Schedule for: 17w5080 - High Dimensional Probability

A welcome drink will be served at the hotel.

We consider infinitely divisible random fields perturbed by an additive independent noise. We investigate admissible perturbations under which the perturbed field, which need not be infinitely divisible, is absolutely continuous with respect to the unperturbed one, and establish the related isomorphism identities. The celebrated Dynkin's isomorphism theorem is an example of such phenomenon, where the local time of a Markov process is the perturbation of a permanental field.

The talk is motivated by the properties surrounding the spectral density of a stationary process and of a random field. We start by presenting a characterization of the spectral density in function of projection operators on sub-sigma fields. We also point out that the limiting distribution of the real and the imaginary part of the Fourier transform of a stationary random field is almost surely an independent vector with Gaussian marginal distributions, whose variance is, up to a constant, the field’s spectral density. The dependence structure of the random field is general and we do not impose any conditions on the speed of convergence to zero of the covariances, or smoothness of the spectral density. The only condition required is that the variables are adapted to a commuting filtration and are regular in some sense. The results go beyond the Bernoulli fields and apply to both short range and long range dependence. The method of proof is based on new probabilistic methods based on martingale approximations and also on borrowed tools from harmonic analysis. Several examples to linear, Volterra and Gaussian random fields will be presented. This is a joint work with Na Zhang.

A Coulomb gas is the canonical Gibbs measure associated with a system of particles in electrostatic interaction. As the number of particles grows to infinity, the empirical measure of a Coulomb gas converges weakly towards an equilibrium measure, characterized by a variational principle. We obtain sub-gaussian concentration inequalities around this equilibrium measure, in the weak and Wasserstein topologies. This yields for instance a concentration inequality at the correct rate for the Ginibre ensemble. The proof relies on new functional inequalities, which are counterparts of Talagrand's transport inequality in the Coulomb interaction setting. Joint work with Djalil Chafaï and Adrien Hardy.

Following Talagrand's concentration results for permutations picked uniformly at random from a symmetric group, Luczak and McDiarmid have generalized it to more general groups of permutations which act suitably `locally'. Here we extend their results by setting transport-entropy inequalities on these permutations groups. Talagrand and Luczak-Mc-Diarmid concentration properties are consequences of these inequalities. The results are also generalised to a larger class of measures including Ewens distributions of arbitrary parameter on the symmetric group. By projection, we derive transport-entropy inequalities for the uniform law on the slice of the discrete hypercube and more generally for the multinomial law. One typical application is deviation bounds for the so-called configuration functions, such as the number of cycles of given lenght in a random permutation.

We present a new representation formula of the variance of a function f under the standard gaussian measure along the Ornstein-Uhlenbeck semi-group. This representation can bee seen as Taylor formula with remainder term. The proof rests on elementary arguments as interpolation, integration by parts and fundamental theorem of analysis. From this representation we can deduce new bound of the variance, in term of norms L^1 and L^2 of partial derivatives, in the same spirit as Talagrand's inequality. The scheme of proof can be also used in the framework of the discrete cube C_n=\{-1,1}^n and permit us to obtain a theorem for the influence of boolean function which can be seen as an extension of the theorem of Kahn-Kalai-Linail at the order 2.

We will present a generalization of a theorem of Rogozin that identifies uniform distributions as extremizers of a class of inequalities, and show how the result can reduce specific random variables questions to geometric ones. In particular, by extending "cube slicing" results of K. Ball, we achieve a unification and sharpening of recent bounds on densities achieved as projections of product measures due to Rudelson and Vershynin, and the bounds on sums of independent random variable due to Bobkov and Chistyakov. Time permitting we will also discuss connections with generalizations of the entropy power inequality.

We shall discuss extrema of the entropy and moments of weighted sums of iid random variables with densities proportional to exp(-|t|^q), subject to the variance being fixed. Based on joint work with A. Eskenazis and P. Nayar.

A problem of efficient estimation of a smooth functional of unknown covariance operator $\Sigma$ of a mean zero Gaussian random variable in a Hilbert space based on a sample of its i.i.d. observations will be discussed. The goal is to find an estimator whose distribution is approximately normal with a minimax optimal variance in a setting when either the dimension of the space, or so called effective rank of the covariance operator are allowed to be large (although much smaller than the sample size). This problem has been recently solved in our joint paper with Loeffler and Nickl in the case of estimation of a linear functional of unknown eigenvector of $\Sigma$ corresponding to its largest eigenvalue (the top principal component). The efficient estimator developed in this paper does not coincide with the naive estimator based on the top principal component of sample covariance which is not efficient due to its large bias. An approach to a more general problem of efficient estimation of a functional $\langle f(\Sigma), B\rangle$ for a given sufficiently smooth function $f:{\mathbb R}\mapsto {\mathbb R}$ and given operator $B$ will be also discussed.

his talk is devoted to strong approximations in the dependent setting. The famous results of Koml\'os, Major and Tusn\'ady (1975-1976) state that it is possible to approximate almost surely the partial sums of size $n$ of i.i.d. centered random variables in ${\mathbb L}^p$ ($p >2$) by a Wiener process with an error term of order $o(n^{1/p})$. In the case of functions of random iterates generated by an iid sequence, we we shall give new dependent conditions, expressed in terms of a natural coupling (in ${\mathbb L}^\infty$ or in ${\mathbb L}^1$), under which the strong approximation result holds with rate $o(n^{1/p})$. The proof is an adaptation of the recent construction given in Berkes, Liu and Wu (2014). As we shall see our conditions are well adapted to a large variety of models, including left random walks on $GL_d({\mathbb R})$, contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains. We shall also provide some examples showing that our ${\mathbb L}^1$-coupling condition is in some sense optimal. This talk is based on a joint work with J. Dedecker and C. Cuny.

I will present various types of uncertainty relations satisfied by Haar distributed random unitary matrices in high dimensions. If time permits I will also discuss their applications to quantum information theory (in particular to the problems of information locking and data hiding) and to special cases of the Dvoretzky theorem.

Estimation of the covariance matrix has attracted significant attention of the statistical research community over the years, partially due to important applications such as Principal Component Analysis. However, frequently used empirical covariance estimator (and its modifications) is very sensitive to outliers, or ``atypical’’ points in the sample. As P. Huber wrote in 1964, “...This raises a question which could have been asked already by Gauss, but which was, as far as I know, only raised a few years ago (notably by Tukey): what happens if the true distribution deviates slightly from the assumed normal one? As is now well known, the sample mean then may have a catastrophically bad performance…” Motivated by Tukey's question, we develop a new estimator of the (element-wise) mean of a random matrix, which includes covariance estimation problem as a special case. Assuming that the entries of a matrix possess only finite second moment, this new estimator admits sub-Gaussian or sub-exponential concentration around the unknown mean in the operator norm. Our arguments rely on generic chaining techniques applied to operator-valued stochastic processes, as well as bounds on the trace moment-generating function. We will discuss extensions of our approach to matrix-valued U-statistics and examples such as matrix completion problem. Part of the talk will be based on a joint work with Xiaohan Wei.

This talk is concerned with growth-fragmentation models, implemented for investigating the growth of a population of cells. From a stochastic point of view, we are dealing with the evolution of a system of particles. The evolution of the system is then driven by two phenomenons. First, particles evolve on a deterministic basis: they age, they grow. Secondly, particles split randomly: a particle of age $a$ or size $x$ splits into two particles (of age $0$, of size at birth $x/2$), at a rate $B$ depending on the age or on the size of the splitting particle. The division rate $B$ is an unknown function, on which we focus our attention. The main goal of this statistical work is to get a nonparametric estimate of the division rate. To do so, various observation schemes may be considered: {\bf (i)} Observation {\it up to a given generation} in the genealogical tree of the population. {\bf (ii)} Continuous time observation {\it up to time $T$}. This scheme, which differs radically from those previously used, entails specific difficulties -- mainly a bias selection. As different as these two schemes may be, I will try to highlight their common features. Mainly, a competition occurs between the growth of the tree (measured by the Malthus parameter) and the convergence to equilibrium of the branching process (measured by some $\rho_B$ say). We still have to deal with open questions in these toy models: adaptativity with respect to the smothness of $B$ - which requires some deviation inequalities (availaible to treat the observation scheme (i) but not (ii)), but also adaptivity with respect to the parameter $\rho_B$ measuring the convergence to equilibrium.

Let $X_1,\ldots,X_n$ be random variables such that there exists a constant $C>1$ satisfying $\|X_i\|_{2p} \leq C \|X_i\|_p$ for every $p \geq 1$. We define random chaos $S=\sum a_{i_1,...,i_d} X_{i_1}\cdots X_{i_d}$. We will show two-sided deterministic bounds on $||S||_p$, with constant depending only on $C$ and $d$ in two cases: 1) $X_1,\ldots,X_n$ are nonnegative and $a_{i_1,...,i_d}\geq 0$. 2) $X_1,\ldots ,X_n$ are symmetric, $d=2$.

Both for random words and random permutations, I will present a panoramic view of recent results on the asymptotic law of the, centered and normalized, length of their longest common (and increasing) subsequences. Tools and results involve concentration inequalities for geodesics of LCSs paths, Stein' s method as well as maximal eigenvalues of some Gaussian random matrices.

Probabilistic models of directed polymers in random environment have received considerable attention in recent years. Much of this attention has focused on integrable models. In this talk, I will describe some new computational tools that do not require integrability. As an application, I will sketch the proof of a subsequential version of the longstanding conjecture that in the low temperature phase, the endpoint distribution is asymptotically localized in a region of stochastically bounded diameter. This is based on joint work with Erik Bates.

In this talk we consider sequences of polynomials that satisfy differential--difference recurrences. Our interest is motivated by the fact that polynomials satisfying such recurrences frequently appear as generating polynomials of integer valued random variables that are of interest in discrete mathematics. It is, therefore, of interest to understand the properties of such polynomials and their probabilistic consequences. We will be primarily interested in the limiting distribution of the corresponding random variables and we give a few examples, leading to a Poisson, normal, and Rayleigh distributions.

By combining Stein's method and Malliavin calculus on the Poisson space with an adaption of the spectral viewpoint highlighted by Ledoux (2012) we provide exact fourth moment bounds for the normal approximation of multiple Wiener-Itô integrals on the Poisson space. In particular, we show that the fourth moment phenomenon first discovered by Nualart and Peccati (2005) also holds true in this discrete framework. This is joint work with Giovanni Peccati.

We study the volume ratio of the projective tensor products $\ell^n_p\otimes_{\pi}\ell_q^n\otimes_{\pi}\ell_r^n$ with $1\leq p\leq q \leq r \leq \infty$. We obtain asymptotic formulas that are sharp in almost all cases. From the Bourgain-Milman bound on the volume ratio of Banach spaces in terms of their cotype 2 constant, we obtain, as a consequence of our estimates, information on the cotype of these 3-fold projective tensor products. Our results naturally generalize to k-fold products.

Let us define, for a compact set $A \subset \R^n$, the Minkowski averages of $A$: $$A(k) = \left\{\frac{a_1+\cdots +a_k}{k} : a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots + A}}\Big).$$ I shall show some monotonicity properties of $A(k)$ towards convexity when considering, the Hausdorff distance, the volume deficit and a non-convexity index of Schneider. For the volume deficit, the monotonicity holds in dimension 1 but fails for $n\ge 12$, thus disproving a conjecture of Bobkov, Madiman and Wang. For Schneider's non-convexity index, a strong form of monotonicity holds. And for the Hausdorff distance, some monotonicity holds for $k$ large enough, depending of the ambient dimension and not on the set $A$. Based on a work in collaboration with Mokshay Madiman, Arnaud Marsiglietti and Artem Zvavitch.

Since the 60's $\gamma$-radonifying operators have been used in connection to Gaussian measures on Banach spaces. In recent years $\gamma$-radonifying operators play an important role in several subfields of (stochastic) analysis, usually when extending results from finite dimensions to infinite dimensions. For example this has played a major role in: harmonic analysis, functional calculus, control theory, stochastic integration theory, Malliavin calculus. In the talk I will give an overview on this and mention several recent results.

I would like to explain the role of chaining method in the comparison of certain functionals of processes - expectations of suprema or tails of suprema. The basic setting for the problem is that we have two processes and we can compare their increments in the sense of moments or tails. Now the question is does it imply that the suprema of these two processes can also be compared and what are the consequences when such a comparison is possible?

For any separable Banach space $(F,\| \cdot \|)$ and independent $F$-valued random vectors $X$ and $Y$ such that ${\mathbf E} \|X\|, {\mathbf E} \|Y\|< \infty$, we have $$\inf_{z \in F} ({\mathbf E}\|X-z\|+{\mathbf E}\|Y-z\|) \leq 3 \cdot {\mathbf E}\| X-Y\|.$$ Indeed, it suffices to consider $z=({\mathbf E} X+{\mathbf E} Y)/2$ and use Jensen's inequality. Assaf Naor asked whether the constant $3$ in the inequality is optimal. We will discuss this and related problems.

In this talk we present the Meyer-Yoeurp decomposition for UMD Banach space-valued martingales. Namely, we prove that $X$ is a UMD Banach space if and only if for any fixed $p\in (1,\infty)$, any $X$-valued martingale $M$ has a unique decomposition $M = M^d + M^c$ such that $M^d$ is a purely discontinuous martingale, $M^c$ is a continuous martingale, $M^c_0=0$, and $\mathbb E \|M^d_{\infty}\|^p + \mathbb E \|M^c_{\infty}\|^p\leq c_p \mathbb E \|M_{\infty}\|^p$. An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingale into a sum of a quasi-left continuous martingale and a martingale with accessible jumps. Meyer-Yoeurp and Yoeurp decompositions play a significant role in stochastic integration theory for càdlàg martingales, For instance one can show sharp estimates for an $L^p$-norm of an $L^q$-valued stochastic integral with respect to a general local martingale. An important tool for obtaining these estimates are the recently proven Burkholder-Rosenthal-type inequalities for discrete $L^q$-valued martingales. This talk is partially based on joint work with Sjoerd Dirksen (RWTH Aachen University).

Higher order concentration results are proved for differentiable functions on Euclidean spaces with LSI type measures provided that they are Lipschitz bounded of order $d\ge 2$ and orthogonal to polynomials of order $d-1$. This is recent joint work with S. Bobkov and H. Sambale. It extends to concentration of measure for functions on discrete spaces subject to higher order $L^2$ type differences uniformly bounded and an appropriate Hoeffding expansion structure. The results yield uniform exponential bounds for $|f|^{2/d}$ extending previous 2nd order results for functions on these spaces. Some applications of these bounds are given. In particular applications of 2nd order concentrations for polynomials on the sphere and spherical averages of 2nd order uncorrelated isotropic vector are discussed, illustrating previous joint results with S.Bobkov and G. Chistyakov.

Initially motivated by the study of the non-asymptotic performance of non-parametric tests based on permutation methods, some concentration inequalities for uniformly permuted sums are derived from the fundamental inequalities for random permutations of Talagrand. The idea is to first obtain a rough inequality for the square root of the permuted sum, and then, iterate the previous analysis and plug this first inequality to obtain a general concentration of permuted sums around their median. Then, concentration inequalities around the mean are deduced. This method allows us to obtain a Bernstein-type inequality. In particular, one recovers the Gaussian behavior of such permuted sums under classical conditions encountered in the literature.

I will explain how to use chaos decomposition technique to analyze the asymptotic behavior of geometric quantities related to the so-called Berry random wave model.