On the Bickel-Rosenblatt test of goodness-of-fit for the residuals of autoregressive processes

We investigate in this paper a Bickel-Rosenblatt test of goodness-of-fit for the density of the noise in an autoregressive model. Since the seminal work of Bickel and Rosenblatt, it is well-known that the integrated squared error of the Parzen-Rosenblatt density estimator, once correctly renormalized, is asymptotically Gaussian for independent and identically distributed (i.i.d.) sequences. We show that the result still holds when the statistic is built from the residuals of general stable and explosive autoregressive processes. In the univariate unstable case, we prove that the result holds when the unit root is located at $-1$ whereas we give further results when the unit root is located at $1$. In particular, we establish that except for some particular asymmetric kernels leading to a non-Gaussian limiting distribution and a slower convergence, the statistic has the same order of magnitude. We also study some common unstable cases, like the integrated seasonal process. Finally we build a goodness-of-fit Bickel-Rosenblatt test for the true density of the noise together with its empirical properties on the basis of a simulation study

Testing the Effect of Relative Pollen Productivity on the REVEALS Model: A Validated Reconstruction of Europe-Wide Holocene Vegetation

Reliable quantitative vegetation reconstructions for Europe during the Holocene are crucial to improving our understanding of landscape dynamics, making it possible to assess the past effects of environmental variables and land-use change on ecosystems and biodiversity, and mitigating their effects in the future. We present here the most spatially extensive and temporally continuous pollen-based reconstructions of plant cover in Europe (at a spatial resolution of 1° × 1°) over the Holocene (last 11.7 ka BP) using the ‘Regional Estimates of VEgetation Abundance from Large Sites’ (REVEALS) model. This study has three main aims. First, to present the most accurate and reliable generation of REVEALS reconstructions across Europe so far. This has been achieved by including a larger number of pollen records compared to former analyses, in particular from the Mediterranean area. Second, to discuss methodological issues in the quantification of past land cover by using alternative datasets of relative pollen productivities (RPPs), one of the key input parameters of REVEALS, to test model sensitivity. Finally, to validate our reconstructions with the global forest change dataset. The results suggest that the RPPs.st1 (31 taxa) dataset is best suited to producing regional vegetation cover estimates for Europe. These reconstructions offer a long-term perspective providing unique possibilities to explore spatial-temporal changes in past land cover and biodiversit

A conditional Berry-Esseen bound and a conditional large deviation result without Laplace transform. Application to hashing with linear probing

We study the asymptotic behavior of a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables. We prove a Berry-Esseen bound in a general setting and a large deviation result when the Laplace trans-form of the underlying distribution is not defined in a neighborhood of zero. Then we present several combinatorial applications. In particular, we prove a large deviation result for the model of hashing with linear probing

Large deviations and Berry-Esseen bounds for hashing with linear probing

We study the asymptotic behavior of a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables. First, we prove a large deviations result in the context of hashing with linear probing. By the way, we establish a large deviations result for triangular arrays when the Laplace transform is not defined in a neighborhood of 0. Second, we prove a Berry-Esseen bound in a general setting

Deviation results for sparse tables in hashing with linear probing

We consider the model of hashing with linear probing and we establish the moderate and large deviations for the total displacement in sparse tables. In this context, Weibull-like-tailed random variables appear. Deviations for sums of such heavy-tailed random variables are studied in \cite{Nagaev69-1,Nagaev69-2}. Here we adapt the proofs therein to deal with conditioned sums of such variables and solve the open question in \cite{TFC12}. By the way, we establish the deviations of the total displacement in full tables, which can be derived from the deviations of empirical processes of i.i.d.\ random variables established in \cite{Wu94}.

Estimating the minimal length of tardos code

Abstract. This paper estimates the minimal length of a binary probabilistic traitor tracing code. We consider the code construction proposed by G. Tardos in 2003, with the symmetric accusation function as improved by B. Skoric et al. The length estimation is based on two pillars. First, we consider the Worst Case Attack that a group of c colluders can lead. This attack minimizes the mutual information between the code sequence of a colluder and the pirated sequence. Second, an algorithm pertaining to the field of rare event analysis is presented in order to estimate the probabilities of error: the probability that an innocent user is framed, and the probabilities that all colluders are missed. Therefore, for a given collusion size, we are able to estimate the minimal length of the code satisfying some error probabilities constraints. This estimation is far lower than the known lower bounds

Testing the Effect of Relative Pollen Productivity on the REVEALS Model: A Validated Reconstruction of Europe-Wide Holocene Vegetation

Reliable quantitative vegetation reconstructions for Europe during the Holocene are crucial to improving our understanding of landscape dynamics, making it possible to assess the past effects of environmental variables and land-use change on ecosystems and biodiversity, and mitigating their effects in the future. We present here the most spatially extensive and temporally continuous pollen-based reconstructions of plant cover in Europe (at a spatial resolution of 1º × 1º) over the Holocene (last 11.7 ka BP) using the "Regional Estimates of VEgetation Abundance from Large Sites" (REVEALS) model. This study has three main aims. First, to present the most accurate and reliable generation of REVEALS reconstructions across Europe so far. This has been achieved by including a larger number of pollen records compared to former analyses, in particular from the Mediterranean area. Second, to discuss methodological issues in the quantification of past land cover by using alternative datasets of relative pollen productivities (RPPs), one of the key input parameters of REVEALS, to test model sensitivity. Finally, to validate our reconstructions with the global forest change dataset. The results suggest that the RPPs.st1 (31 taxa) dataset is best suited to producing regional vegetation cover estimates for Europe. These reconstructions offer a long-term perspective providing unique possibilities to explore spatial-temporal changes in past land cover and biodiversity.This research was funded by the TERRANOVA Project, H2020 Marie Sklodowska-Curie grant agreement no. 81390