Contributions published at Quantitative Finance (Ashkan Nikeghbali)

In this paper, we relate the framework of mod-φ convergence to the construction of approximation schemes for lattice-distributed random variables. The point of view taken here is the one of Fourier analysis in the Wiener algebra, allowing the computation of asymptotic equivalents of the local, Kolmogorov and total variation distances. By using signed measures instead of probability measures, we are able to construct better approximations of discrete lattice distributions than the standard Poisson approximation. This theory applies to various examples arising from combinatorics and number theory: number of cycles in permutations, number of prime divisors of a random integer, number of irreducible factors of a random polynomial, etc. Our approach allows us to deal with approximations in higher dimensions as well. In this setting, we bring out the influence of the correlations between the components of the random vectors in our asymptotic formulas.

It is known that a unitary matrix can be decomposed into a product of reflections, one for each dimension, and that the Haar measure on the unitary group pushes forward to independent uniform measures on the reflections. We consider the sequence of unitary matrices given by successive products of random reflections. In this coupling, we show that powers of the sequence of matrices converge in a suitable sense to a flow of operators, which acts on a random vector space. The vector space has an explicit description as a subspace of the space of sequences of complex numbers. The eigenvalues of the matrices converge almost surely to the eigenvalues of the flow, which are distributed in law according to a sine-kernel point process. The eigenvectors of the matrices converge almost surely to vectors, which are distributed in law as Gaussian random fields on a countable set.

We prove local limit theorems for mod-φconvergent sequences of ran-dom variables,φbeing a stable distribution. In particular, we give two new proofsof the local limit theorem stated inDelbaen et al.(2015): one proof based on thenotion ofzone of controlintroduced inFéray et al.(2019+a), and one proof basedon the notion ofmod-φconvergence inL1(iR). These new approaches allow usto identify the infinitesimal scales at which the stable approximation is valid. Wecomplete our analysis with a large variety of examples to which our results ap-ply, and which stem from random matrix theory, number theory, combinatorics orstatistical mechanics.

We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the Gaussian Unitary Ensemble. In fact, the result is the by-product of a general limit theorem for the convergence of random entire functions whose zeros present a simple regularity property.

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It is known that a unitary matrix can be decomposed into a product of complex reflections, one for each dimension, and that these reflections are independent and uniformly distributed on the space where they live if the initial matrix is Haar-distributed. If we take an infinite sequence of such reflections, and consider their successive products, then we get an infinite sequence of unitary matrices of increasing dimension, all of them following the circular unitary ensemble. In this coupling, we show that the eigenvalues of the matrices converge almost surely to the eigenvalues of the flow, which are distributed according to a sine-kernel point process, and we get some estimates of the rate of convergence. Moreover, we also prove that the eigenvectors of the matrices converge almost surely to vectors which are distributed as Gaussian random fields on a countable set.

We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. Our scaling is performed at the so-called “microscopic” level, that is we consider the characteristic polynomial at points whose distance to $1$ has order $1 / n$. We prove that the rescaled characteristic polynomial does not even have a moment of order one, hence making the classical techniques of random matrix theory difficult to apply. In order to deal with this issue, we couple all the dimensions $n$ on a single probability space, in such a way that almost sure convergence occurs when $n$ goes to infinity. The strong convergence results in this setup provide us with a new approach to ratios: we are able to solve open problems about the limiting distribution of ratios of characteristic polynomials evaluated at points of the form exp($2i\pi\alpha/n$) and related objects (such as the logarithmic derivative). We also explicitly describe the dependence relation for the logarithm of the characteristic polynomial evaluated at several points on the microscopic scale. On the number theory side, inspired by the work by Keating and Snaith, we conjecture some new limit theorems for the value distribution of the Riemann zeta function on the critical line at the level of stochastic processes.

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Using Fourier analysis, we study local limit theorems in weak-convergence problems. Among many applications, we discuss random matrix theory, some probabilistic models in number theory, the winding number of complex Brownian motion and the classical situation of the central limit theorem, and a conjecture concerning the distribution of values of the Riemann zeta function on the critical line.

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