1,132,578 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

### 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

### Median inverse problem and approximating the number of $k$-median inverses of a permutation

We introduce the "Median Inverse Problem" for metric spaces. In particular,
having a permutation $\pi$ in the symmetric group $S_n$ (endowed with the
breakpoint distance), we study the set of all $k$-subsets
$\{x_1,...,x_k\}\subset S_n$ for which $\pi$ is a breakpoint median. The set of
all $k$-tuples $(x_1,...,x_k)$ with this property is called the $k$-median
inverse of $\pi$. Finding an upper bound for the cardinality of this set, we
provide an asymptotic upper bound for the probability that $\pi$ is a
breakpoint median of $k$ permutations $\xi_1^{(n)},...,\xi_k^{(n)}$ chosen
uniformly and independently at random from $S_n$

### 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

- …