Conjunctive Bayesian networks (CBNs) are graphical models that describe the accumulation of events which are constrained in the order of their occurrence. A CBN is given by a partial order on a (finite) set of events. CBNs generalize the oncogenetic tree models of Desper et al. by allowing the occurrence of an event to depend on more than one predecessor event. The present paper studies the statistical and algebraic properties of CBNs. We determine the maximum likelihood parameters and present a combinatorial solution to the model selection problem. Our method performs well on two datasets where the events are HIV mutations associated with drug resistance. Concluding with a study of the algebraic properties of CBNs, we show that CBNs are toric varieties after a coordinate transformation and that their ideals possess a quadratic Gröbner basis.

This paper is concerned with the estimation of the period of an unknown periodic function in Gaussian white noise. A class of estimators of the period is constructed by means of a penalized maximum likelihood method. A second-order asymptotic expansion of the risk of these estimators is obtained. Moreover, the minimax problem for the second-order term is studied and an estimator of the preceding class is shown to be second order efficient.

We investigate the moment estimation for an ergodic diffusion process with unknown trend coefficient. We consider nonparametric and parametric estimation. In each case, we present a lower bound for the risk and then construct an asymptotically efficient estimator of the moment type functional or of a parameter which has a one-to-one correspondence to such a functional. Next, we clarify a higher order property of the moment type estimator by the Edgeworth expansion of the distribution function.

We exploit a result by Nerman which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs, where we obtain asymptotic normality for the number of vertices with a given degree in the random graph G(n, m) with a fixed number of edges from the corresponding result for the random graph G(n, p) with independent edges. We also give some simple applications to random allocations and to spacings. Finally, inspired by these results, but logically independent of them, we investigate whether a one-sided version of the Cramér–Wold theorem holds. We show that such a version holds under a weak supplementary condition, but not without it.

In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this model and also show how our model generalizes many other probabilistic models, including random walk in random environment on trees, recursive distributional equations and multi-type branching random walk on ℝ.

We give optimal convergence rates in the central limit theorem for a large class of martingale difference sequences with bounded third moments. The rates depend on the behaviour of the conditional variances and, for stationary sequences, the rate n^−1/2log n is reached.

In the same spirit as Tsybakov, we define the optimality of an aggregation procedure in the problem of classification. Using an aggregate with exponential weights, we obtain an optimal rate of convex aggregation for the hinge risk under the margin assumption. Moreover, we obtain an optimal rate of model selection aggregation under the margin assumption for the excess Bayes risk.

Let B^H, K={B^H, K(t), t∈ℝ₊} be a bifractional Brownian motion in ℝ^d. We prove that B^H, K is strongly locally non-deterministic. Applying this property and a stochastic integral representation of B^H, K, we establish Chung’s law of the iterated logarithm for B^H, K, as well as sharp Hölder conditions and tail probability estimates for the local times of B^H, K.

We also consider the existence and regularity of the local times of the multiparameter bifractional Brownian motion B^H̅, K̅={B^H̅, K̅(t), t∈ℝ₊^N} in ℝ^d using the Wiener–Itô chaos expansion.

We prove that a positive self-similar Markov process (X, ℙ) that hits 0 in a finite time admits a self-similar recurrent extension that leaves 0 continuously if and only if the underlying Lévy process satisfies Cramér’s condition.

In this paper, we establish the Bahadur–Kiefer representation for sample quantiles for a class of weakly dependent linear processes. The rate of approximation is the same as for i.i.d. sequences and is thus optimal.

We establish bounds for the covariance of a large class of functions of infinite variance stable random variables, including unbounded functions such as the power function and the logarithm. These bounds involve measures of dependence between the stable variables, some of which are new. The bounds are also used to deduce the central limit theorem for unbounded functions of stable moving average time series. This result extends the earlier results of Tailen Hsing and the authors on central limit theorems for bounded functions of stable moving averages. It can be used to show asymptotic normality of wavelet-based estimators of the self-similarity parameter in fractional stable motions.

Given n independent random marked d-vectors (points) X_i distributed with a common density, define the measure ν_n=∑_iξ_i, where ξ_i is a measure (not necessarily a point measure) which stabilizes; this means that ξ_i is determined by the (suitably rescaled) set of points near X_i. For bounded test functions f on R^d, we give weak and strong laws of large numbers for ν_n(f). The general results are applied to demonstrate that an unknown set A in d-space can be consistently estimated, given data on which of the points X_i lie in A, by the corresponding union of Voronoi cells, answering a question raised by Khmaladze and Toronjadze. Further applications are given concerning the Gamma statistic for estimating the variance in nonparametric regression.

The aim of this paper it to establish sufficient conditions for consistency of moving block bootstrap for non-stationary time series with periodic and almost periodic structure. The parameter of the study is the mean value of the expectation function. Consistency holds in quite general situations: if all joint distributions of the series are periodic, then it suffices to assume the central limit theorem and strong mixing property, together with summability of the autocovariance function. In the case where the mean function is almost periodic, we additionally need uniform boundedness of the fourth moments of the root statistics. It is shown that these theoretical results can be applied in statistical inference concerning the Fourier coefficients of periodically (PC) and almost periodically (APC) correlated time series. A simulation example shows how to use a graphical diagnostic test for significant frequencies and stationarity within these classes of time series.

In this paper, the asymptotic distributions of estimators for the regularized functional canonical correlation and variates of the population are derived. The method is based on the possibility of expressing these regularized quantities as the maximum eigenvalue and the corresponding eigenfunctions of an associated pair of regularized operators, similar to the Euclidean case. The known weak convergence of the sample covariance operator, coupled with a delta-method for analytic functions of covariance operators, yields the weak convergence of the pair of associated operators. From the latter weak convergence, the limiting distributions of the canonical quantities of interest can be derived with the help of some further perturbation theory.