Testing the epidemic change in nearly nonstationary autoregressive processes

Some tests for an epidemic type change in a first order nearly nonstationary autoregressive process are investigated. Limit distributions of the tests are found under no change. Consistencyis examined under short epidemics in the mean of innovations

Une Topologie Préhilbertienne Sur L'espace Des Mesures A Signes Bornées

International audienceLet X a metric space and B its Borel o-algebra. Let M the space of bounded signed measures on B. We construct a scalar product on M so that M can be imbedded in a reproducing kernel space H„ . We give hilbertian bases of . Weak topology and prehilbertian topology are compared on M . Compactness criteria in H„ and M are given.Soit B la tribu borélienne d’un espace métrique X. Soit M l’espace des mesures à signes bornées sur B. On construit un produit scalaire permettant de plonger M dans un espace fonctionnel autoreproduisant . On donne des bases hilbertiennes de H„ . On compare la topologie faible et la trace de la topologie hilbertienne sur M . On donne des critères de relative compacité dans H„ et dans M

Convergences Stochastiques De Suites De Mesures Aléatoires Signées Considérées Comme Variables Aléatoires Hilbertiennes

International audienceLet X be a metric space and B its Borel cr-algebra. Let AI the space of bounded signed measures on B. In order to construct the signed random measures as hilbertian random variables we use the embedding of AI in a separable reproducing kernel space Hk. That construction is effectively possible when X is compact or locally compact or separable. Mesurability problems due to embedding are resolved in these three cases. Real random variables naturally associated with a so defined random measure //* are ingrains of Hr elements with respect to /i*. For random sequences and their associated random variables, three modes of convergence are studied (almost surely, in probability, in law)Soit B la tribu borélienne d’un espace métrique X. On utilise le plongement de l’espace M. des mesures signées bornées sur B dans un espace de Hilbert autoreproduisant séparable Hk afin de construire les mesures aléatoires signées comme variables aléatoires hilbertiennes. Cette construction est effective- ment possible dans les trois cas où X est compact, localement compact, séparable. Les problèmes de mesurabilité dus à ce plongement sont résolus dans ces 3 cas. Les variables aléatoires réelles naturellement associées à une mesure aléatoire ainsi définie sont les intégrales des éléments de Hk par rapport à cette mesure. On étudie la convergence presque sûre, en probabilité et en loi des suites de mesures aléatoires et de leurs v.a.r. associées

Weak Hölder convergence of processes with application to the perturbed empirical process

We consider stochastic processes as random elements in some spaces of Hölder functions vanishing at infinity. The corresponding scale of spaces $C^{α,0}_0$ is shown to be isomorphic to some scale of Banach sequence spaces. This enables us to obtain some tightness criterion in these spaces. As an application, we prove the weak Hölder convergence of the convolution-smoothed empirical process of an i.i.d. sample $(X_1,...,X_n)$ under a natural assumption about the regularity of the marginal distribution function F of the sample. In particular, when F is Lipschitz, the best possible bound α<1/2 for the weak α-Hölder convergence of such processes is achieved

Hölderian invariance principle for Hilbertian linear processes

Let $(\xi_n)_{n\ge 1}$ be the polygonal partial sums processes built on the linear processes $X_n=\sum_{i\ge 0}a_i(\epsilon_{n-i})$, n ≥ 1, where $(\epsilon_i)_{i\in\mathbb{Z}}$ are i.i.d., centered random elements in some separable Hilbert space $\mathbb{H}$ and the ai's are bounded linear operators $\mathbb{H}\to \mathbb{H}$, with $\sum_{i\ge 0}\lVert a_i\rVert<\infty$. We investigate functional central limit theorem for $\xi_n$ in the Hölder spaces $\mathrm{H}^o_\rho(\mathbb{H})$ of functions $x:[0,1]\to\mathbb{H}$ such that ||x(t + h) - x(t)|| = o(p(h)) uniformly in t, where p(h) = hαL(1/h), 0 ≤ h ≤ 1 with 0 ≤ α ≤ 1/2 and L slowly varying at infinity. We obtain the $\mathrm{H}^o_\rho(\mathbb{H})$ weak convergence of $\xi_n$ to some $\mathbb{H}$ valued Brownian motion under the optimal assumption that for any c>0, $tP(\lVert \epsilon_0\rVert>ct^{1/2}\rho(1/t))=o(1)$ when t tends to infinity, subject to some mild restriction on L in the boundary case α = 1/2. Our result holds in particular with the weight functions p(h) = h1/2lnβ(1/h), β > 1/2>

Processus stochastiques dans les espaces de Besov

LILLE1-BU (590092102) / SudocSudocFranceF

On Bernstein–Kantorovich invariance principle in Hölder spaces and weighted scan statistics

Let ξn be the polygonal line partial sums process built on i.i.d. centered random variables Xi, i ≥ 1. The Bernstein-Kantorovich theorem states the equivalence between the finiteness of E|X1|max(2,r) and the joint weak convergence in C[0, 1] of n−1∕2ξn to a Brownian motion W with the moments convergence of E∥n−1/2ξn∥∞r E∥n-1∕2ξn∥∞r to E∥W∥∞r E∥W∥∞r . For 0 < α < 1∕2 and p (α) = (1 ∕ 2 - α) -1, we prove that the joint convergence in the separable Hölder space Hαo Hαo of n−1∕2ξn to W jointly with the one of E∥n−1∕2ξn∥αr E∥n-1∕2ξn∥αr to E∥W∥αr E∥W∥αr holds if and only if P(|X1| > t) = o(t−p(α)) when r 1∕2 − 1∕p with an appropriate normalization

On the asymptotic of the maximal weighted increment of a random walk with regularly varying jumps: the boundary case

Let (Xi)i1 be i.i.d. random variables with EX1 = 0, regularly varying with exponent a > 2 and taP(jX1j > t) L(t) slowly varying as t ! 1. We give the limit distribution of Tn( )=max0j 0. We prove that c1 n (Tn( a) n), converges in distribution to some random variable Z if and only if L has a limit a 2 [0;1] at infinity. In such case, there are A > 0, B 2 R such that Z = AVa;; + B in distribution, where for 0 < < 1, Va;; := max(T( a); Ya) with T( a) and Ya independent and Va;;0 := T( a), Va;;1 := Ya. When < 1, a possible choice for the normalization is cn = n1=a and n = 0, with Z = Va;; . We also build an example where L has no limit at infinity and (Tn( ))n1 has for each 2 [0;1] a subsequence converging after normalization to Va;;

Estimating a changed segment in a sample

Abstract In the paper we consider a changed segment model for sample distributions. We generalize Dümbgen&apos;s [6] change point estimator and obtain optimal rates of convergence of estimators of the begining and the length of the changed segment