A transformation from Hausdorff to Stieltjes moment sequences

Berg, Christian; Durán, Antonio J.

doi:10.1007/bf02385478

Cited by 32 publications

(46 citation statements)

References 21 publications

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“…While our arguments are probabilistic, Berg and Duran [1] obtain similar results by analytic methods.…”

Section: Introduction and Main Resultssupporting

confidence: 78%

“…with W a standard BM (1). Here the point of the representation (2) rather than (1) is that R (ν) 2 is the diffusion with the self-similarity (or semi-stability) property used by Lamperti [15] in his main result, Theorem 4.1, part of which may informally be stated as follows: any 1-self-similar strictly positive and 'nice' Markov process is a time-change of the exponential of a Lévy process; see (5) below.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…where Z (1) = n 1 X i, (1) as usual, andP t (z, dỹ) is the well understood transition function for the one-dimensional 1-semi-stable Markov process Z resulting from the one-dimensional Lamperti representation of the Lévy processξ = ξ i , cf. the discussion of the agglomeration property in Corollary 5 below.…”

Section: The Multi-self-similarity Property; Proofs Of Theorems 1 2 mentioning

confidence: 99%

“…Thus, in general, the transition function for X may be found from the knowledge of the transition functions from state 1 in the one-dimensional case and an understanding of the conditional law of X (1) s given Z (1) s for all s. Note that by Dynkin's criterion (see e.g. Pitman and Rogers [16]) the discussion leading to (20) shows that Z = n 1 X i is in fact a Markov process with respect to the filtration generated by X.…”

Section: The Multi-self-similarity Property; Proofs Of Theorems 1 2 mentioning

confidence: 99%

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Multi-self-similar Markov processes on ℝ+ n and their Lamperti representations

Jacobsen¹,

Yor

2003

Probab. Theory Relat. Fields

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A classical result, due to Lamperti, establishes a one-to-one correspondence between a class of strictly positive Markov processes that are self-similar, and the class of one-dimensional Lévy processes. This correspondence is obtained by suitably time-changing the exponential of the Lévy process. In this paper we generalise Lamperti's result to processes in n dimensions. For the representation we obtain, it is essential that the same time-change be applied to all coordinates of the processes involved. Also for the statement of the main result we need the proper concept of self-similarity in higher dimensions, referred to as multi-self-similarity in the paper.The special case where the Lévy process ξ is standard Brownian motion in n dimensions is studied in detail. There are also specific comments on the case where ξ is an n-dimensional compound Poisson process with drift.Finally, we present some results concerning moment sequences, obtained by studying the multi-self-similar processes that correspond to n-dimensional subordinators.

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“…While our arguments are probabilistic, Berg and Duran [1] obtain similar results by analytic methods.…”

Section: Introduction and Main Resultssupporting

confidence: 78%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: The Multi-self-similarity Property; Proofs Of Theorems 1 2 mentioning

confidence: 99%

Section: The Multi-self-similarity Property; Proofs Of Theorems 1 2 mentioning

confidence: 99%

See 2 more Smart Citations

Multi-self-similar Markov processes on ℝ+ n and their Lamperti representations

Jacobsen¹,

Yor

2003

Probab. Theory Relat. Fields

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“…This class was introduced and extensively studied by S. Bernstein [5] and D. Widder [30] in connection with the so-called completely (or absolutely) monotonic analytic functions (see the definition in Section 3.2). We only mention a deep penetration of the both classes into complex analysis, inequalities analysis [2], special functions [25], probability theory [19], radial-function interpolation [29], harmonic analysis on semigroups [3] (for further discussion and references, see recent survey [4]). …”

Section: ])mentioning

confidence: 99%

Length functions of lemniscates

Kuznetsova

Tkachev²

2003

manuscripta mathematica

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Abstract. We study metric and analytic properties of generalized lemniscates Et(f ) = {z ∈ C : ln |f (z)| = t}, where f is an analytic function. Our main result states that the length function |Et(f )| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln |Et(f )| is convex on any interval free of critical points of ln |f |. As another application we deduce explicit formulae of the length function in some special cases.

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Non‐classical Tauberian and Abelian type criteria for the moment problem

Patie

Vaidyanathan

2022

Mathematische Nachrichten

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The aim of this paper is to provide some new criteria for the determinacy problem of the Stieltjes moment problem. We first give a Tauberian type criterion for moment indeterminacy that is expressed purely in terms of the asymptotic behavior of the moment sequence (and its extension to imaginary lines). Under an additional assumption this provides a converse to the classical Carleman's criterion, thus yielding an equivalent condition for moment determinacy. We also provide a criterion for moment determinacy that only involves the large asymptotic behavior of the distribution (or of the density if it exists), which can be thought of as an Abelian counterpart to the previous Tauberian type result. This latter criterion generalizes Hardy's condition for determinacy, and under some further assumptions yields a converse to the Pedersen's refinement of the celebrated Krein's theorem. The proofs utilize non-classical Tauberian results for moment sequences that are analogues to the ones developed in [8] and [3] for the bi-lateral Laplace transforms in the context of asymptotically parabolic functions. We illustrate our results by studying the time-dependent moment problem for the law of log-Lévy processes viewed as a generalization of the log-normal distribution. Along the way, we derive the large asymptotic behavior of the density of spectrally-negative Lévy processes having a Gaussian component, which may be of independent interest.

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A transformation from Hausdorff to Stieltjes moment sequences

Cited by 32 publications

References 21 publications

Multi-self-similar Markov processes on ℝ+ n and their Lamperti representations

Multi-self-similar Markov processes on ℝ+ n and their Lamperti representations

Length functions of lemniscates

Non‐classical Tauberian and Abelian type criteria for the moment problem

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