Celebratio Mathematica

A. Nikeghbali and M. Yor: “A definition and some characteristic properties of pseudo-stopping times,” Ann. Probab. 33 : 5 (2005), pp. 1804–1824. MR 2165580 Zbl 1083.60035 ArXiv math/0406459 article

A. Nikeghbali and M. Yor: “Doob’s maximal identity, multiplicative decompositions and enlargements of filtrations,” pp. 791–814 in Joseph Doob: A collection of mathematical articles in his memory, published as Ill. J. Math. 50 : 1–4. Issue edited by J. Burkholder. University of Illinois at Urbana-Champaign, 2006. MR 2247846 Zbl 1101.60059 incollection

In the theory of progressive enlargements of filtrations, the supermartingale \[ Z_t = \mathbf{P}(g\gt t \mid \mathcal{F}_t) \] associated with an honest time \( g \), and its additive (Doob–Meyer) decomposition, play an essential role. In this paper, we propose an alternative approach, using a multiplicative representation for the supermartingale \( Z_t \), based on Doob’s maximal identity. We thus give new examples of progressive enlargements. Moreover, we give, in our setting, a proof of the decomposition formula for martingales, using initial enlargement techniques, and use it to obtain some path decompositions given the maximum or minimum of some processes.

Joseph L. Doob	Related	Volume
Ashkan Nikeghbali	Related

C. P. Hughes, A. Nikeghbali, and M. Yor: “An arithmetic model for the total disorder process,” Probab. Theory Relat. Fields 141 : 1–2 (May 2008), pp. 47–59. MR 2372965 Zbl 1144.60020 article

P. Bourgade, C. P. Hughes, A. Nikeghbali, and M. Yor: “The characteristic polynomial of a random unitary matrix: A probabilistic approach,” Duke Math. J. 145 : 1 (2008), pp. 45–69. MR 2451289 Zbl 1155.15025 ArXiv 0706.0333 article

In this article, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin–Fourier transform of such a random polynomial, first obtained by Keating and Snaith in [2000] using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith in [2000] is now obtained from the classical central limit theorems of probability theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm-type results.

A. Nikeghbali and M. Yor: “The Barnes \( G \) function and its relations with sums and products of generalized gamma convolution variables,” Electron. Commun. Probab. 14 (2009), pp. 396–411. Article no. 39. MR 2545290 Zbl 1189.60073 ArXiv 0707.3187 article

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. This leads us to prove some non standard type of limit theorems for the logarithmic mean of the so called generalized gamma convolutions.

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