86 research outputs found

### Universality of the mean number of real zeros of random trigonometric polynomials under a weak Cramer condition

We investigate the mean number of real zeros over an interval $[a,b]$ of a
random trigonometric polynomial of the form $\sum_{k=1}^n a_k \cos(kt)+b_k
\sin(kt)$ where the coefficients are i.i.d. random variables. Under mild
assumptions on the law of the entries, we prove that this mean number is
asymptotically equivalent to $\frac{n(b-a)}{\pi\sqrt{3}}$ as $n$ goes to
infinity, as in the known case of standard Gaussian coefficients. Our principal
requirement is a new Cramer type condition on the characteristic function of
the entries which does not only hold for all continuous distributions but also
for discrete ones in a generic sense. To our knowledge, this constitutes the
first universality result concerning the mean number of zeros of random
trigonometric polynomials. Besides, this is also the first time that one makes
use of the celebrated Kac-Rice formula not only for continuous random variables
as it was the case so far, but also for discrete ones. Beyond the proof of a
non asymptotic version of Kac-Rice formula, our strategy consists in using
suitable small ball estimates and Edgeworth expansions for the Kolmogorov
metric under our new weak Cramer condition, which both constitute important
byproducts of our approach

### Generalization of the Nualart-Peccati criterion

The celebrated Nualart-Peccati criterion [Ann. Probab. 33 (2005) 177-193]
ensures the convergence in distribution toward a standard Gaussian random
variable $N$ of a given sequence $\{X_n\}_{n\ge1}$ of multiple Wiener-It\^{o}
integrals of fixed order, if $\mathbb {E}[X_n^2]\to1$ and $\mathbb
{E}[X_n^4]\to \mathbb {E}[N^4]=3$. Since its appearance in 2005, the natural
question of ascertaining which other moments can replace the fourth moment in
the above criterion has remained entirely open. Based on the technique recently
introduced in [J. Funct. Anal. 266 (2014) 2341-2359], we settle this problem
and establish that the convergence of any even moment, greater than four, to
the corresponding moment of the standard Gaussian distribution, guarantees the
central convergence. As a by-product, we provide many new moment inequalities
for multiple Wiener-It\^{o} integrals. For instance, if $X$ is a normalized
multiple Wiener-It\^{o} integral of order greater than one, $\forall
k\ge2,\qquad \mathbb {E}\bigl[X^{2k}\bigr]>\mathbb {E}
\bigl[N^{2k}\bigr]=(2k-1)!!.$Comment: Published at http://dx.doi.org/10.1214/14-AOP992 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Stein's method, Malliavin calculus, Dirichlet forms and the fourth moment theorem

The fourth moment theorem provides error bounds of the order $\sqrt{{\mathbb
E}(F^4) - 3}$ in the central limit theorem for elements $F$ of Wiener chaos of
any order such that ${\mathbb E}(F^2) = 1$. It was proved by Nourdin and
Peccati (2009) using Stein's method and the Malliavin calculus. It was also
proved by Azmoodeh, Campese and Poly (2014) using Stein's method and Dirichlet
forms. This paper is an exposition on the connections between Stein's method
and the Malliavin calculus and between Stein's method and Dirichlet forms, and
on how these connections are exploited in proving the fourth moment theorem

### Convergence in distribution norms in the CLT for non identical distributed random variables

We study the convergence in distribution norms in the Central Limit Theorem
for non identical distributed random variables that is $\varepsilon_{n}(f):={\mathbb{E}}\Big(f\Big(\frac 1{\sqrt
n}\sum_{i=1}^{n}Z_{i}\Big)\Big)-{\mathbb{E}}\big(f(G)\big)\rightarrow 0$
where $Z_{i}$ are centred independent random variables and $G$ is a Gaussian
random variable. We also consider local developments (Edgeworth expansion).
This kind of results is well understood in the case of smooth test functions
$f$. If one deals with measurable and bounded test functions (convergence in
total variation distance), a well known theorem due to Prohorov shows that some
regularity condition for the law of the random variables $Z_{i}$, $i\in
{\mathbb{N}}$, on hand is needed. Essentially, one needs that the law of $Z_{i}$ is locally lower bounded by the Lebesgue measure (Doeblin's condition).
This topic is also widely discussed in the literature. Our main contribution is
to discuss convergence in distribution norms, that is to replace the test
function $f$ by some derivative $\partial_{\alpha }f$ and to obtain upper
bounds for $\varepsilon_{n}(\partial_{\alpha }f)$ in terms of the infinite norm
of $f$. Some applications are also discussed: an invariance principle for the
occupation time for random walks, small balls estimates and expected value of
the number of roots of trigonometric polynomials with random coefficients

### Fourth Moment Theorems for Markov Diffusion Generators

Inspired by the insightful article arXiv:1210.7587, we revisit the
Nualart-Peccati-criterion arXiv:math/0503598 (now known as the Fourth Moment
Theorem) from the point of view of spectral theory of general Markov diffusion
generators. We are not only able to drastically simplify all of its previous
proofs, but also to provide new settings of diffusive generators (Laguerre,
Jacobi) where such a criterion holds. Convergence towards gamma and beta
distributions under moment conditions is also discussed.Comment: 15 page

### Classical and free Fourth Moment Theorems: universality and thresholds

Let $X$ be a centered random variable with unit variance, zero third moment,
and such that $E[X^4] \ge 3$. Let $\{F_n : n\geq 1\}$ denote a normalized
sequence of homogeneous sums of fixed degree $d\geq 2$, built from independent
copies of $X$. Under these minimal conditions, we prove that $F_n$ converges in
distribution to a standard Gaussian random variable if and only if the
corresponding sequence of fourth moments converges to $3$. The statement is
then extended (mutatis mutandis) to the free probability setting. We shall also
discuss the optimality of our conditions in terms of explicit thresholds, as
well as establish several connections with the so-called universality
phenomenon of probability theory. Both in the classical and free probability
frameworks, our results extend and unify previous Fourth Moment Theorems for
Gaussian and semicircular approximations. Our techniques are based on a fine
combinatorial analysis of higher moments for homogeneous sums.Comment: 26 page

### Stein's method on the second Wiener chaos : 2-Wasserstein distance

In the first part of the paper we use a new Fourier technique to obtain a
Stein characterizations for random variables in the second Wiener chaos. We
provide the connection between this result and similar conclusions that can be
derived using Malliavin calculus. We also introduce a new form of discrepancy
which we use, in the second part of the paper, to provide bounds on the
2-Wasserstein distance between linear combinations of independent centered
random variables. Our method of proof is entirely original. In particular it
does not rely on estimation of bounds on solutions of the so-called Stein
equations at the heart of Stein's method. We provide several applications, and
discuss comparison with recent similar results on the same topic

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