New Checkable Conditions for Moment Determinacy of Probability Distributions

Stoyanov, Jordan; Lin, Gwo Dong; Kopanov, Peter

doi:10.1137/s0040585x97t990083

Cited by 22 publications

(20 citation statements)

References 11 publications

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“…These methods can be applied to the Markov moment problem as well (see [35], Theorem 2 and [2], Corollary 12.29). Sufficient criteria for determinacy and indeterminacy are also under attention (see [2,20]); (3) We review our earlier results on polynomial approximation on unbounded subsets, and its applications to the vector-valued Markov moment problem, recently published, completed, and generalized in [38] (see also [36]). To this aim, we essentially use the notion of a moment-determinate (M− determinate) measure (see [2,13,20]); (4) Two results on truncated scalar-valued Markov moment problem [37] are briefly discussed.…”

Section: Methodsmentioning

confidence: 99%

“…The symbol used below has the usual meaning of "monotone increasing". [20], Theorem 1; Hamburger case). Assume that the density f of Ψ is symmetrical on R, and continuous and strictly positive outside an interval (−t 0 , t 0 ), t 0 > 1, such that the following conditions hold:…”

Section: Theorem 8 (Seementioning

confidence: 99%

“…Theorem 13. (See [20], Theorem 2; Stieltjes case). Assume that the density g of Φ is continuous and strictly positive on [a, ∞) for some a > 1, such that the following conditions hold:…”

Section: Theorem 12 (Seementioning

confidence: 99%

“…For general knowledge in analysis and functional analysis related to the present paper, see [1][2][3][4][5][6][7][8][9][10]. Various aspects of the moment problem are discussed in [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. For other applications, results on Markov moment problem and connections with other fields of analysis see [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42].…”

Section: Introductionmentioning

confidence: 99%

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On the Moment Problem and Related Problems

Olteanu

2021

Mathematics

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Firstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence (yj)j∈ℕn of real numbers and a closed subset F⊆ℝn, n∈{1,2,…}, find a positive regular Borel measure μ on F such that ∫Ftjdμ=yj, j∈ℕn. This is the full moment problem. The existence, uniqueness, and construction of the unknown solution μ are the focus of attention. The numbers yj, j∈ℕn are called the moments of the measure μ. When a sandwich condition on the solution is required, we have a Markov moment problem. Secondly, we study the existence and uniqueness of the solutions to some full Markov moment problems. If the moments yj are self-adjoint operators, we have an operator-valued moment problem. Related results are the subject of attention. The truncated moment problem is also discussed, constituting the third aim of this work.

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Section: Methodsmentioning

confidence: 99%

Section: Theorem 8 (Seementioning

confidence: 99%

“…Theorem 13. (See [20], Theorem 2; Stieltjes case). Assume that the density g of Φ is continuous and strictly positive on [a, ∞) for some a > 1, such that the following conditions hold:…”

Section: Theorem 12 (Seementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Moment Problem and Related Problems

Olteanu

2021

Mathematics

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“…= ∞ for distributions on R) is not necessary for M-determinacy, this condition has been successfully used for characterization of probability distributions. A series of interesting results can be found, e.g., in the papers [4], [5], [15,16]. Note also that Carleman's condition may play a key role in the above Fréchet-Shohat Theorem when proving limit theorems for branching random walks (see [8], [10]).…”

Section: Though Carleman's Condition (1) (And Its Analoguementioning

confidence: 99%

On Conditions for a Probability Distribution to Be Uniquely Determined by Its Moments

Yarovaya¹,

Stoyanov²,

Kostyashin³

2020

Theory Probab. Appl.

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We study the relationship between the well-known Carleman's condition guaranteeing that a probability distribution is uniquely determined by its moments, and a recent easily checkable condition on the rate of growth of the moments. We use asymptotic methods in theory of integrals and involve properties of the Lambert W -function to show that the quadratic rate of growth of the ratios of consecutive moments, as a sufficient condition for uniqueness, is more restrictive than Carleman's condition. We derive a series of statements, one of them showing that Carleman's condition does not imply Hardy's condition, although the inverse implication is true. Related topics are also discussed.

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