Showing papers in "ALEA-Latin American Journal of Probability and Mathematical Statistics in 2020"

Multiscale functional inequalities in probability: Concentration properties

Abstract: In a companion article we have introduced a notion of multiscale functional inequalities for functions $X(A)$ of an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. These inequalities are multiscale weighted versions of standard Poincare, covariance, and logarithmic Sobolev inequalities. They hold for all the examples of fields $A$ arising in the modelling of heterogeneous materials in the applied sciences whereas their standard versions are much more restrictive. In this contribution we first investigate the link between multiscale functional inequalities and more standard decorrelation or mixing properties of random fields. Next, we show that multiscale functional inequalities imply fine concentration properties for nonlinear functions $X(A)$. This constitutes the main stochastic ingredient to the quenched large-scale regularity theory for random elliptic operators by the second author, Neukamm, and Otto, and to the corresponding quantitative stochastic homogenization results.

Delayed and rushed motions through time change

Abstract: We consider time changes given by subordinators and their inverse processes. Our analysis shows that, quite surprisingly, inverse processes are not necessarily leading to delayed processes.

Optimal Gamma Approximation on Wiener Space

Abstract: Nourdin and Peccati (2009a) established a neat characterization of Gamma approximation on a fixed Wiener chaos in terms of convergence of only the third and fourth cumulants. In this paper, we provide an optimal rate of convergence in the d2-distance in terms of the maximum of the third and fourth cumulants analogous to the result for normal approximation in Nourdin and Peccati (2015). In order to achieve our goal, we introduce a novel operator theory approach to Stein’s method. The recent development in Stein’s method for the Gamma distribution of Dobler and Peccati (2018) plays a pivotal role in our analysis. Several examples in the context of quadratic forms are considered to illustrate our optimal bound.

Analytic-geometric methods for finite Markov chains with applications to quasi-stationarity

Abstract: For a relatively large class of well-behaved absorbing (or killed) finite Markov chains, we give detailed quantitative estimates regarding the behavior of the chain before it is absorbed (or killed). Typical examples are random walks on box-like finite subsets of the square lattice $\mathbb Z^d$ absorbed (or killed) at the boundary. The analysis is based on Poincare, Nash, and Harnack inequalities, moderate growth, and on the notions of John and inner-uniform domains.

High-dimensional sample covariance matrices with Curie-Weiss entries

Abstract: We study the limiting spectral distribution of sample covariance matrices $XX^T$, where $X$ are $p\times n$ random matrices with correlated entries, for the cases $p/n\to y\in [0,\infty)$. If $y>0$, we obtain the Mar\v{c}enko-Pastur distribution and in the case $y=0$ the semicircle distribution (after appropriate rescaling). The entries we consider are Curie-Weiss spins, which are correlated random signs, where the degree of the correlation is governed by an inverse temperature $\beta>0$. The model exhibits a phase transition at $\beta=1$. The correlation between any two entries decays at a rate of $O(np)$ for $\beta \in (0,1)$, $O(\sqrt{np}$) for $\beta=1$, and for $\beta>1$ the correlation does not vanish in the limit. In our proofs we use Stieltjes transforms and concentration of random quadratic forms.

On the anisotropic stable JCIR process

Abstract: We investigate the anisotropic stable JCIR process which is a multi-dimensional extension of the stable JCIR process but also a multi-dimensional analogue of the classical JCIR process. We prove that the heat kernel of the anisotropic stable JCIR process exists and it satisfies an a-priori bound in a weighted anisotropic Besov norm. Based on this regularity result we deduce the strong Feller property and prove, for the subcritical case, exponential ergodicity in total variation. Also, we show that in the one-dimensional case the corresponding heat kernel is smooth.

Hydrodynamics for ssep with non-reversible slow boundary dynamics: part ii, below the critical regime

Abstract: The purpose of this article is to provide a simple proof of the hydrodynamic and hydrostatic behavior of the SSEP in contact with reservoirs which inject and remove particles in a finite size windows at the extremities of the bulk. More precisely, the reservoirs inject/remove particles at/from any point of a window of size K placed at each extremity of the bulk and particles are injected/removed to the first open/occupied position in that window. The reservoirs have slow dynamics, in the sense that they intervene at speed N −θ w.r.t. the bulk dynamics. In the first part of this article, [4], we treated the case θ > 1 for which the entropy method can be adapted. We treat here the case where the boundary dynamics is too fast for the Entropy Method to apply. We prove using duality estimates inspired by [2, 3] that the hydrodynamic limit is given by the heat equation with Dirichlet boundary conditions, where the density at the boundaries is fixed by the parameters of the model.

Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces

Abstract: We consider non-linear evolution equations arising from mean-field limits of particle systems on discrete spaces. We investigate a notion of curvature bounds for these dynamics based on convexity of the free energy along interpolations in a discrete transportation distance related to the gradient flow structure of the dynamics. This notion extends the one for linear Markov chain dynamics studied by Erbar and Maas. We show that positive curvature bounds entail several functional inequalities controlling the convergence to equilibrium of the dynamics. We establish explicit curvature bounds for several examples of mean-field limits of various classical models from statistical mechanics.

On Nonlinear Rough Paths

Abstract: In this paper, we establish the theory of nonlinear rough paths. We give the definition of nonlinear rough paths, and develop the integrals. Then, we study differential equations driven by nonlinear rough paths. Afterwards, we compare the nonlinear rough paths and classic theory of linear rough paths. Finally, we apply this theory to rough partial differential equations.

Kinetic walks for sampling

Abstract: The persistent walk is a classical model in kinetic theory, which has also been studied as a toy model for MCMC questions. Its continuous limit, the telegraph process, has recently been extended to various velocity jump processes (Bouncy Particle Sampler, Zig-Zag process, etc.) in order to sample general target distributions on $\mathbb R^d$. This paper studies, from a sampling point of view, general kinetic walks that are natural discrete-time (and possibly discrete-space) counterparts of these continuous-space processes. The main contributions of the paper are the definition and study of a discrete-space Zig-Zag sampler and the definition and time-discretisation of hybrid jump/diffusion kinetic samplers for multi-scale potentials on $\mathbb R^d$.

Anisotropic Gaussian wave models

Abstract: Let $d$ be an integer greater or equal to 2 and let $\mathbf k$ be a $d$-dimensional random vector. We call Gaussian wave model with random wavevector $\mathbf k$ any stationary Gaussian random field defined on $\mathbb{R}^d$ with covariance function $t\mapsto \mathbb{E}[\cos(\mathbf k.t)]$. Any stationary Gaussian random field on $\mathbb{R}^d$ can be studied as a random wave. The purpose of the present paper is to link properties of the random wave with the distribution of the random wavevector, with a focus on geometric properties. We mainly concentrate on random waves such that the distribution of the norm of the wavevector and the one of its direction are independent. In the planar case, we prove that the expected length of the nodal lines is decreasing as the anisotropy of the wavevector is increasing, and we study the direction that maximizes the expected length of the crest lines. We illustrate our results on two specific models: a generalization of Berry's monochromatic planar waves and a spatiotemporal sea wave model whose random wavevector is supported by the Airy surface in $\mathbb{R}^3$. According to a general theorem, these two Gaussian fields are anisotropic almost sure solutions of partial differential equations that involve the Laplacian operator: $\Delta f+\kappa^2f=0$ (where $\kappa=\|\mathbf k\|$) for the former, $\Delta f+\partial^4_tf=0$ for the latter.

Some remarks on associated random fields, random measures and point processes

Abstract: In this paper, we first show that for a countable family of random elements taking values in a partially ordered Polish space (POP), association (both positive and negative) of all finite dimensional marginals implies that of the infinite sequence. Our proof proceeds via Strassen's theorem for stochastic domination and thus avoids the assumption of normally ordered on the product space as needed for positive association in [Lindqvist 1988]. We use these results to show on Polish spaces that finite dimensional negative association implies negative association of the random measure and negative association is preserved under weak convergence of random measures. The former provides a simpler proof in the most general setting of Polish spaces complementing the recent proofs in [Poinas et al. 2017] and [Lyons 2014] which restrict to point processes in Euclidean spaces and locally compact Polish spaces respectively. We also provide some examples of associated random measures which shall illustrate our results as well.

Adaptive recursive kernel conditional density estimators under censoring data

Abstract: Let (T,X) be independent identically distributed pairs of random variables and denote f (t|x) the conditional density of T given X = x, we consider that the random variable T is subject to random censoring by another random variable C. In this paper, we propose and investigate an adaptive recursive kernel conditional density estimation under censored data, which allows us to circumvent the weak performances of Kaplan-Meier estimator (Kaplan and Meier, 1958) in the right-tail of the distribution. The first aim of this paper is to study the properties of the proposed adaptive recursive estimators and compare it with the non-recursive estimator of f (t|x). It turns out that, with an adequate selected bandwidth and a special stepsize, the proposed recursive estimators often provides better results compared to the non-recursive one in terms of estimation error and much better in terms of computational costs. We corroborated these theoretical results through some simulation study.

Hermite density deconvolution

Abstract: We consider the additive model: Z = X + e, where X and e are independent. We construct a new estimator of the density of X from n observations of Z. We propose a projection method which exploits the specific properties of the Hermite basis. We study the quality of the resulting estimator by proving a bound on the integrated quadratic risk. We then propose an adaptive estimation procedure, that is a method of selecting a relevant model. We check that our estimator reaches the classical convergence speeds of deconvolution. Numerical simulations are proposed and a comparison with the results of the method proposed in Comte and Lacour (2011) is performed.

Conditional expectations through Boolean cumulantsand subordination – towards a better understandingof the Lukacs property in free probability

Abstract: We use here a recent idea of studying functions of free random variables using Boolean cumulants. We develop idea of explicit calculations of conditional expectation using Boolean cumulants. We demonstrate Boolean cumulants approach allows to calculate explicitly some conditional expectations of functions in free random variables. We present how Boolean cumulants together with subordination simplify proofs of some results which are known in research literature.

Anisotropic oriented percolation in high dimensions

Abstract: In this paper we study anisotropic oriented percolation on $\mathbb{Z}^d$ for $d\geq 4$ and show that the local condition for phase transition is closely related to the mean-field condition. More precisely, we show that if the sum of the local probabilities is strictly greater than one and each probability is not too large, then percolation occurs.

On a limiting point process related to modified permutation matrices

Abstract: We consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle. For each of these two ensembles of matrices, rescaling properly the eigenangles provides a limiting point process as the size of the matrices goes to infinity. If $J$ is an interval of $\mathbb{R}$, we show that, as the length of $J$ tends to infinity, the number of points lying in $J$ of the limiting point process related to modified permutation matrices is asymptotically normal. Moreover, for permutation matrices without modification, if $a$ and $a+b$ denote the endpoints of $J$, we still have an asymptotic normality for the number of points lying in $J$, in the two following cases: [$a$ fixed and $b \to \infty$] and [$a,b \to \infty$ with $b$ proportional to $a$].

Transience and Recurrence of Markov Processeswith Constrained Local Time

Abstract: We study Markov processes conditioned so that their local time must grow slower than a prescribed function. Building upon recent work on Brownian motion with constrained local time in [5] and [33], we study transience and recurrence for a broad class of Markov processes. In order to understand the distribution of the local time, we determine the distribution of a non-decreasing L\\'evy process (the inverse local time) conditioned to remain above a given level which varies in time. We study a time-dependent region, in contrast to previous works in which a process is conditioned to remain in a fixed region (e.g. [21,27]), so we must study boundary crossing probabilities for a family of curves, and thus obtain uniform asymptotics for such a family. Main results include necessary and sufficient conditions for transience or recurrence of the conditioned Markov process. We will explicitly determine the distribution of the inverse local time for the conditioned process, and in the transient case, we explicitly determine the law of the conditioned Markov process. In the recurrent case, we characterise the \"entropic repulsion envelope\" via necessary and sufficient conditions.

Finding a planted clique by adaptive probing

Abstract: We consider a variant of the planted clique problem where we are allowed unbounded computational time but can only investigate a small part of the graph by adaptive edge queries. We determine (up to logarithmic factors) the number of queries necessary both for detecting the presence of a planted clique and for finding the planted clique. Specifically, let $G \sim G(n,1/2,k)$ be a random graph on $n$ vertices with a planted clique of size $k$. We show that no algorithm that makes at most $q = o(n^2 / k^2 + n)$ adaptive queries to the adjacency matrix of $G$ is likely to find the planted clique. On the other hand, when $k \geq (2+\epsilon) \log_2 n$ there exists a simple algorithm (with unbounded computational power) that finds the planted clique with high probability by making $q = O( (n^2 / k^2) \log^2 n + n \log n)$ adaptive queries. For detection, the additive $n$ term is not necessary: the number of queries needed to detect the presence of a planted clique is $n^2 / k^2$ (up to logarithmic factors).

F-KPP Scaling limit and selection principle for a Brunet-Derrida type particle system

Abstract: Fil: Groisman, Pablo Jose. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Ciudad Universitaria. Instituto de Investigaciones Matematicas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matematicas "Luis A. Santalo"; Argentina

Localisation in a growth model with interaction : arbitrary graphs.

Abstract: This paper concerns the long term behaviour of a growth model describing a random sequential allocation of particles on a finite graph. The probability of allocating a particle at a vertex is proportional to a log-linear function of numbers of existing particles in a neighbourhood of a vertex. When this function depends only on the number of particles in the vertex, the model becomes a special case of the generalised Polya urn model. In this special case all but finitely many particles are allocated at a single random vertex almost surely. In our model interaction leads to the fact that, with probability one, all but finitely many particles are allocated at vertices of a maximal clique.

A CLT for the total energy of the two-dimensional critical Ising model

Abstract: Consider the Ising model on $([1,2N]\times[1,2M])\cap\mathbb{Z}^2$ at critical temperature with periodic boundary condition in the horizontal direction and free boundary condition in the vertical direction. Let $E_{M,N}$ be its total energy (or Hamiltonian). Suppose $M$ is a function of $N$ satisfying $M\geq N/(\ln N)^{\alpha}$ for some $\alpha\in[0,1)$. In particular, one may take $M=N$. We prove that \begin{equation*} \frac{E_{M,N}+4\sqrt{2}M N-(4/\pi)N\ln N}{\sqrt{(32/\pi)MN\ln N}} \end{equation*} converges weakly to a standard Gaussian distribution as $N\rightarrow\infty$.

Differentiability of the speed of biased random walks on Galton-Watson trees

Abstract: We prove that the speed of a $\lambda$-biased random walk on a supercritical Galton-Watson tree is differentiable for $\lambda$ such that the walk is ballistic and obeys a central limit theorem, and give an expression of the derivative using a certain $2$-dimensional Gaussian random variable. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres.

Skorohod and rough integration with respect to the non-commutative fractional Brownian motion

Abstract: We pursue our investigations, initiated in [8], about stochastic integration with respect to the non-commutative fractional Brownian motion (NC-fBm). Our main objective in this paper is to compare the pathwise constructions of [8] with a Skorohod-type interpretation of the integral. As a first step, we provide details on the basic tools and properties associated with non-commutative Malliavin calculus, by mimicking the presentation of Nualart's celebrated treatise [14]. Then we check that, just as in the classical (commutative) situation, Skorohod integration can indeed be considered in the presence of the NC-fBm, at least for a Hurst index H > 1 4. This finally puts us in a position to state and prove the desired comparison result, which can be regarded as an Ito-Stratonovich correction formula for the NC-fBm.

Distribution-Free, Size Adaptive SubmatrixDetection with Acceleration

Abstract: Given a large matrix containing independent data entries, we consider the problem of detecting a submatrix inside the data matrix that contains larger-than-usual values. Different from previous literature, we do not have exact information about the dimension of the potential elevated submatrix. We propose a Bonferroni type testing procedure based on permutation tests, and show that our proposed test loses no first-order asymptotic power compared to tests with full knowledge of potential elevated submatrix. In order to speed up the calculation during the test, an approximation net is constructed and we show that Bonferroni type permutation test on the approximation net loses no power on the first order asymptotically.

Existence of the anchored isoperimetric profile in supercritical bond percolation in dimension two and higher

Abstract: Let d ≥ 2. We consider an i.i.d. supercritical bond percolation on Z^d , every edge is open with a probability p > p_c (d), where p_c (d) denotes the critical point. We condition on the event that 0 belongs to the infinite cluster C_∞ and we consider connected subgraphs of C_∞ having at most n^d vertices and containing 0. Among these subgraphs, we are interested in the ones that minimize the open edge boundary size to volume ratio. These minimizers properly rescaled converge towards a translate of a deterministic shape and their open edge boundary size to volume ratio properly rescaled converges towards a deterministic constant.

A reversible allelic partition process and Pitman sampling formula

Abstract: We introduce a continuous-time Markov chain describing dynamic allelic partitions which extends the branching process construction of the Pitman sampling formula in Pitman (2006) and the birth-and-death process with immigration studied in Karlin and McGregor (1967), in turn related to the celebrated Ewens sampling formula. A biological basis for the scheme is provided in terms of a population of individuals grouped into families, that evolves according to a sequence of births, deaths and immigrations. We investigate the asymptotic behaviour of the chain and show that, as opposed to the birth-and-death process with immigration, this construction maintains in the temporal limit the mutual dependence among the multiplicities. When the death rate exceeds the birth rate, the system is shown to have reversible distribution identified as a mixture of Pitman sampling formulae, with negative binomial mixing distribution on the population size. The population therefore converges to a stationary random configuration, characterised by a finite number of families and individuals.