In this paper, we consider the random plane forest uniformly drawn from all possible plane forests with a given degree sequence. Under suitable conditions on the degree sequences, we consider the limit of a sequence of such forests with the number of vertices tends to infinity in terms of Gromov–Hausdorff–Prokhorov topology. This work falls into the general framework of showing convergence of random combinatorial structures to certain Gromov–Hausdorff scaling limits, described in terms of the Brownian Continuum Random Tree (BCRT), pioneered by the work of Aldous (Ann. Probab. 19 (1991) 1–28; In Stochastic Analysis (Durham, 1990) (1991) 23–70 Cambridge Univ. Press; Ann. Probab. 21 (1993) 248–289). In fact, we identify the limiting random object as a sequence of random real trees encoded by excursions of some first passage bridges reflected at minimum. We establish such convergence by studying the associated Lukasiewicz walk of the degree sequences. In particular, our work is closely related to and uses the results from the recent work of Broutin and Marckert (Random Structures Algorithms 44 (2014) 290–316) on scaling limit of random trees with prescribed degree sequences, and the work of Addario-Berry (Random Structures Algorithms 41 (2012) 253–261) on tail bounds of the height of a random tree with prescribed degree sequence.

A Walsh diffusion on Euclidean space moves along each ray from the origin, as a solution to a stochastic differential equation with certain drift and diffusion coefficients, as long as it stays away from the origin. As it hits the origin, it instantaneously chooses a new direction according to a given probability law, called the spinning measure. A special example is a real-valued diffusion with skew reflections at the origin. This process continuously (in the weak sense) depends on the spinning measure. We determine a stationary measure for such process, explore long-term convergence to this distribution and establish an explicit rate of exponential convergence.

The multiplicative coalescent is a mean-field Markov process in which any pair of blocks coalesces at rate proportional to the product of their masses. In Aldous and Limic (Electron. J. Probab. 3 (1998) Paper no. 3) each extreme eternal version of the multiplicative coalescent was described in three different ways, one of which matched its (marginal) law to that of the ordered excursion lengths above past minima of a certain Lévy-type process.

Using a modification of the breadth-first-walk construction from Aldous (Ann. Probab. 25 (1997) 812–854) and Aldous and Limic (Electron. J. Probab. 3 (1998) Paper no. 3), and some new insight from the thesis by Uribe Bravo (Markovian bridges, Brownian excursions, and stochastic fragmentation and coalescence (2007) UNAM), this work settles an open problem (3) from Aldous (Ann. Probab. 25 (1997) 812–854) in the more general context of Aldous and Limic (Electron. J. Probab. 3 (1998) Paper no. 3). Informally speaking, each eternal version is entirely encoded by its Lévy-type process, and contrary to Aldous’ original intuition, the time for the multiplicative coalescent does correspond to the linear increase in the constant part of the drift of the Lévy-type process. In the “standard multiplicative coalescent” context of Aldous (Ann. Probab. 25 (1997) 812–854), this result was first announced by Armendáriz in 2001, while its first published proof is due to Broutin and Marckert (Probab. Theory Related Fields 166 (2016) 515–552), who simultaneously account for the process of excess (or surplus) edge counts.

Regularly varying space-time processes have proved useful to study extremal dependence in space-time data. We propose a semiparametric estimation procedure based on a closed form expression of the extremogram to estimate parametric models of extremal dependence functions. We establish the asymptotic properties of the resulting parameter estimates and propose subsampling procedures to obtain asymptotically correct confidence intervals. A simulation study shows that the proposed procedure works well for moderate sample sizes and is robust to small departures from the underlying model. Finally, we apply this estimation procedure to fitting a max-stable process to radar rainfall measurements in a region in Florida. Complementary results and some proofs of key results are presented together with the simulation study in the supplement [Buhl et al. (2018)].

We derive tight non-asymptotic bounds for the Kolmogorov distance between the probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimension-free and depend on the nuclear (Schatten-one) norm of the difference between the covariance operators of the elements and on the norm of the mean shift. The obtained bounds significantly improve the bound based on Pinsker’s inequality via the Kullback–Leibler divergence. We also establish an anti-concentration bound for a squared norm of a non-centered Gaussian element in Hilbert space. The paper presents a number of examples motivating our results and applications of the obtained bounds to statistical inference and to high-dimensional CLT.

Under the sublinear expectation $\mathbb{E}[\cdot]:=\mathop{\mathrm{sup}}_{\theta\in\Theta}E_{\theta}[\cdot]$ for a given set of linear expectations $\{E_{\theta}:\theta\in\Theta\}$, we establish a new law of large numbers and a new central limit theorem with rate of convergence. We present some interesting special cases and discuss a related statistical inference problem. We also give an approximation and a representation of the $G$-normal distribution, which was used as the limit in Peng’s (Law of large numbers and central limit theorem under nonlinear expectations (2007) Preprint) central limit theorem, in a probability space.

Consider a Poisson point process with unknown support boundary curve $g$, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form $\int\Phi(g(x))\,dx$. Following a nonparametric maximum-likelihood approach, we construct an estimator which is UMVU over Hölder balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on $\Phi$ which are satisfied for $\Phi(u)=|u|^{p}$, $p\ge1$. As an application, we consider the problem of estimating the $L^{p}$-norm and derive the minimax separation rates in the corresponding nonparametric hypothesis testing problem. Structural differences to results for regular nonparametric models are discussed.

The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the empirical measure obtained from $n$ independent samples from $\mu$ approaches $\mu$ in the Wasserstein distance of any order. We prove sharp asymptotic and finite-sample results for this rate of convergence for general measures on general compact metric spaces. Our finite-sample results show the existence of multi-scale behavior, where measures can exhibit radically different rates of convergence as $n$ grows.

We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of the descendants at birth depends on the trait of the mother and on the number of descendants. In this article, we explicitly describe the penalized Markov process, named auxiliary process, corresponding to the dynamic of the trait of a “typical” individual by giving its associated infinitesimal generator. We prove a Many-to-One formula and a Many-to-One formula for forks. Furthermore, we prove that this auxiliary process characterizes exactly the process of the trait of a uniformly sampled individual in a large population approximation. We detail three examples of growth-fragmentation models: the linear growth model, the exponential growth model and the parasite infection model.

We consider inference in the scalar diffusion model $\,\mathrm{d}X_{t}=b(X_{t})\,\mathrm{d}t+\sigma(X_{t})\,\mathrm{d}W_{t}$ with discrete data $(X_{j\Delta_{n}})_{0\leq j\leq n}$, $n\to\infty$, $\Delta_{n}\to0$ and periodic coefficients. For $\sigma$ given, we prove a general theorem detailing conditions under which Bayesian posteriors will contract in $L^{2}$-distance around the true drift function $b_{0}$ at the frequentist minimax rate (up to logarithmic factors) over Besov smoothness classes. We exhibit natural nonparametric priors which satisfy our conditions. Our results show that the Bayesian method adapts both to an unknown sampling regime and to unknown smoothness.

We introduce a Benamou–Brenier formulation for the continuous-time martingale optimal transport problem as a weak length relaxation of its discrete-time counterpart. By the correspondence between classical martingale problems and Fokker–Planck equations, we obtain an equivalent PDE formulation for which basic properties such as existence, duality and geodesic equations can be analytically studied, yielding corresponding results for the stochastic formulation. In the one dimensional case, sufficient conditions for finiteness of the cost are also given and a link between geodesics and porous medium equations is partially investigated.

The generalized Dickman distribution $\mathcal{D}_{\theta}$ with parameter $\theta>0$ is the unique solution to the distributional equality $W=_{d}W^{*}$, where \begin{align} W^∗=_d U^1/θ(W+1),\tag{1} \end{align} with $W$ non-negative with probability one, $U\sim\mathcal{U}[0,1]$ independent of $W$, and $=_{d}$ denoting equality in distribution. These distributions appear in number theory, stochastic geometry, perpetuities and the study of algorithms. We obtain bounds in Wasserstein type distances between $\mathcal{D}_{\theta}$ and the distribution of \begin{eqnarray*}W_{n}=\frac{1}{n}\sum_{i=1}^{n}Y_{k}B_{k},\end{eqnarray*} where $B_{1},\ldots,B_{n},Y_{1},\ldots,Y_{n}$ are independent with $B_{k}$ distributed $\operatorname{Ber}(1/k)$ or $\mathcal{P}(\theta/k)$, $E[Y_{k}]=k$ and $\operatorname{Var}(Y_{k})=\sigma_{k}^{2}$, and provide an application to the minimal directed spanning tree in $\mathbb{R}^{2}$. We also provide bounds with optimal rates for the Dickman convergence of weighted sums, arising in probabilistic number theory, of the form \begin{eqnarray*}S_{n}=\frac{1}{\log(p_{n})}\sum_{k=1}^{n}X_{k}\log(p_{k}),\end{eqnarray*} where $(p_{k})_{k\ge1}$ is an enumeration of the prime numbers in increasing order and $X_{k}$ is geometric with parameter $(1-1/p_{k})$, Bernoulli with success probability $1/(1+p_{k})$ or Poisson with mean $\lambda_{k}$.

Lastly, we broaden the class of generalized Dickman distributions by studying the fixed points of the transformation \begin{eqnarray*}s(W^{*})=_{d}U^{1/\theta}s(W+1)\end{eqnarray*} generalizing (1), that allows the use of non-identity utility functions $s(\cdot)$ in Vervaat perpetuities. We obtain distributional bounds for recursive methods that can be used to simulate from this family.

Let $(X_{i},\mathcal{F}_{i})_{i\geq 1}$ be a sequence of martingale differences. Set $S_{n}=\sum_{i=1}^{n}X_{i}$ and $[S]_{n}=\sum_{i=1}^{n}X_{i}^{2}$. We prove a Cramér type moderate deviation expansion for $\mathbf{P}(S_{n}/\sqrt{[S]_{n}}\geq x)$ as $n\to +\infty $. Our results partly extend the earlier work of Jing, Shao and Wang (Ann. Probab. 31 (2003) 2167–2215) for independent random variables.

We provide a Lyapunov type bound in the multivariate central limit theorem for sums of independent, but not necessarily identically distributed random vectors. The error in the normal approximation is estimated for certain classes of sets, which include the class of measurable convex sets. The error bound is stated with explicit constants. The result is proved by means of Stein’s method. In addition, we improve the constant in the bound of the Gaussian perimeter of convex sets.

We consider in this paper the problem of sampling a high-dimensional probability distribution $\pi$ having a density w.r.t. the Lebesgue measure on $\mathbb{R}^{d}$, known up to a normalization constant $x\mapsto\pi(x)=\mathrm{e}^{-U(x)}/\int_{\mathbb{R}^{d}}\mathrm{e}^{-U(y)}\,\mathrm{d}y$. Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that $U$ is continuously differentiable, $\nabla U$ is globally Lipschitz and $U$ is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order $2$ and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of these bounds is explicit. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims.

Gaussian process (GP) models have become a well-established framework for the adaptive design of costly experiments, and notably of computer experiments. GP-based sequential designs have been found practically efficient for various objectives, such as global optimization (estimating the global maximum or maximizer(s) of a function), reliability analysis (estimating a probability of failure) or the estimation of level sets and excursion sets. In this paper, we study the consistency of an important class of sequential designs, known as stepwise uncertainty reduction (SUR) strategies. Our approach relies on the key observation that the sequence of residual uncertainty measures, in SUR strategies, is generally a supermartingale with respect to the filtration generated by the observations. This observation enables us to establish generic consistency results for a broad class of SUR strategies. The consistency of several popular sequential design strategies is then obtained by means of this general result. Notably, we establish the consistency of two SUR strategies proposed by Bect, Ginsbourger, Li, Picheny and Vazquez (Stat. Comput. 22 (2012) 773–793) – to the best of our knowledge, these are the first proofs of consistency for GP-based sequential design algorithms dedicated to the estimation of excursion sets and their measure. We also establish a new, more general proof of consistency for the expected improvement algorithm for global optimization which, unlike previous results in the literature, applies to any GP with continuous sample paths.

The main result of this paper is the rate of convergence to Hermite-type distributions in non-central limit theorems. To the best of our knowledge, this is the first result in the literature on rates of convergence of functionals of random fields to Hermite-type distributions with ranks greater than 2. The results were obtained under rather general assumptions on the spectral densities of random fields. These assumptions are even weaker than in the known convergence results for the case of Rosenblatt distributions. Additionally, Lévy concentration functions for Hermite-type distributions were investigated.

We consider the random field \begin{equation*}M(t)=\mathop{\mathrm{sup}}_{n\geq1}\{-\log A_{n}+X_{n}(t)\},\qquad t\in T,\end{equation*} for a set $T\subset\mathbb{R}^{m}$, where $(X_{n})$ is an i.i.d. sequence of centered Gaussian random fields on $T$ and $0<A_{1}<A_{2}<\cdots$ are the arrivals of a general renewal process on $(0,\infty)$, independent of $(X_{n})$. In particular, a large class of max-stable random fields with Gumbel marginals have such a representation. Assume that one needs $c(d)=c(\{t_{1},\ldots,t_{d}\})$ function evaluations to sample $X_{n}$ at $d$ locations $t_{1},\ldots,t_{d}\in T$. We provide an algorithm which samples $M(t_{1}),\ldots,M(t_{d})$ with complexity $O(c(d)^{1+o(1)})$ as measured in the $L_{p}$ norm sense for any $p\ge1$. Moreover, if $X_{n}$ has an a.s. converging series representation, then $M$ can be a.s. approximated with error $\delta$ uniformly over $T$ and with complexity $O(1/(\delta\log(1/\delta))^{1/\alpha})$, where $\alpha$ relates to the Hölder continuity exponent of the process $X_{n}$ (so, if $X_{n}$ is Brownian motion, $\alpha=1/2$).

This paper deals with suitable quantifications in approximating a probability measure by an “empirical” random probability measure $\hat{\mathfrak{p}}_{n}$, depending on the first $n$ terms of a sequence $\{\tilde{\xi}_{i}\}_{i\geq1}$ of random elements. Section 2 studies the range of oscillation near zero of the Wasserstein distance $\mathrm{d}^{(p)}_{[\mathbb{S}]}$ between $\mathfrak{p}_{0}$ and $\hat{\mathfrak{p}}_{n}$, assuming the $\tilde{\xi}_{i}$’s i.i.d. from $\mathfrak{p}_{0}$. In Theorem 2.1 $\mathfrak{p}_{0}$ can be fixed in the space of all probability measures on $(\mathbb{R}^{d},\mathscr{B}(\mathbb{R}^{d}))$ and $\hat{\mathfrak{p}}_{n}$ coincides with the empirical measure $\tilde{\mathfrak{e}}_{n}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{\tilde{\xi}_{i}}$. In Theorem 2.2 (Theorem 2.3, respectively), $\mathfrak{p}_{0}$ is a $d$-dimensional Gaussian distribution (an element of a distinguished statistical exponential family, respectively) and $\hat{\mathfrak{p}}_{n}$ is another $d$-dimensional Gaussian distribution with estimated mean and covariance matrix (another element of the same family with an estimated parameter, respectively). These new results improve on allied recent works by providing also uniform bounds with respect to $n$, meaning the finiteness of the $p$-moment of $\mathop{\mathrm{sup}}_{n\geq1}b_{n}\mathrm{d}^{(p)}_{[\mathbb{S}]}(\mathfrak{p}_{0},\hat{\mathfrak{p}}_{n})$ is proved for some diverging sequence $b_{n}$ of positive numbers. In Section 3, assuming the $\tilde{\xi}_{i}$’s exchangeable, one studies the range of oscillation near zero of the Wasserstein distance between the conditional distribution – also called posterior – of the directing measure of the sequence, given $\tilde{\xi}_{1},\ldots,\tilde{\xi}_{n}$, and the point mass at $\hat{\mathfrak{p}}_{n}$. Similarly, a bound for the approximation of predictive distributions is given. Finally, Theorems from 3.3 to 3.5 reconsider Theorems from 2.1 to 2.3, respectively, according to a Bayesian perspective.

Upper and lower bounds are derived for the Gaussian mean width of a convex hull of $M$ points intersected with a Euclidean ball of a given radius. The upper bound holds for any collection of extreme points bounded in Euclidean norm. The upper bound and the lower bound match up to a multiplicative constant whenever the extreme points satisfy a one sided Restricted Isometry Property.

An appealing aspect of the upper bound is that no assumption on the covariance structure of the extreme points is needed. This aspect is especially useful to study regression problems with anisotropic design distributions. We provide applications of this bound to the Lasso estimator in fixed-design regression, the Empirical Risk Minimizer in the anisotropic persistence problem, and the convex aggregation problem in density estimation.

The family of goodness-of-fit tests based on $\Phi$-divergences is known to be optimal for detecting signals hidden in high-dimensional noise data when the heterogeneous normal mixture model is underlying. This test family includes Tukey’s popular higher criticism test and the famous Berk–Jones test. In this paper we address the open question whether the tests’ optimality is still present beyond the prime normal mixture model. On the one hand, we transfer the known optimality of the higher criticism test for different models, for example, for the heteroscedastic normal, general Gaussian and exponential-$\chi^{2}$-mixture models, to the whole test family. On the other hand, we discuss the optimality for new model classes based on exponential families including the scale exponential, the scale Fréchet and the location Gumbel models. For all these examples we apply a general machinery which might be used to show the tests’ optimality for further models/model classes in future.

In this article, we consider the hyperbolic and parabolic Anderson models in arbitrary space dimension $d$, with constant initial condition, driven by a Gaussian noise which is white in time. We consider two spatial covariance structures: (i) the Fourier transform of the spectral measure of the noise is a non-negative locally-integrable function; (ii) $d=1$ and the noise is a fractional Brownian motion in space with index $1/4<H<1/2$. In both cases, we show that there is striking similarity between the Laplace transforms of the second moment of the solutions to these two models. Building on this connection and the recent powerful results of [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 1305–1340] for the parabolic model, we compute the second order (upper) Lyapunov exponent for the hyperbolic model. In case (i), when the spatial covariance of the noise is given by the Riesz kernel, we present a unified method for calculating the second order Lyapunov exponents for the two models.

In this paper, we use the semi-group method and an adaptation of the $L^{2}$-method of Hörmander to establish some $\Phi$-entropy inequalities and asymmetric covariance estimates for the strictly convex measures in $\mathbb{R}^{n}$. These inequalities extends the ones for the strictly log-concave measures to more general setting of convex measures. The $\Phi$-entropy inequalities are turned out to be sharp in the special case of Cauchy measures. Finally, we show that the similar inequalities for log-concave measures can be obtained from our results in the limiting case.

We establish general conditions under which Markov chains produced by the Hamiltonian Monte Carlo method will and will not be geometrically ergodic. We consider implementations with both position-independent and position-dependent integration times. In the former case, we find that the conditions for geometric ergodicity are essentially a gradient of the log-density which asymptotically points towards the centre of the space and grows no faster than linearly. In an idealised scenario in which the integration time is allowed to change in different regions of the space, we show that geometric ergodicity can be recovered for a much broader class of tail behaviours, leading to some guidelines for the choice of this free parameter in practice.

In this paper, we study high-dimensional random projections of $\ell_{p}^{n}$-balls. More precisely, for any $n\in\mathbb{N}$ let $E_{n}$ be a random subspace of dimension $k_{n}\in\{1,\ldots,n\}$ and $X_{n}$ be a random point in the unit ball of $\ell_{p}^{n}$. Our work provides a description of the Gaussian fluctuations of the Euclidean norm $\|P_{E_{n}}X_{n}\|_{2}$ of random orthogonal projections of $X_{n}$ onto $E_{n}$. In particular, under the condition that $k_{n}\to\infty$ it is shown that these random variables satisfy a central limit theorem, as the space dimension $n$ tends to infinity. Moreover, if $k_{n}\to\infty$ fast enough, we provide a Berry–Esseen bound on the rate of convergence in the central limit theorem. At the end, we provide a discussion of the large deviations counterpart to our central limit theorem.

A well-known drawback of $\ell_{1}$-penalized estimators is the systematic shrinkage of the large coefficients towards zero. A simple remedy is to treat Lasso as a model-selection procedure and to perform a second refitting step on the selected support. In this work, we formalize the notion of refitting and provide oracle bounds for arbitrary refitting procedures of the Lasso solution. One of the most widely used refitting techniques which is based on Least-Squares may bring a problem of interpretability, since the signs of the refitted estimator might be flipped with respect to the original estimator. This problem arises from the fact that the Least-Squares refitting considers only the support of the Lasso solution, avoiding any information about signs or amplitudes. To this end, we define a sign consistent refitting as an arbitrary refitting procedure, preserving the signs of the first step Lasso solution and provide Oracle inequalities for such estimators. Finally, we consider special refitting strategies: Bregman Lasso and Boosted Lasso. Bregman Lasso has a fruitful property to converge to the Sign-Least-Squares refitting (Least-Squares with sign constraints), which provides with greater interpretability. We additionally study the Bregman Lasso refitting in the case of orthogonal design, providing with simple intuition behind the proposed method. Boosted Lasso, in contrast, considers information about magnitudes of the first Lasso step and allows to develop better oracle rates for prediction. Finally, we conduct an extensive numerical study to show advantages of one approach over others in different synthetic and semi-real scenarios.