Homogeneous Sums Research Articles

High-Dimensional Central Limit Theorems for Homogeneous Sums

This paper develops a quantitative version of de Jong’s central limit theorem for homogeneous sums in a high-dimensional setting. More precisely, under appropriate moment assumptions, we establish an upper bound for the Kolmogorov distance between a multi-dimensional vector of homogeneous sums and a Gaussian vector so that the bound depends polynomially on the logarithm of the dimension and is governed by the fourth cumulants and the maximal influences of the components. As a corollary, we obtain high-dimensional versions of fourth-moment theorems, universality results and Peccati–Tudor-type theorems for homogeneous sums. We also sharpen some existing (quantitative) central limit theorems by applications of our result.

Global Gaussian Estimates for the Heat Kernel of Homogeneous Sums of Squares

Let ${\mathscr{H}}={\sum }_{j=1}^{m}{X_{j}^{2}}-\partial _{t}$ be a heat-type operator in $\mathbb {R}^{n+1}$ , where X = {X1,…,Xm} is a system of smooth Hörmander’s vector fields in $\mathbb {R}^{n}$ , and every Xj is homogeneous of degree 1 with respect to a family of non-isotropic dilations in $\mathbb {R}^{n}$ , while no underlying group structure is assumed. In this paper we prove global (in space and time) upper and lower Gaussian estimates for the heat kernel Γ(t,x;s,y) of ${\mathscr{H}}$ , in terms of the Carnot-Carathéodory distance induced by X on $\mathbb {R}^{n}$ , as well as global upper Gaussian estimates for the t- or X-derivatives of any order of Γ. From the Gaussian bounds we derive the unique solvability of the Cauchy problem for a possibly unbounded continuous initial datum satisfying exponential growth at infinity. Also, we study the solvability of the ${\mathscr{H}}$ -Dirichlet problem on an arbitrary bounded domain. Finally, we establish a global scale-invariant Harnack inequality for non-negative solutions of ${\mathscr{H}}u=0$ .

Almost sure convergence on chaoses

We present several new phenomena about almost sure convergence on homogeneous chaoses that include Gaussian Wiener chaos and homogeneous sums in independent random variables. Concretely, we establish the fact that almost sure convergence on a fixed finite sum of chaoses forces the almost sure convergence of each chaotic component. Our strategy uses “extra randomness” and a simple conditioning argument. These ideas are close to the spirit of Stein’s method of exchangeable pairs. Some natural questions are left open in this note.

Multidimensional limit theorems for homogeneous sums: A survey and a general transfer principle

The aim of the present paper is to establish the multidimensional counterpart of the \textit{fourth moment criterion} for homogeneous sums in independent leptokurtic and mesokurtic random variables (that is, having positive and zero fourth cumulant, respectively), recently established in \cite{NPPS} in both the classical and in the free setting. As a consequence, the transfer principle for the Central limit Theorem between Wiener and Wigner chaos can be extended to a multidimensional transfer principle between vectors of homogeneous sums in independent commutative random variables with zero third moment and with non-negative fourth cumulant, and homogeneous sums in freely independent non-commutative random variables with non-negative fourth cumulant.

A four moments theorem for Gamma limits on a Poisson chaos

This paper deals with sequences of random variables belonging to a fixed chaos of order q generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of the third and fourth moment of such a suitably normalized sequence to the third and fourth moment of a centred Gamma law implies convergence in distribution of the involved random variables. A positive answer is obtained for q = 2 and q = 4. The proof of this four moments theorem is based on a number of new estimates for contraction norms. Applications concern homogeneous sums and U-statistics on the Poisson space.

Classical and Free Fourth Moment Theorems: Universality and Thresholds

Let \(X\) be a centered random variable with unit variance and zero third moment, and such that \(\mathrm{IE}[X^4] \ge 3\). Let \(\{F_n {:}\, n\ge 1\}\) denote a normalized sequence of homogeneous sums of fixed degree \(d\ge 2\), 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.

Malliavin-Stein method for variance-gamma approximation on Wiener space

We combine Malliavin calculus with Stein's method to derive bounds for the Variance-Gamma approximation of functionals of isonormal Gaussian processes, in particular of random variables living inside a fixed Wiener chaos induced by such a process. The bounds are presented in terms of Malliavin operators and norms of contractions. We show that a sequence of distributions of random variables in the second Wiener chaos converges to a Variance-Gamma distribution if and only if their moments of order two to six converge to that of a Variance-Gamma distributed random variable (six moment theorem). Moreover, simplified versions for Laplace or symmetrized Gamma distributions are presented. Also multivariate extensions and a universality result for homogeneous sums are considered.

Invariance principles for homogeneous sums of free random variables

We extend, in the free probability framework, an invariance principle for multilinear homogeneous sums with low influences recently established in [E. Mossel, R. O'Donnell and K. Oleszkiewicz (2010). Noise stability of functions with low influences: invariance and optimality. {\it Ann. Math.} {\bf 171}, no. 1, 295-341]. To do so, a hypercontractivity property for those homogeneous sums is necessary, and to prove it has turned out to be our main task. Finally, we deduce from our extension several universality phenomenons, in the spirit of the paper [I. Nourdin, G. Peccati and G. Reinert (2010). Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. {\it Ann. Probab.} {\bf 38}, no. 5, 1947-1985].

Universal Gaussian fluctuations on the discrete Poisson chaos

We prove that homogeneous sums inside a fixed discrete Poisson chaos are universal with respect to normal approximations. This result parallels some recent findings, in a Gaussian context, by Nourdin, Peccati and Reinert (Ann. Probab. 38 (2010) 1947–1985). As a by-product of our analysis, we provide some refinements of the CLTs for random variables on the Poisson space proved by Peccati, Sole, Taqqu and Utzet (Ann. Probab. 38 (2010) 443–478) and by Peccati and Zheng (Electron. J. Probab. 15 (2010) 1487–1527).

Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos

We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. In particular, we show that chaotic random variables enjoy the following form of universality: (a) the normal and chi-square approximations of any homogenous sum can be completely characterized and assessed by first switching to its Wiener chaos counterpart, and (b) the simple upper bounds and convergence criteria available on the Wiener chaos extend almost verbatim to the class of homogeneous sums.

On reciprocity formulae for inhomogeneous and homogeneous Dedekind sums

A number of authors, including Apostol [1], Carlitz [2], Mikolás [5] and Rademacher [9] have obtained linear relations for the Dedekind sums(the inhomogeneous sum) and the homogeneous sumHere denotes the periodic extension into ℝ of the Bernoulli polynomial Bm(X) on [0, 1] given by the relationwith the exception that we define