Regular variation provides a convenient theoretical framework for studying large events. In the multivariate setting, the spectral measure characterizes the dependence structure of the extremes. This measure gathers information on the localization of extreme events and often has sparse support since severe events do not simultaneously occur in all directions. However, it is defined through weak convergence, which does not provide a natural way to capture this sparsity structure. In this paper, we introduce the notion of sparse regular variation, which makes it possible to better learn the dependence structure of extreme events. This concept is based on the Euclidean projection onto the simplex, for which efficient algorithms are known. We prove that under mild assumptions sparse regular variation and regular variation are equivalent notions, and we establish several results for sparsely regularly varying random vectors.

It is well known and readily seen that the maximum of n independent and uniformly on [0, 1] distributed random variables, suitably standardised, converges in total variation distance, as n increases, to the standard negative exponential distribution. We extend this result to higher dimensions by considering copulas. We show that the strong convergence result holds for copulas that are in a differential neighbourhood of a multivariate generalised Pareto copula. Sklar’s theorem then implies convergence in variational distance of the maximum of n independent and identically distributed random vectors with arbitrary common distribution function and (under conditions on the marginals) of its appropriately normalised version. We illustrate how these convergence results can be exploited to establish the almost-sure consistency of some estimation procedures for max-stable models, using sample maxima.

Let X = (Xt)t≥0 be a stochastic process issued from $x \in \mathbb{R}$ that admits a marginal stationary measure v, i.e. vPtf = vf for all t ≥ 0, where $\textbf{P}_t\,f(x)= \mathbb{E}_x[f(\textbf{X}_t)]$ . In this paper, we introduce the (resp. biorthogonal) spectral projections correlation functions which are expressed in terms of projections.” Also, update first published online date, if available. into the eigenspaces of Pt (resp. and of its adjoint in the weighted Hilbert space L2 (v)). We obtain closed-form expressions involving eigenvalues, the condition number and/or the angle between the projections in the following different situations: when X = X with X = (Xt)t ≥ 0 being a Markov process, X is the subordination of X in the sense of Bochner, and X is a non-Markovian process which is obtained by time-changing X with an inverse of a subordinator. It turns out that these spectral projections correlation functions have different expressions with respect to these classes of processes which enables to identify substantial and deep properties about their dynamics. This interesting fact can be used to design original statistical tests to make inferences, for example, about the path properties of the process (presence of jumps), distance from symmetry (self-adjoint or non-self-adjoint) and short-to-long-range dependence. To reveal the usefulness of our results, we apply them to a class of non-self-adjoint Markov semigroups studied in Patie and Savov (to appear, Mem. Amer. Math. Soc., 179p), and then time-change by subordinators and their inverses.

Component graphs $\unicode[STIX]{x1D6E4}_{0}(F)$ are defined for arrays of sets $F$, and, in particular, for arrays of path components for Vietoris–Rips complexes and Lesnick complexes. The path components of $\unicode[STIX]{x1D6E4}_{0}(F)$ are the stable components of the array $F$. The stable components for the system of Lesnick complexes $\{L_{s,k}(X)\}$ for a finite data set $X$ decompose into layers, which are themselves path components of a graph. Combinatorial scoring functions are defined for layers and stable components.

Consider a random vector $\textbf{U}$ whose distribution function coincides in its upper tail with that of an Archimedean copula. We report the fact that the conditional distribution of $\textbf{U}$ , conditional on one of its components, has under a mild condition on the generator function independent upper tails, no matter what the unconditional tail behavior is. This finding is extended to Archimax copulas.

In this paper we investigate the link between the joint law of a d-dimensional random vector and the law of some of its multivariate marginals. We introduce and focus on a class of distributions, that we call projective, for which we give detailed properties. This allows us to obtain necessary conditions for a given construction to be projective. We illustrate our results by proposing some theoretical projective distributions, as elliptical distributions or a new class of distribution having given bivariate margins. In the case where the data does not necessarily correspond to a projective distribution, we also explain how to build proper distributions while checking that the distance to the prescribed projections is small enough.

Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.

The aging of population is perhaps the most important problem that developed countries must face in the near future. Dependency can be seen as a consequence of the process of gradual aging. In a health context, this contingency is defined as a lack of autonomy in performing basic activities of daily living that requires the care of another person or significant help. In Europe in general and in Spain in particular, this phenomena represents a problem with economic, political, social and demographic implications. The prevalence of dependency in the population, as well as its intensity and evolution over the course of a person’s life are issues of greatest importance that should be addressed. The aim of this work is the estimation of life expectancy free of dependency (LEFD) based on functional trajectories to enhance the regular estimation of health expectancy. Using information from the Spanish survey EDAD 2008, we estimate the number of years spent free of dependency for disabled people according to gender, dependency degree (moderate, severe, major) and the earlier or later onset of dependency compared to a central trend. The main findings are as follows: first, we show evidence that to estimate LEFD ignoring the information provided by the functional trajectories may lead to non-representative LEFD estimates; second, in general, dependency-free life expectancy is higher for women than for men. However, its intensity is higher in women with later onset on dependency; Third, the loss of autonomy is higher (and more abrupt) in men than in women. Finally, the diversity of patterns observed at later onset of dependency tends to a dependency extreme-pattern in both genders.

We give a simple conceptual proof of the consistency of a test for multivariate uniformity in a bounded set K ⊂ ℝd that is based on the maximal spacing generated by independent and identically distributed points X1, . . ., Xn in K, i.e. the volume of the largest convex set of a given shape that is contained in K and avoids each of these points. Since asymptotic results for the d > 1 case are only availabe under uniformity, a key element of the proof is a suitable coupling. The proof is general enough to cover the case of testing for uniformity on compact Riemannian manifolds with spacings defined by geodesic balls.

Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The main focus of the paper is on weak recovery of the community based on the graph G, with o(K) misclassified vertices on average, in the sublinear regime n1-o(1) ≤ K ≤ o(n). A critical parameter is the effective signal-to-noise ratio λ = K2(p - q)2 / ((n - K)q), with λ = 1 corresponding to the Kesten–Stigum threshold. We show that a belief propagation (BP) algorithm achieves weak recovery if λ > 1 / e, beyond the Kesten–Stigum threshold by a factor of 1 / e. The BP algorithm only needs to run for log*n + O(1) iterations, with the total time complexity O(|E|log*n), where log*n is the iterated logarithm of n. Conversely, if λ ≤ 1 / e, no local algorithm can asymptotically outperform trivial random guessing. Furthermore, a linear message-passing algorithm that corresponds to applying a power iteration to the nonbacktracking matrix of the graph is shown to attain weak recovery if and only if λ > 1. In addition, the BP algorithm can be combined with a linear-time voting procedure to achieve the information limit of exact recovery (correctly classify all vertices with high probability) for all K ≥ (n / logn) (ρBP + o(1)), where ρBP is a function of p / q.

Based on the ratio of two block maxima, we propose a large sample test for the length of memory of a stationary symmetric α-stable discrete parameter random field. We show that the power function converges to 1 as the sample-size increases to ∞ under various classes of alternatives having longer memory in the sense of Samorodnitsky (2004). Ergodic theory of nonsingular ℤd-actions plays a very important role in the design and analysis of our large sample test.

Let X1, X2, . . . be independent copies of a random vector X with values in ℝd and a continuous distribution function. The random vector Xn is a complete record, if each of its components is a record. As we require X to have independent components, crucial results for univariate records clearly carry over. But there are substantial differences as well. While there are infinitely many records in the d = 1 case, they occur only finitely many times in the series if d ≥ 2. Consequently, there is a terminal complete record with probability 1. We compute the distribution of the random total number of complete records and investigate the distribution of the terminal record. For complete records, the sequence of waiting times forms a Markov chain, but unlike the univariate case now the state at ∞ is an absorbing element of the state space.

In many applications, the Gaussian convolution is approximately computed by means of recursive filters, with a significant improvement of computational efficiency. We are interested in theoretical and numerical issues related to such an use of recursive filters in a three-dimensional variational data assimilation (3Dvar) scheme as it appears in the software OceanVar. In that context, the main numerical problem consists in solving large linear systems with high efficiency, so that an iterative solver, namely the conjugate gradient method, is equipped with a recursive filter in order to compute matrix-vector multiplications that in fact are Gaussian convolutions. Here we present an error analysis that gives effective bounds for the perturbation on the solution of such linear systems, when is computed by means of recursive filters. We first prove that such a solution can be seen as the exact solution of a perturbed linear system. Then we study the related perturbation on the solution and we demonstrate that it can be bounded in terms of the difference between the two linear operators associated to the Gaussian convolution and the recursive filter, respectively. Moreover, we show through numerical experiments that the error on the solution, which exhibits a kind of edge effect, i.e. most of the error is localized in the first and last few entries of the computed solution, is due to the structure of the difference of the two linear operators.