Abstract
Purpose
In this work the author gathers several methods and techniques to construct systematically Stieltjes classes for densities defined on R+.
Design/methodology/approach
The author uses complex integration to obtain integrable functions with vanishing moments sequence, and then the author considers some operators defined on the vanishing moments subspace.
Findings
The author gather several methods and techniques to construct systematically Stieltjes classes for densities defined on
Originality/value
The author computes the Hilbert transform of powers of
Keywords
Citation
López-García, M. (2022), "Operators on the vanishing moments subspace and Stieltjes classes for M-indeterminate probability distributions", Arab Journal of Mathematical Sciences, Vol. 28 No. 2, pp. 229-242. https://doi.org/10.1108/AJMS-04-2021-0083
Publisher
:Emerald Publishing Limited
Copyright © 2021, Marcos López-García
License
Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
Consider the subspace
The vanishing moments subspace
We also consider the subspace
First, we introduce a method to get functions in
We use complex integration to show that
Then we introduce some operators mapping the subspace
Suppose now that f is a probability density function (we use further just density) of a random variable X such that all moments are finite, i.e.,
In the survey [3] the author revisited recent developments on the checkable moment-(in)determinacy criteria including Cramér's condition, Carleman's condition, Hardy's condition, Krein's condition and the growth rate of moments. In this survey the author analyzes Hamburger and Stieltjes cases.
In this work we only focus in the Stieltjes case, i.e we consider distributions supported on
Clearly, S(f, h) is a family of densities all having the same moment sequence as f.
If X ∼ f is M-determinate, then the perturbation h = 0, and the Stieltjes class consists of a single element, the center f.
The main aim of this work is to find perturbations for Stieltjes classes with center at a density f > 0. To do this, the basic idea is take a function
When X ∼ f with a density f in
In [5, Theorem 1.2] the author proved that if f is a density in
In particular, here we compute
In order to test our approach we apply the developed methods to the generalized gamma (GG) distribution (see Examples 4, 8, 11 and 12), powers of the generalized inverse gaussian (GIG) distribution (see Examples 3, 14 and 16), powers of the half-logistic distribution (see Example 7) and to the generalized lognormal (GLN) distribution (see Examples 5, 6, 19 and 21).
This work is organized as follows. In Section 2 we give the precise conditions on g to prove (1), and we apply this result to get functions in
2. Functions with vanishing moments
In this section we use complex integration to obtain functions in
Let
Let 0 < α < 1,
Then
We pick 0 < ɛ < A < ∞, Cauchy's theorem implies that
Since
Let 0 < α < 1. Suppose that g ∈ hol(Sα) satisfies conditions (5), (6) with γ = (n + 1)/α,
By setting γ = (n + 1)/α for all
We recall an inequality that will be useful to get our estimates: since ex ≥ x for all x > 0 we have
Throughout this work the constant K will be a normalizing constant to produce a density function in each case.
For all b1, b2 > 0, 0 < c1, c2 < 1/2 and
Indeed, we just apply Theorem 2 with g(z) = zβ exp(−ρ1zλ − ρ2z−1) for any
From (9) we have that g satisfies condition (8) for all 0 < α < 1,
From Theorem 2 we have
For all 0 < α < 1/2, a, b > 0 we have
Indeed, consider g(z) = zβ exp(−ρz) for any β > − 1/α, ρ > 0. From (9) we have that g satisfies condition (8) for all
Theorem 2 implies that
Recall that ⌊⋅⌋ is the floor function and ⌈⋅⌉ is the ceiling function. For
For all 0 < α < 1/2, b > 0,
To see this, consider g(z) = exp(−ρz(Log z)m) for any ρ > 0, here Log z stands for the principal branch of the logarithm function. For
Clearly I2 < ∞, and (9) implies that
Clearly I1 < ∞ when m is even. Assume that m is odd, thus
Hence g satisfies condition (8) for all
Since the real part is an additive function, we have for A > e and t ∈ [0, πα] that
On the other hand, there is C > 0 such that
Theorem 2 implies that
For all
Consider g(z) = zβ exp(−ρ(Log z)2m) for any
As in (15) and using (13) we can see that
Theorem 2 implies that
For all a > 0, 0 < α < 1/2 we have
Consider
Theorem 2 implies that
Notice that
3. Operators on the vanishing moment subspace
For
The binomial formula implies that
The case m = 1 was considered in [8, Lemma 1].
For a, b > 0 and 0 < α < 1 we have
Let
If
Let (J, μ) be a measure space. Assume that
Fubini's theorem implies that
Let μ be a positive bounded measure on
We consider the function
Clearly
We apply the last result to obtain new Stieltjes classes with center at f(x) = K exp(−xα), x > 0, as follows.
By (12) we have
From (17) we get
Since (x/e)α ≤ x/e for all x ≥ e, there exist a constant 0 < C < 1 such that xα − x ≤ −Cx for all x ≥ e, thus
Now, let p be a polynomial with real coefficients, with p(0) = 0 and p′ > 0 on
Let 0 < α < 1/2, a, b > 0 be fixed and 1 ≤ n < (2α)−1,
Let Λ ≠ ∅. Assume that
Let 0 < c1, c2 < 1/2, b1, b2 > 0,
we proceed as in Example 12 and use Remark 13 to get that
Once again, we obtain new perturbations for strong Stieltjes classes with center at
4. Krein criterion and the Hilbert transform
In this section we use a different technique to construct Stieltjes classes. This method involves the computation of the Hilbert transform of ln f, where f is a density
a) For any constant
b) Let 0 < |γ| < 1. Then
c)
As a consequence we obtain a Stieltjes class with center at a generalized inverse Gaussian density.
Let
Thus
As before, we can see that h(x) = sin(πa − b tan(πα)xα), x > 0, is a perturbation for the Stieltjes class with center at the density f(x) = Kxa−1 exp(−bxα), x > 0, provided that 0 < α < 1/2, a, b > 0. This is the case n = 1 in Example 12.
Finally, in the last examples we get two Stieltjes classes that we could not obtain by the method of complex integration given in Section 2. The densities involved are special cases of generalized log-normal densities, see [9]. In order to construct these examples we need to find out the Hilbert transform of | ln x|n, x > 0,
For all
We introduce the following constants
First we compute the Hilbert transform of even powers of | ln x|.
For
Let t > 0 fixed and ɛ > 0 small enough. Since the geometric series with ratio r = x2/t2 converges uniformly for x ∈ [0, t −ɛ], and by using (19), we get that
Multiplying the last equality by t and using that
Similarly, we can obtain that
As before, we multiply the last equality by t to get
By the other hand,
Finally, we use that
L'Hôpital's rule implies that the last limit is equal to zero, and the result follows.□
Similar to (9), now we give a basic estimate for the logarithm function: since xs ≤ exp(xs) for all x, s > 0, we have
For
Finally, we compute the Hilbert transform of odd powers of | ln x|. The computations are very similar to those in the proof of Lemma 18.
For
For t > 1 we have
We just make a sketch of the proof. Let t ∈ (0, 1) fixed. We have the following equalities
Therefore we obtain
The last limit is equal to zero and the result follows. When t > 1 a change of variables shows that
For
In this setting, we also can use the functions in
5. Conclusion
We gather several methods and techniques to construct systematically Stieltjes classes for M-indeterminate probability densities defined on
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Acknowledgements
Funding: This research received no specific grant from any funding agency.