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13
Exponential functionals of Lévy processes
 Probabilty Surveys
, 2005
"... Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0 ..."
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Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0
On powers of Stieltjes moment sequences
, 2005
"... For a Bernstein function f the sequence sn = f(1)·...·f(n) is a Stieltjes moment sequence with the property that all powers s c n, c> 0 are again Stieltjes moment sequences. We prove that s c n is Stieltjes determinate for c ≤ 2, but it can be indeterminate for c> 2 as is shown by the moment s ..."
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For a Bernstein function f the sequence sn = f(1)·...·f(n) is a Stieltjes moment sequence with the property that all powers s c n, c> 0 are again Stieltjes moment sequences. We prove that s c n is Stieltjes determinate for c ≤ 2, but it can be indeterminate for c> 2 as is shown by the moment sequence (n!) c, corresponding to the Bernstein function f(s) = s. Nevertheless there always exists a unique product convolution semigroup (ρc)c>0 such that ρc has moments s c n. We apply the indeterminacy of (n!) c for c> 2 to prove that the distribution of the product of p independent identically distributed normal random variables is indeterminate if and only if p ≥ 3.
Permanents, Transportation Polytopes and Positive Definite Kernels on Histograms
, 2007
"... For two integral histograms r =(r1,...,rd) and c = (c1,...,cd) of equal sum N, the MongeKantorovich distance dMK(r, c) between r and c parameterized by a d × d distance matrix T is the minimum of all costs <F,T>taken over matrices F of the transportation polytope U(r, c). Recent results sugge ..."
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Cited by 4 (0 self)
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For two integral histograms r =(r1,...,rd) and c = (c1,...,cd) of equal sum N, the MongeKantorovich distance dMK(r, c) between r and c parameterized by a d × d distance matrix T is the minimum of all costs <F,T>taken over matrices F of the transportation polytope U(r, c). Recent results suggest that this distance is not negative definite, and hence, through Schoenberg’s wellknown result, exp( − 1 t dMK) may not be a positive definite kernel for all t> 0. Rather than using directly dMK to define a similarity between r and c,wepropose in this paper to investigate kernels on r and c based on the whole transportation polytope U(r, c). We prove that when r and c have binary counts, which is equivalent to stating that r and c represent clouds of points of equal size, the permanent of an adequate Gram matrix induced by the distance matrix T is a positive definite kernel under favorable conditions on T. We also show that the volume of the polytope U(r, c), that is the number of integral transportation plans, is a positive definite quantity in r and c through the RobinsonSchenstedKnuth correspondence between transportation matrices and Young Tableaux. We follow by proposing a family of positive definite kernels related to the generating function of the polytope through recent results obtained separately by A. Barvinok on the one hand, and C.Berg and A.J. Duran on the other hand. We finally present preliminary results led on a subset of the MNIST database to compare clouds of points through the permanent kernel.
Iteration of the rational function z − 1/z and a Hausdorff moment sequence
, 2008
"... In a previous paper we considered a positive function f, uniquely determined for s> 0 by the requirements f(1) = 1, log(1/f) is convex and the functional equation ..."
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Cited by 3 (3 self)
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In a previous paper we considered a positive function f, uniquely determined for s> 0 by the requirements f(1) = 1, log(1/f) is convex and the functional equation
On powers of Stieltjes moment sequences, II
"... We consider the set of Stieltjes moment sequences, for which every positive power is again a Stieltjes moment sequence, we and prove an integral representation of the logarithm of the moment sequence in analogy to the LévyKhintchine representation. We use the result to construct product convolution ..."
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We consider the set of Stieltjes moment sequences, for which every positive power is again a Stieltjes moment sequence, we and prove an integral representation of the logarithm of the moment sequence in analogy to the LévyKhintchine representation. We use the result to construct product convolution semigroups with moments of all orders and to calculate their Mellin transforms. As an application we construct a positive generating function for the orthonormal Hermite polynomials.
1 Introduction In [13] Bertoin et al. studied the distribution Iq of the exponential functional
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On a generalized Gamma . . .
, 2003
"... We discuss a probability distribution Iq depending on a parameter 0 < q < 1 and determined by its moments n!/(q; q)n. The treatment is purely analytical. The distribution has been discussed recently by Bertoin, Biane and Yor in connection with a study of exponential functionals of Lévy ..."
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We discuss a probability distribution Iq depending on a parameter 0 < q < 1 and determined by its moments n!/(q; q)n. The treatment is purely analytical. The distribution has been discussed recently by Bertoin, Biane and Yor in connection with a study of exponential functionals of Lévy