Multiplicative decompositions and frequency of vanishing of nonnegative submartingales

In this paper, we establish a multiplicative decomposition formula for nonnegative local martingales and use it to characterize the set of continuous local submartingales Y of the form Y=N+A, where the measure dA is carried by the set of zeros of Y. In particular, we shall see that in the set of all local submartingales with the same martingale part in the multiplicative decomposition, these submartingales are the smallest ones. We also study some integrability questions in the multiplicative decomposition and interpret the notion of saturated sets in the light of our results.Comment: Typos corrected. Close to the published versio

Doob's maximal identity, multiplicative decompositions and enlargements of filtrations

In the theory of progressive enlargements of filtrations, the supermartingale $Z_{t}=\mathbf{P}(g>t\mid \mathcal{F}_{t})$ 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.Comment: Typos correcte

The zeros of random polynomials cluster uniformly near the unit circle

In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erdos and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomial.Comment: Corrects some typos and strengthens Theorem

Mod-discrete expansions

In this paper, we consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law cannot be expected. The setting is one in which the simplest approximation to the $n$'th random variable $X_n$ is by a particular member $R_n$ of a given family of distributions, whose variance increases with $n$. The basic assumption is that the ratio of the characteristic function of $X_n$ and that of R_n$ converges to a limit in a prescribed fashion. Our results cover a number of classical examples in probability theory, combinatorics and number theory

Mod-Gaussian convergence and the value distribution of ζ(1/2+it)\zeta(1/2+it) and related quantities

In the context of mod-Gaussian convergence, as defined previously in our work with J. Jacod, we obtain lower bounds for local probabilities for a sequence of random vectors which are approximately Gaussian with increasing covariance. This is motivated by the conjecture concerning the density of the set of values of the Riemann zeta function on the critical line. We obtain evidence for this fact, and derive unconditional results for random matrices in compact classical groups, as well as for certain families of L-functions over finite fields.Comment: 26 pages, 2 figures, v3: stronger quantitative statements and other change

The zeros of random polynomials cluster uniformly near the unit circle

In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erdos and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomia

Circular Jacobi ensembles and deformed Verblunsky coefficients

Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: $$c_{\delta,\beta}^{(n)} \prod_{1\leq k -1/2$. If$e$is a cyclic vector for a unitary$n\times n$matrix$U$, the spectral measure of the pair$(U,e)$is well parameterized by its Verblunsky coefficients$(\alpha_0, ..., \alpha_{n-1})$. We introduce here a deformation$(\gamma_0, >..., \gamma_{n-1})$of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product$r(\gamma_0)... r(\gamma_{n-1})$of elementary reflections parameterized by these coefficients. If$\gamma_0, ..., \gamma_{n-1}$are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above. These deformed Verblunsky coefficients also allow to prove that, in the regime$\delta = \delta(n)$with$\delta(n)/n \to \dd$, the spectral measure and the empirical spectral distribution weakly converge to an explicit nontrivial probability measure supported by an arc of the unit circle. We also prove the large deviations for the empirical spectral distribution. Formula with Formula . If e is a cyclic vector for a unitary n x n matrix U, the spectral measure of the pair (U, e) is well parameterized by its Verblunsky coefficients ({alpha}0, ..., {alpha}n-1). We introduce here a deformation ({gamma}0, ..., {gamma}n-1) of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product r({gamma}0)··· r({gamma}n-1) of elementary reflections parameterized by these coefficients. If {gamma}0, ..., {gamma}n-1 are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above. These deformed Verblunsky coefficients also allow us to prove that, in the regime {delta} = {delta} (n) with {delta} (n)/ n -> β d/2, the spectral measure and the empirical spectral distribution weakly converge to an explicit nontrivial probability measure supported by an arc of the unit circle. We also prove the large deviations for the empirical spectral distributio

The zeros of random polynomials cluster uniformly near the unit circle

In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erdos and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomia