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The barnes G function and its relations with sums and products of generalized Gamma variables, in preparation
"... Abstract. We give a probabilistic interpretation for the Barnes G-function which appears in random matrix theory and in analytic number theory in the important moments conjecture due to Keating-Snaith for the Riemann zeta function, via the analogy with the characteristic polynomial of random unitary ..."
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Abstract. We give a probabilistic interpretation for the Barnes G-function which appears in random matrix theory and in analytic number theory in the important moments conjecture due to Keating-Snaith for the Riemann zeta function, via the analogy with the characteristic polynomial of random unitary matrices. We show that the Mellin transform of the characteristic polynomial of random unitary matrices and the Barnes G-function are intimately related with products and sums of gamma, beta and log-gamma variables. In particular, we show that the law of the modulus of the characteristic polynomial of random unitary matrices can be expressed with the help of products of gamma or beta variables, and that the reciprocal of the Barnes G-function has a Lévy-Khintchin type representation. These results lead us to introduce the so called generalized gamma convolution variables.
Optimal Lp-Metric for Minimizing Powered Deviations in Regression
"... Minimizations by least squares or by least absolute deviations are well known criteria in regression modeling. In this work the criterion of generalized mean by powered deviations is suggested. If the parameter of the generalized mean equals one or two, the fitting corresponds to the least absolute ..."
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Minimizations by least squares or by least absolute deviations are well known criteria in regression modeling. In this work the criterion of generalized mean by powered deviations is suggested. If the parameter of the generalized mean equals one or two, the fitting corresponds to the least absolute or the least squared deviations, respectively. Varying the power parameter yields an optimum value for the objective with a minimum possible residual error. Estimation of a most favorable value of the generalized mean parameter shows that it almost does not depend on data. The optimal power always occurs to be close to 1.7, so these powered deviations should be used for a better regression fit.
Complete monotonicity of a polygamma function plus the square of another polygamma function
, 2009
"... For m, n ∈ N, let ..."
Bounds for the logarithm of the Euler gamma function and its derivatives
, 2015
"... We consider differences between log Γ(x) and truncations of certain clas-sical asymptotic expansions in inverse powers of x − λ whose coefficients are expressed in terms of Bernoulli polynomials Bn(λ), and we obtain conditions under which these differences are strictly completely monotonic. In the s ..."
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We consider differences between log Γ(x) and truncations of certain clas-sical asymptotic expansions in inverse powers of x − λ whose coefficients are expressed in terms of Bernoulli polynomials Bn(λ), and we obtain conditions under which these differences are strictly completely monotonic. In the sym-metric cases λ = 0 and λ = 1/2, we recover results of Sonin, Nörlund and Alzer. Also we show how to derive these asymptotic expansions using the functional equation of the logarithmic derivative of the Euler gamma function, the representation of 1/x as a difference F (x + 1) − F (x), and a backward induction. 1
ANZIAM J. 44(2003), 609–623 INEQUALITIES FOR THE BETA FUNCTION OF n VARIABLES
, 2001
"... We present various inequalities for Euler’s beta function of n variables. One of our theorems states that the inequalities an 1Qn iD1 xi − B.x1; : : : ; xn / bn () hold for all xi 1 (i D 1; : : : ; n; n 3) with the best possible constants an D 0 and bn D 1 − 1=.n − 1/W. This extends a recently ..."
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We present various inequalities for Euler’s beta function of n variables. One of our theorems states that the inequalities an 1Qn iD1 xi − B.x1; : : : ; xn / bn () hold for all xi 1 (i D 1; : : : ; n; n 3) with the best possible constants an D 0 and bn D 1 − 1=.n − 1/W. This extends a recently published result of Dragomir et al., who investigated () for the special case n D 2. 1.
INEQUALITIES FOR THE GAMMA FUNCTION AND ESTIMATES FOR THE VOLUME OF SECTIONS OF B n p
, 2000
"... Abstract. Let Bn p = {(xi) ∈ Rn; ∑n 1 |xi | p ≤ 1} and let E be a k-dimensional subspace of Rn. We prove that |E ∩ Bn p | 1/k k ≥ |Bn p | 1/n n, for 1 ≤ k ≤ (n − 1)/2 and k = n − 1 whenever 1 < p < 2. We also consider 0 < p < 1 and other related cases. We obtain sharp inequalities invol ..."
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Abstract. Let Bn p = {(xi) ∈ Rn; ∑n 1 |xi | p ≤ 1} and let E be a k-dimensional subspace of Rn. We prove that |E ∩ Bn p | 1/k k ≥ |Bn p | 1/n n, for 1 ≤ k ≤ (n − 1)/2 and k = n − 1 whenever 1 < p < 2. We also consider 0 < p < 1 and other related cases. We obtain sharp inequalities involving Gamma function in order to get these results.
Universite de Tunis El Manar, Faculte des Sciences de Tunis
"... Dedicated to the memory of Professor Marc Yor Abstract. We establish a link between the distribution of an exponential func-tional, I; and the undershoots of a subordinator, which is given in terms of the associated harmonic potential measure. This allows us to give a necessary and sucient condition ..."
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Dedicated to the memory of Professor Marc Yor Abstract. We establish a link between the distribution of an exponential func-tional, I; and the undershoots of a subordinator, which is given in terms of the associated harmonic potential measure. This allows us to give a necessary and sucient condition in terms of the Levy measure for the exponential functional to be multiplicative innitely divisible. We then provide a formula for the moment generating functions of log I and logR where R is the so-called remainder random variable associated to I. We provide a realization of the remainder random variable R as an innite product involving independent last position random variables of the subordinator. Some properties of harmonic measures are obtained and some examples are provided. 1. Introduction and
