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We introduce a class of discrete time stationary trawl processes taking real or integer values and written as sums of past values of independent 'seed' processes on shrinking intervals ('trawl heights'). Related trawl processes in continuous time were studied in Barndorff-Nielsen (2011) and Barndorff-Nielsen et al. (2014), however in our case the i.i.d. seed processes can be very general and need not be infinitely divisible. In the case when the trawl height decays with the lag as j −α for some 1 < α < 2, the trawl process exhibits long memory and its covariance decays as j 1−α . We show that under general conditions on generic seed process, the normalized partial sums of such trawl process may tend either to a fractional Brownian motion or to an α-stable Lévy process. 60G22 Fractional processes, including fractional Brownian motion 60G51 Processes with independent increments; Lévy processes 60G99 Trawl process 60K99 Long range memory process

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Genetic data are frequently categorical and have complex dependence structures that are not always well understood. For this reason, clustering and classification based on genetic data, while highly relevant, are challenging statistical problems. Here we consider a highly versatile U-statistics based approach built on dissimilarities between pairs of data points for nonparametric clustering. In this work we propose statistical tests to assess group homogeneity taking into account the multiple testing issues, and a clustering algorithm based on dissimilarities within and between groups that highly speeds up the homogeneity test. We also propose a test to verify classification significance of a sample in one of two groups. A Monte Carlo simulation study is presented to evaluate power of the classification test, considering different group sizes and degree of separation. Size and power of the homogeneity test are also analyzed through simulations that compare it to competing methods. Finally, the methodology is applied to three different genetic datasets: global human genetic diversity, breast tumor gene expression and Dengue virus serotypes. These applications showcase this statistical framework's ability to answer diverse biological questions while adapting to the specificities of the different datatypes.

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Here we present a theoretical study on the main properties of Fractionally Integrated Exponential Generalized Autoregressive Conditional Heteroskedastic (FIEGARCH) processes. We analyze the conditions for the existence, the invertibility, the stationarity and the ergodicity of these processes. We prove that, if {Xt}t∈Z is a FIEGARCH(p, d, q) process then, under mild conditions, {ln(X 2 t )}t∈Z is an ARFIMA(q, d, 0), that is, an autoregressive fractionally integrated moving average process. The convergence order for the polynomial coefficients that describes the volatility is presented and results related to the spectral representation and to the covariance structure of both processes {ln(X 2 t )}t∈Z and {ln(σ 2 t )}t∈Z are also discussed. Expressions for the kurtosis and the asymmetry measures for any stationary FIEGARCH (p, d, q) process are also derived. The h-step ahead forecast for the processes {Xt}t∈Z, {ln(σ 2 t )}t∈Z and {ln(X 2 t )}t∈Z are given with their respective mean square error forecast. The work also presents a Monte Carlo simulation study showing how to generate, estimate and forecast based on six different FIEGARCH models. The forecasting performance of six models belonging to the class of autoregressive conditional heteroskedastic models (namely, ARCH-type models) and radial basis models is compared through an empirical application to Brazilian stock market exchange index.

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In this work we derive the copulas related to Manneville-Pomeau processes. We examine both bidimensional and multidimensional cases and derive some properties for the related copulas. Computational issues, approximations and random variate generation problems are also addressed and simple numerical experiments to test the approximations developed are also perform. In particular, we propose an approximation to the copula which we show to converge uniformly to the true copula. To illustrate the usefulness of the theory, we derive a fast procedure to estimate the underlying parameter in Manneville-Pomeau processes.where A t,k 's are given by (3.3).Proof: The result follows easily from what was just discussed and from the fact that the intervals A t,k 's are (pairwise) disjoints.As for the copulas related to MP processes, in view of the stationarity of the MP process, the following result follows easily.Proposition 3.1. Let {X n } n∈N be an MP process with parameter s ∈ (0, 1) and ϕ ∈ L 1 (µ s ) be an almost everywhere increasing function. Then, for any t, h ∈ N, C Xt,X t+h (u, v) = C X 0 ,X h (u, v), everywhere in I 2 .