We introduce a family of convex (concave) functions called sup (inf) of powers, which areused as generator functions for a special type of quasi-arithmetic means. Using thesemeans, we generalize the large deviation result on self-normalized statistics that wasobtained in the homogeneous case by [Q.-M. Shao, Self-normalized large deviations.Ann. Probab. 25 (1997) 285–328]. Furthermore, in thehomogenous case, we derive the Bahadur exact slope for tests using self-normalizedstatistics.

We prove the local asymptotic normality for the full parameters of the normal inverseGaussian Lévy process X, when we observe high-frequency data XΔn,X2Δn,...,XnΔnwith sampling mesh Δn → 0 and the terminalsampling time nΔn → ∞. Therate of convergence turns out to be (√nΔn, √nΔn, √n, √n) for the dominating parameter(α,β,δ,μ), where α stands for the heaviness of thetails, β the degree of skewness, δ the scale, andμ the location. The essential feature in our study is that the suitablynormalized increments of X in small time is approximatelyCauchy-distributed, which specifically comes out in the form of the asymptotic Fisherinformation matrix.

In this article, our aim is to estimate the successive derivatives of the stationarydensity f of a strictly stationary and β-mixing process (Xt)t≥0. This process is observed at discrete timest = 0,Δ,...,nΔ. Thesampling interval Δ can be fixed or small. We use a penalizedleast-square approach to compute adaptive estimators. If the derivativef(j) belongs to the Besov space \hbox{$\rond{B}_{2,\infty}^{\alpha}$}$B_{\mathrm{2}\mathit{,}\mathrm{\infty }}^{\mathit{\alpha }}$, then our estimator converges at rate (nΔ)−α/(2α+2j+1). Then we consider a diffusion with known diffusioncoefficient. We use the particular form of the stationary density to compute an adaptiveestimator of its first derivative f′. When the sampling intervalΔ tends to 0, and when the diffusion coefficient is known, theconvergence rate of our estimator is (nΔ)−α/(2α+1). When the diffusion coefficient is known, we alsoconstruct a quotient estimator of the drift for low-frequency data.

Equip the edges of the lattice ℤ2 with i.i.d. random capacities. A law oflarge numbers is known for the maximal flow crossing a rectangle in ℝ2 when theside lengths of the rectangle go to infinity. We prove that the lower large deviations areof surface order, and we prove the corresponding large deviation principle from below.This extends and improves previous large deviations results of Grimmett and Kesten [9] obtained for boxes of particular orientation.

We study the convergence rate of randomly truncated stochastic algorithms, which consistin the truncation of the standard Robbins–Monro procedure on an increasing sequence ofcompact sets. Such a truncation is often required in practice to ensure convergence whenstandard algorithms fail because the expected-value function grows too fast. In this work,we give a self contained proof of a central limit theorem for this algorithm under localassumptions on the expected-value function, which are fairly easy to check in practice.

We derive a central limit theorem for triangular arrays of possibly nonstationary random variables satisfying a condition of weak dependence in the sense of Doukhan and Louhichi [Stoch. Proc. Appl. 84 (1999) 313–342]. The proof uses a new variant of the Lindeberg method: the behavior of the partial sums is compared to that of partial sums of dependent Gaussian random variables. We also discuss a few applications in statistics which show that our central limit theorem is tailor-made for statistics of different type.

Multistable processes, that is, processes which are, at each “time”, tangent to a stableprocess, but where the index of stability varies along the path, have been recentlyintroduced as models for phenomena where the intensity of jumps is non constant. In thiswork, we give further results on (multifractional) multistable processes related to theirlocal structure. We show that, under certain conditions, the incremental moments display ascaling behaviour, and that the pointwise Hölder exponent is, as expected, related to thelocal stability index. We compute the precise value of the almost sure Hölder exponent inthe case of the multistable Lévy motion, which turns out to reveal an interestingphenomenon.

We study the almost sure asymptotic behaviour of stochastic approximation algorithms for the search of zero of a real function. The quadratic strong law of large numbers is extended to the powers greater than one. In other words, the convergence of moments in the almost sure central limit theorem (ASCLT) is established. As a by-product of this convergence, one gets another proof of ASCLT for stochastic approximation algorithms. The convergence result is applied to several examples as estimation of quantiles and recursive estimation of the mean.

Consider a strong Markov process in continuous time, taking values in some Polish statespace. Recently, Douc et al. [Stoc. Proc. Appl.119, (2009) 897–923] introduced verifiable conditions in terms ofa supermartingale property implying an explicit control of modulated moments of hittingtimes. We show how this control can be translated into a control of polynomial moments ofabstract regeneration times which are obtained by using the regeneration method ofNummelin, extended to the time-continuous context. As a consequence, if ap-th moment of the regeneration times exists, we obtain non asymptoticdeviation bounds of the form

\begin{equation*} P_{\nu} \left(\left|\frac1t\int_0^tf(X_s){\rm d}s-\mu(f)\right|\geq\ge\right)\leqK(p)\frac1{t^{p- 1}}\frac 1{\ge^{2(p-1)}}\|f\|_\infty^{2(p-1)} ,\quad p \geq 2.\end{equation*}${\mathit{P}}_{\mathit{\nu }}\left(\left|\frac{\mathrm{1}}{\mathit{t}}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{t}}\mathit{f}\mathrm{\left(}{\mathit{X}}_{\mathit{s}}\mathrm{\right)}\mathrm{d}\mathit{s}\mathrm{-}\mathit{\mu }\mathrm{\left(}\mathit{f}\mathrm{\right)}\right|\mathrm{\ge }\mathit{\epsilon }\right)\mathrm{\le }\mathit{K}\mathrm{\left(}\mathit{p}\mathrm{\right)}\frac{\mathrm{1}}{{\mathit{t}}^{\mathit{p}\mathrm{-}\mathrm{1}}}\frac{\mathrm{1}}{{\mathit{\epsilon }}^{\mathrm{2}\mathrm{(}\mathit{p}\mathrm{-}\mathrm{1}\mathrm{)}}}\mathrm{\parallel }\mathit{f}{\mathrm{\parallel }}_{\mathrm{\infty }}^{\mathrm{2}\mathrm{(}\mathit{p}\mathrm{-}\mathrm{1}\mathrm{)}}\mathit{,}\mathit{p}\mathrm{\ge }\mathrm{2.}$

We define multifractional Brownian fields indexed by a metric space, such as a manifold with its geodesic distance, when the distance is of negative type. This construction applies when the Brownian field indexed by the metric space exists, in particular for spheres, hyperbolic spaces and real trees.

Consider testingH0 : F ∈ ω0against H1 : F ∈ ω1for a random sampleX1, ..., Xnfrom F, where ω0 andω1 are two disjoint sets of cdfs onℝ = (−∞, ∞). Two non-local types of efficiencies, referred to as thefixed-α and fixed-β efficiencies, are introduced forthis two-hypothesis testing situation. Theoretical tools are developed to evaluate theseefficiencies for some of the most usual goodness of fit tests (including theKolmogorov–Smirnov tests). Numerical comparisons are provided using several examples.

This paper deals with the problem of estimating the level setsL(c) =  {F(x) ≥ c},with c ∈ (0,1), of an unknown distribution function F on ℝ+2. A plug-inapproach is followed. That is, given a consistent estimatorFn of F, we estimateL(c) byLn(c) =  {Fn(x) ≥ c}.In our setting, non-compactness property is a priori required for thelevel sets to estimate. We state consistency results with respect to the Hausdorffdistance and the volume of the symmetric difference. Our results are motivated byapplications in multivariate risk theory. In particular we propose a new bivariate versionof the conditional tail expectation by conditioning the two-dimensional random vector tobe in the level set L(c). We also present simulated andreal examples which illustrate our theoretical results.

We consider, in the continuous time version, γ independent random walks on Z+ in random environment in Sinai’s regime. Let Tγ be the first meeting time of one pair of the γ random walks starting at different positions. We first show that the tail of the quenched distribution of Tγ, after a suitable rescaling, converges in probability, to some functional of the Brownian motion. Then we compute the law of this functional. Eventually, we obtain results about the moments of this meeting time. Being Eω the quenched expectation, we show that, for almost all environments ω, Eω[Tγc] is finite for c < γ(γ − 1) / 2 and infinite for c > γ(γ − 1) / 2.

In this note we prove that the local martingale part of a convex functionf of a d-dimensional semimartingaleX = M + A can be written in terms ofan Itô stochastic integral∫H(X)dM, whereH(x) is some particular measurable choice ofsubgradient \hbox{$\sub$}$\overline{\mathrm{\nabla }}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ off at x, and M is the martingale partof X. This result was first proved by Bouleau in [N. Bouleau, C.R. Acad. Sci. Paris Sér. I Math. 292 (1981) 87–90]. Here wepresent a new treatment of the problem. We first prove the result for\hbox{$\widetilde{X}=X+\epsilon B$}$\begin{array}{}{}_{\mathrm{􏽥}}\\ \mathit{X}\end{array}\mathrm{=}\mathit{X}\mathrm{+}\mathit{ϵB}$,ϵ > 0, where B is a standardBrownian motion, and then pass to the limit as ϵ → 0, using results in[M.T. Barlow and P. Protter, On convergence of semimartingales. In Séminaire deProbabilités, XXIV, 1988/89, Lect. Notes Math., vol. 1426.Springer, Berlin (1990) 188–193; E. Carlen and P. Protter, Illinois J. Math.36 (1992) 420–427]. The former paper concerns convergence ofsemimartingale decompositions of semimartingales, while the latter studies a special caseof converging convex functions of semimartingales.

We investigate here the central limit theorem of the increment ratio statistic of amultifractional Brownian motion, leading to a CLT for the time varying Hurst index. Theproofs are quite simple relying on Breuer–Major theorems and an original freezingof time strategy. A simulation study shows the goodness of fit of thisestimator.

This paper deals with the problem of estimating a regression function f,in a random design framework. We build and study two adaptive estimators based on modelselection, applied with warped bases. We start with a collection of finite dimensionallinear spaces, spanned by orthonormal bases. Instead of expanding directly the targetfunction f on these bases, we rather consider the expansion ofh = f ∘ G-1, whereG is the cumulative distribution function of the design, followingKerkyacharian and Picard [Bernoulli 10 (2004) 1053–1105].The data-driven selection of the (best) space is done with two strategies: we use both apenalization version of a “warped contrast”, and a model selection device in the spirit ofGoldenshluger and Lepski [Ann. Stat. 39 (2011) 1608–1632].We propose by these methods two functions, ĥl(l = 1, 2), easier to compute than least-squares estimators. Weestablish nonasymptotic mean-squared integrated risk bounds for the resulting estimators, \hbox{$\hat{f}_l=\hat{h}_l\circ G$}f̂l = ĥl°G if G is known, or\hbox{$\hat{f}_l=\hat{h}_l\circ\hat{G}$}f̂l = ĥl°Ĝ(l = 1,2) otherwise, where Ĝ is theempirical distribution function. We study also adaptive properties, in case the regressionfunction belongs to a Besov or Sobolev space, and compare the theoretical and practicalperformances of the two selection rules.

We deal with a subject in the interplay between nonparametric statistics and geometricmeasure theory. The measure L0(G) of theboundary of a set G ⊂ ℝd (withd ≥ 2) can be formally defined, via a simple limit, bythe so-called Minkowski content. We study the estimation ofL0(G) from a sample of random points insideand outside G. The sample design assumes that, for each sample point, weknow (without error) whether or not that point belongs to G. Under thisdesign we suggest a simple nonparametric estimator and investigate its consistencyproperties. The main emphasis in this paper is on generality. So we are especiallyconcerned with proving the consistency of our estimator under minimal assumptions on theset G. In particular, we establish a mild shape condition onG under which the proposed estimator is consistent inL2. Roughly speaking, such condition establishes that theset of “very spiky” points at the boundary of G must be “small”. This isformalized in terms of the Minkowski content of such set. Several examples arediscussed.

We study the scenario of graph-based clustering algorithms such as spectral clustering. Given a set of data points, one first has to construct a graph on the data points and then apply a graph clustering algorithm to find a suitable partition of the graph. Our main question is if and how the construction of the graph (choice of the graph, choice of parameters, choice of weights) influences the outcome of the final clustering result. To this end we study the convergence of cluster quality measures such as the normalized cut or the Cheeger cut on various kinds of random geometric graphs as the sample size tends to infinity. It turns out that the limit values of the same objective function are systematically different on different types of graphs. This implies that clustering results systematically depend on the graph and can be very different for different types of graph. We provide examples to illustrate the implications on spectral clustering.

The transition kernel of the well-known Metropolis-Hastings (MH) algorithm has a point mass at the chain’s current position, which prevent direct smoothness properties to be derived for the successive densities of marginals issued from this algorithm. We show here that under mild smoothness assumption on the MH algorithm “input” densities (the initial, proposal and target distributions), propagation of a Lipschitz condition for the iterative densities can be proved. This allows us to build a consistent nonparametric estimate of the entropy for these iterative densities. This theoretical study can be viewed as a building block for a more general MCMC evaluation tool grounded on such estimates.

In a recent paper Fay and Philippe [ESAIM: PS 6 (2002) 239–258] proposed a goodness-of-fit test for long-range dependent processes which uses the logarithmic contrast as information measure. These authors established asymptotic normality under the null hypothesis and local alternatives. In the present note we extend these results and show that the corresponding test statistic is also normally distributed under fixed alternatives.