Studying the shape of phylogenetic trees under different random models is an important issue in evolutionary biology. In this paper, we propose a general framework for deriving detailed statistical results for patterns in phylogenetic trees under the Yule-Harding model and the uniform model, two of the most fundamental random models considered in phylogenetics. Our framework will unify several ...

A general moment bound for sums of products of Gaussian vector’s functions extending the moment bound in Taqqu (1977, Lemma 4.5) is established. A general central limit theorem for triangular arrays of nonlinear functionals of multidimensional non-stationary Gaussian sequences is proved. This theorem extends the previous results of Breuer and Major (1981), Arcones (1994) and others. A Berry-Ess...

∞ i=1 aiεn−i, where the εi are i.i.d. with mean 0 and at least finite second moment, and the ai are assumed to satisfy |ai| = O(i) with β > 1/2. When 1/2 < β < 1, Xn is usually called a long-range dependent or long-memory process. For a certain class of Borel functions K(x1, . . . , xd+1), d≥ 0, from R to R, which includes indicator functions and polynomials, the stationary sequence K(Xn,Xn+1, ...

In this paper, we study the inner and outer boundary densities of some sets with self-similar boundary having Minkowski dimension s > d−1 in R. These quantities turn out to be crucial in some problems of set estimation, as we show here for the Voronoi approximation of the set with a random input constituted by n iid points in some larger bounded domain. We prove that some classes of such sets h...

We consider the problem of providing nonparametric confidence guarantees for undirected graphs under weak assumptions. In particular, we do not assume sparsity, incoherence or Normality. We allow the dimension D to increase with the sample size n. First, we prove lower bounds that show that if we want accurate inferences with low assumptions then there are limitations on the dimension as a func...

Abstract Let $\mu $ be a probability measure on $\mathrm {GL}_d(\mathbb {R})$ , and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$ are i.i.d. with law . Under assumptions that has finite exponential moment generates proximal strongly irreducible semigroup, we prove Berry–Esseen bound optimal rate $O(1/\sqrt n)$ for coefficients of $S_n$ settling long-standing...

Many statistics used to test for association between pairs (yi, zi) of multivariate observations, sampled from n individuals in a population, are based on comparing the similarity aij of each pair (i, j) of individuals, as evidenced by the values yi and yj , with their similarity bij based on the values zi and zj . A common strategy is to compute the sample correlation between these two sets of...