]0, ∞[ such that, for each u, f(uv)f(u/v) is completely monotone as a function of w = v + v^{−1}. Using this result Bondesson gave a simple proof that (1.2) belongs to the class B, see [11],[12]. The density (1.3) also belongs to the class B and is therefore a generalized gamma convolution and in particular infinitely divisible. In fact the probability density

C(n, c)x^{c−1}\{(−q^{\frac{1}{2} −c}x, −q^{\frac{1}{2} +c}/x; q)_{n}\}^{−1} = C(n, c)x^{c+n−1}\{(−q^{\frac{1}{2} −c}x, −q^{\frac{1}{2} −c−n}x; q)_{n}\}^{−1}

has the right form and converges pointwise to ω_{c}(x).

The set V of measures with the log-normal moments therefore contains infinitely many infinitely divisible measures, and the assertion follows. □

By the same method as above we get:

Theorem 2.8 The q-Laguerre density (1.5) belongs to the class B and is in particular infinitely divisible.

Since the same method does not apply to the densities (1.6), except when c = α + 1, we do not know if the set V of solutions with q-Laguerre moments (1.4) contains one or infinitely may infinitely divisible distributions.

3 A conjecture

If μ and ν are measures with moments of any order then the convolution μ ∗ ν has also moments of any order and

s_{n}(μ ∗ ν) = ∑_{k=0}^{n} \binom{n}{k}s_{k}(μ)s_{n−k}(ν).  (3.1)

It was proved by Devinatz, cf.[14],[5], that if ν ≠ 0 and μ is indeterminate, then μ ∗ ν is indeterminate. On the other hand it is possible that μ is determinate but μ ∗ μ is indeterminate, cf. [5],[13].

Let (μ_{t})_{t>0} be a convolution semigroup with moments of any order. If it is known that μ_{t} is indeterminate it follows from the result of Devinatz that μ_{t} ∗ μ_{s} is indeterminate for any s > 0.

However, we do not need the full result of Devinatz to prove this, since μ̂_{s} does not vanish because of (2.4). We can simplify the argument as follows: If σ has the same moments as μ_{t}, then σ ∗ μ_{s} has the same moments as μ_{t} ∗ μ_{s} by (3.1), and if σ ∗ μ_{s} = μ_{t} ∗ μ_{s}, we get by Fourier transformation and cancelation that σ = μ_{t}.

The question is if it can happen that μ_{t} is determinate for 0 < t < t_{0} and indeterminate for t_{0} < t for some number 0 < t_{0} < ∞. By Theorem 2.5 this would require the Lévy measure to be determinate. As noticed in [5] the Carleman condition cannot be satisfied for the measures μ_{t}, t < t_{0}.

Determinacy of a measure μ depends on the behaviour of the smallest eigenvalue λ_{n} of the Hankel matrix

(s_{i+j}(μ))_{0≤i,j≤n}.  (3.2)

It was proved in [8] that μ is determinate if and only if lim λ_{n} = 0. In the special case of μ = μ_{t} the Hankel matrix consists of polynomials in t of degree at most 2n, and since the smallest eigenvalue is the minimum of the Hankel form on the unit sphere in ℂ^{n+1}, it is easily seen that the smallest eigenvalue λ_{n}(t) is a strictly positive continuous function.

The existence of a number t_{0} as above would require that λ_{n}(t) converges to 0 for t < t_{0} and to a strictly positive limit for t > t_{0}. Because of the polynomial behaviour of the moments we do not think that this is possible and hence we formulate:

Conjecture 3.1 For a convolution semigroup (μ_{t})_{t>0} of measures with moments of any order there are only the possibilities:

1. All the measures μ_{t} are determinate.

2. All the measures μ_{t} are indeterminate.

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Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark