Almost sure convergence of the kT-occupation time density estimator

In this Note, we introduce an extension of the k-nearest neighbor estimator in continuous time, the $k_T$-occupation time estimator, and we give sufficient conditions for its existence. Then, we show the almost sure convergence for α-mixing and bounded processes in two cases, the superoptimal case (when parametric rates are reached) and the optimal case (when i.i.d. rates of density estimation are reached). To cite this article: B. Labrador, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A note on uniform strong convergence of bivariate density estimates

In this paper we consider a class of estimates of a bivariate density function f based on an independent sample of size n. Under the assumption that f is uniformly continuous, the uniform strong consistency of such estimates was first proved by Nadaraya (1970) for a large class of kernel functions. In this note we show that the assumption of the uniform continuity of f is necessary for this type of convergence.     

Rates of strong uniform consistency for multivariate kernel density estimators

Let f_n denote the usual kernel density estimator in several dimensions. It is shown that if {a_n} is a regular band sequence, K is a bounded square integrable kernel of several variables, satisfying some additional mild conditions ((K₁) below), and if the data consist of an i.i.d. sample from a distribution possessing a bounded density f with respect to Lebesgue measure on R^d, then for some absolute constant C that depends only on d. With some additional but still weak conditions, it is proved that the above sequence of normalized suprema converges a.s. to . Convergence of the moment generating functions is also proved. Neither of these results require f to be strictly positive. These results improve upon, and extend to several dimensions, results by Silverman [13] for univariate densities.     

Rates of uniform convergence of extreme order statistics

Summary Bounds for the convergence uniformly over all Borel sets of the largest order statistic as well as of the joint distribution of extremes are established which reveal in which way these rates are determined by the distance of the underlying density from the density of the corresponding generalized Pareto distribution. The results are highlighted by several examples among which there is a bound for the rate at which the joint distribution of thek largest order statistics from a normal distribution converges uniformly to its limit.     

The rate of strong uniform consistency for the multivariate product-limit estimator

The general asymptotic order of magnitude is determined for the maximal deviation of the multivariate product-limit estimate from the estimated survival function on R^k. This order depends on the joint behavior of the censoring and censored distributions in a well-defined way. Corresponding to specific joint behaviors, several lim sup results are deduced generalizing everything that is known in the univariate case. The results are also extended for the variable censoring model.     

On the uniform complete convergence of density function estimates

Letf be a uniformly continuous density function. LetW be a non-negative weight function which is continuous on its compact support [a, b] and ∫ _a ^b W(x)dx=1. The complete convergence of $$\mathop {\sup }\limits_{ - \infty< s< \infty } \left| {\frac{1}{{nb\left( n \right)}}\sum\limits_{k - 1}^n {W\left( {\frac{{s - X_k }}{{b\left( n \right)}}} \right)} - f\left( s \right)} \right|$$ to zero is obtained under varying conditions on the bandwidthsb(n), support or moments off, and smoothness ofW. For example, one choice of the weight function for these results is Epanechnikov's optimal function andnb ²(n)>n ^δ for some δ>0. The uniform complete convergence of the mode estimate is also considered.     

A strong uniform time for random transpositions

A random permutation ofN items generated by a sequence ofK random transpositions is considered. The method of strong uniform times is used to give an upper bound on the variation distance between the distributions of the random permutation generated and a uniformly distributed permutation. The strong uniform time is also used to find the asymptotic distribution of the number of fixed points of the generated permutation. This is used to give a lower bound on the same variation distance. Together these bounds give a striking demonstration of the threshold phenomenon in the convergence of rapidly mixing Markov chains to stationarity.     

Arzelà's Theorem and strong uniform convergence on bornologies

In 1883 Arzelà (1983/1984) [2] gave a necessary and sufficient condition via quasi-uniform convergence for the pointwise limit of a sequence of real-valued continuous functions on a compact interval to be continuous. Arzelà's work paved the way for several outstanding papers. A milestone was the P.S. Alexandroff convergence introduced in 1948 to tackle the question for a sequence of continuous functions from a topological space (not necessarily compact) to a metric space. In 2009, in the realm of metric spaces, Beer and Levi (2009) [10] found another necessary and sufficient condition through the novel notion of strong uniform convergence on finite sets. We offer a direct proof of the equivalence of Arzelà, Alexandroff and Beer-Levi conditions. The proof reveals the internal gear of these important convergences and sheds more light on the problem. We also study the main properties of the topology of strong uniform convergence of functions on bornologies, initiated in Beer and Levi (2009) [10].     

Rates of convergence in the strong law of large numbers for degenerate U-statistics

The problem of rates of convergence in the strong law of large numbers for degenerate U-statistics is discussed. These results are similar to those known for non-degenerate U-statistics.     

On the strong convergence of derivatives in a time optimal problem

We consider a time optimal problem for a system described by a differential inclusion, whose right hand side is not necessarily convex valued. Under the assumption of strict convexity of the map obtained by convexifying the original, non-convex valued map, we obtain the strong convergence of the derivatives of any uniformly converging minimizing sequence. The assumptions required by this result are satisfied, for instance, by the classical brachystochrone problem and by Fermat’s principle.     

Pointwise and uniform convergence of kernel density estimators using random bandwidths

We obtain the rates of pointwise and uniform convergence of kernel density estimators using random bandwidths under i.i.d. as well as strongly mixing dependence assumptions. Pointwise rates are faster and not affected by the tail of the density.     

Rates of convergence in the functional CLT for multidimensional continuous time martingales

New rates of convergence in the multidimensional functional CLT are given by means of the Prokhorov's distance between a brownian motion and a continuous time martingale, with no further assumption than square integrability. The results are completely and simply expressed with distances of predictable characteristics which naturally occur in various statements of CLT for martingales.     

Almost sure convergence of the Bartlett estimator

Summary We study the almost sure convergence of the Bartlett estimator for the asymptotic variance of the sample mean of a stationary weekly dependent process. We also study the a.\ s.\ behavior of this estimator in the case of long-range dependent observations. In the weakly dependent case, we establish conditions under which the estimator is strongly consistent. We also show that, after appropriate normalization, the estimator converges a.s. in the long-range dependent case as well. In both cases, our conditions involve fourth order cumulants and assumptions on the rate of growth of the truncation parameter appearing in the definition of the Bartlett estimator.     

About the statistical uniform convergence

In this work we study the concept of statistical uniform convergence. We generalize some results of uniform convergence in double sequences to the case of statistical convergence. We also prove a basic matrix theorem with statistical convergence.     

On the uniform complete convergence of estimates for multivariate density functions and regression curves

Let (X ₁,Y ₁),...(X _n ,Y _n ) be a random sample from the (k+1)-dimensional multivariate density functionf ^*(x,y). Estimates of thek-dimensional density functionf(x)=∫f ^*(x,y)dy of the form $$\hat f_n (x) = \frac{1}{{nb_1 (n) \cdots b_k (n)}}\sum\limits_{i = 1}^n W \left( {\frac{{x_1 - X_{i1} }}{{b_1 (n)}}, \cdots ,\frac{{x_k - X_{ik} }}{{b_k (n)}}} \right)$$ are considered whereW(x) is a bounded, nonnegative weight function andb ₁ (n),...,b _k (n) and bandwidth sequences depending on the sample size and tending to 0 asn→∞. For the regression function $$m(x) = E(Y|X = x) = \frac{{h(x)}}{{f(x)}}$$ whereh(x)=∫y(f) ^* (x, y)dy , estimates of the form $$\hat h_n (x) = \frac{1}{{nb_1 (n) \cdots b_k (n)}}\sum\limits_{i = 1}^n {Y_i W} \left( {\frac{{x_1 - X_{i1} }}{{b_1 (n)}}, \cdots ,\frac{{x_k - X_{ik} }}{{b_k (n)}}} \right)$$ are considered. In particular, unform consistency of the estimates is obtained by showing that \(||\hat f_n (x) - f(x)||_\infty \) and \(||\hat m_n (x) - m(x)||_\infty \) converge completely to zero for a large class of “good” weight functions and under mild conditions on the bandwidth sequencesb _k (n)'s.     

Hamiltonians on discrete structures: jumps of the integrated density of states and uniform convergence

We study equivariant families of discrete Hamiltonians on amenable geometries and their integrated density of states (IDS). We prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support. The size of a jump of the IDS is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions. From this we deduce uniform convergence (w.r.t. the spectral parameter) of the finite volume approximants of the IDS. Our framework includes quasiperiodic operators on Delone sets, periodic and random operators on quasi-transitive graphs, and operators on percolation graphs.