Phantom distribution functions for some stationary sequences

The notion of a phantom distribution function (phdf) was introduced by O'Brien (1987). We show that the existence of a phdf is a quite common phenomenon for stationary weakly dependent sequences. It is proved that any $\alpha$-mixing stationary sequence with continuous marginals admits a continuous phdf. Sufficient conditions are given for stationary sequences exhibiting weak dependence, what allows the use of attractive models beyond mixing. The case of discontinuous marginals is also discussed for $\alpha$-mixing. Special attention is paid to examples of processes which admit a continuous phantom distribution function while their extremal index is zero. We show that Asmussen (1998) and Roberts et al. (2006) provide natural examples of such processes. We also construct a non-ergodic stationary process of this type

A functional central limit theorem for interacting particle systems on transitive graphs

A finite range interacting particle system on a transitive graph is considered. Assuming that the dynamics and the initial measure are invariant, the normalized empirical distribution process converges in distribution to a centered diffusion process. As an application, a central limit theorem for certain hitting times, interpreted as failure times of a coherent system in reliability, is derived.Comment: 35 page

Dependent Lindeberg central limit theorem and some applications

In this paper, a very useful lemma (in two versions) is proved: it simplifies notably the essential step to establish a Lindeberg central limit theorem for dependent processes. Then, applying this lemma to weakly dependent processes introduced in Doukhan and Louhichi (1999), a new central limit theorem is obtained for sample mean or kernel density estimator. Moreover, by using the subsampling, extensions under weaker assumptions of these central limit theorems are provided. All the usual causal or non causal time series: Gaussian, associated, linear, ARCH($\infty$), bilinear, Volterra processes,$...$, enter this frame

Evaluation for moments of a ratio with application to regression estimation

Many problems involve ratios in probability or in statistical applications. We aim at approximating the moments of such ratios under specific assumptions. Using ideas from Collomb (1977) we propose sharper bounds for the moments of randomly weighted sums which also may appear as a ratio of two random variables. Suitable applications are given in more detail here in the fields of functional estimation, in finance and for censored data analysis. Several weak dependence dependence situations are considered

Test du processus de Poisson homogène par la statistique de Ripley

International audienceLa statistique de Ripley a été introduite pour mesurer l'agrégation des points d'un processus ponctuel. Elle a été très fréquemment utilisée notamment par les écologues, mais peu de résultats théoriques ont été obtenus; nous étudions les propriétés (biais, variance) de cet estimateur sous l'hypothèse d'un processus de Poisson homogène et prouvons un théorème de la limite centrale lorsque les données sont obtenues sur un carré de taille nxn avec n tendant vers l'infini. Nous en déduisons un test asymptotiquement gaussien de l'hypothèse de processus de Poisson homogène

Strong mixing condition for Hawkes processes and application to Whittle estimation from count data

This paper focuses on the time series generated by the event counts of stationary Hawkes processes. When the exact locations of points are not observed, but only counts over time intervals of fixed size, existing methods of estimation are not applicable. We first establish a strong mixing condition with polynomial decay rate for Hawkes processes, from their Poisson cluster structure. This allows us to propose a spectral approach to the estimation of Hawkes processes, based on Whittle's method, which provides consistent and asymptotically normal estimates under common regularity conditions on their reproduction kernels. Simulated datasets and a case-study illustrate the performances of the estimation, notably of the Hawkes reproduction mean and kernel when time intervals are relatively large.Comment: 29 pages, 10 figure

Estimation of long memory in the presence of a smooth nonparametric trend

We consider semi parametric estimation of the long-memory parameter of a stationary process in the presence of an additive nonparametric mean function. We use a semi parametric Whittle type estimator, applied to the tapered, differenced series. Since the mean function is not necessarily a polynomial of finite order, no amount of differencing will completely remove the mean. We establish a central limit theorem for the estimator of the memory parameter, assuming that a slowly increasing number of low frequencies are trimmed from the estimator's objective function. We find in simulations that tapering and trimming are essential for the good performance of the estimator in practice.Statistics Working Papers Serie

Structure of a translocation signal domain mediating conjugative transfer by Type IV secretion systems

Relaxases are proteins responsible for the transfer of plasmid and chromosomal DNA from one bacterium to another during conjugation. They covalently react with a specific phosphodiester bond within DNA origin of transfer sequences, forming a nucleo-protein complex which is subsequently recruited for transport by a plasmid-encoded type IV secretion system. In previous work we identified the targeting translocation signals presented by the conjugative relaxase TraI of plasmid R1. Here we report the structure of TraI translocation signal TSA. In contrast to known translocation signals we show that TSA is an independent folding unit and thus forms a bona fide structural domain. This domain can be further divided into three sub-domains with striking structural homology with helicase sub-domains of the SF1B family. We also show that TSA is part of a larger vestigial helicase domain which has lost its helicase activity but not its single-stranded DNA binding capability. Finally, we further delineate the binding site responsible for translocation activity of TSA by targeting single residues for mutations. Overall, this study provides the first evidence that translocation signals can be part of larger structural scaffolds, overlapping with translocation-independent activities