Narrow Results

We consider the dynamical system given by an algebraic ergodic automorphism T on a torus. We study a Central Limit Theorem for the empirical process associated to the stationary process (f◦Tⁱ)_i∈ℕ, where f is a given ℝ-valued function. We give a sufficient condition on f for this Central Limit Theorem to hold.

In the second part, we prove that the distribution function of a Morse function is continuously differentiable if the dimension of the manifold is at least three and Hölder continuous if the dimension is one or two. As a consequence, the Morse functions satisfy the empirical invariance principle, which is therefore generically verified.

We study the behavior of the empirical distribution function of iterates of intermittent maps in the Hilbert space of square integrable functions with respect to Lebesgue measure. In the long-range dependent case, we prove that the empirical distribution function, suitably normalized, converges to a degenerate stable process, and we give the corresponding almost sure result. We apply the results to the convergence of the Wasserstein distance between the empirical measure and the invariant measure. We also apply it to obtain the asymptotic distribution of the corresponding Cramér–von-Mises statistic.

The authors establish the Hilbertian invariance principle for the empirical process of a stationary Markov process, by extending the forward-backward martingale decomposition of Lyons-Meyer-Zheng to the Hilbert space valued additive functionals associated with general non-reversible Markov processes.

Results of Norwegian Elite Division soccer games are studied for the year 2003. Previous writers have modelled the number of goals a given team scores in a game and then moved on to evaluating the probabilities of a win, a tie and a loss. However in this work the probabilities of win, tie and loss are modelled directly. There are attempts to improve the fit by including various explanatories.

We review gene mapping, or inference for quantitative trait loci, in the context of recent research in semi-parametric and non-parametric inference for mixture models. Gene mapping studies the relationship between a phenotypic trait and inherited genotype. Semi-parametric gene mapping using the exponential tilt covers most standard exponential families and improves estimation of genetic effects. Non-parametric gene mapping, including a generalized Hodges-Lehmann shift estimator and Kaplan-Meier survival curve, provide a general framework for model selection for the influence of genotype on phenotype. Examples and summaries of reported simulations show the power of these methods when data are far from normal.

Suppose that (X_i, Y_i,), i = 1, 2, … , n, are iid. random vectors with uniform marginals and a certain joint distribution F_ρ, where ρ is a parameter with ρ = ρ_o corresponds to the independence case. However, the X's and Y's are observed separately so that the pairing information is missing. Can ρ be consistently estimated? This is an extension of a problem considered in DeGroot and Goel (1980) which focused on the bivariate normal distribution with ρ being the correlation. In this paper we show that consistent discrimination between two distinct parameter values ρ₁ and ρ₂ is impossible if the density f_ρ of F_ρ is square integrable and the second largest singular value of the linear operator , h ∈ L²[0, 1], is strictly less than 1 for ρ = ρ₁ and ρ₂. We also consider this result from the perspective of a bivariate empirical process which contains information equivalent to that of the broken sample.