The mean distance for the occupation times of a Gaussian process

One investigates the question of the asymptotic behavior of the quantity E_q(N)= E_fE_qæ _q ² (P_f, P_q), where P is a probability measure in R^N, satisfying a natural normalization condition, the linear functionals f and g are selected independently with respect to the standard Gaussian measure, while æ_q is the distance in L_q between distribution functions. One proves the inequalities E₁(N)cln(N+1), E_q(N)c_q, for q(1,2].Translated from Zapiski Nauchnykh Seminrov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp 98–108, 1985.The author is grateful to V. N. Sudakov for the formulation of the problem and for his interest in the paper.     

Monotonicity of the matrix geometric mean

An attractive candidate for the geometric mean of m positive definite matrices A ₁, . . . , A _m is their Riemannian barycentre G. One of its important operator theoretic properties, monotonicity in the m arguments, has been established recently by Lawson and Lim. We give an elementary proof of this property using standard matrix analysis and some counting arguments. We derive some new inequalities for G. One of these says that, for any unitarily invariant norm, ||| G ||| is not bigger than the geometric mean of |||A ₁|||, . . . , |||A _m |||.     

Monotonicity of the mean order of subtrees

This paper is concerned with the average number of nodes in certain families of subtrees of a tree. It is shown that this average increases when the underlying tree is enlarged and decreases if the family itself is enlarged within the same underlying tree.     

Counterexample to the Conjecture on Monotonicity of a Gaussian Integral

It is shown that, for the Kantorovich metric on probability measures, the integral

Dimension dependent hypercontractivity for Gaussian kernels

We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace estimates. They imply classical bounds on the Ornstein?CUhlenbeck semigroup and a dimensional and refined (transportation) Talagrand inequality when applied to the Hamilton?CJacobi equation. Hypercontractive bounds on the Ornstein?CUhlenbeck semigroup driven by a non-diffusive Lévy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.     

Sharp power mean bounds for the Gaussian hypergeometric function

Sharp inequalities are established between the Gaussian hypergeometric function and the power mean. These results extend known inequalities involving the complete elliptic integral and the hypergeometric mean.     

Change-point in the mean of dependent observations

We prove the consistency of a family of CUSUM-type estimators of the point of change in the mean of dependent observations and derive the rates of convergence. The result is valid under weak assumptions on the dependence structure.     

Characterizing the round sphere by mean distance

We discuss the measure theoretic metric invariants extent, rendezvous number and mean distance of a general compact metric space X and relate these to classical metric invariants such as diameter and radius. In the final section we focus attention to the category of Riemannian manifolds. The main result of this paper is Theorem 4 stating that the round sphere of constant curvature 1 has maximal mean distance among Riemannian n-manifolds with Ricci curvature Ric?n−1, and that such a manifold is diffeomorphic to a sphere if the mean distance is close to .     

Regular simplices and Gaussian samples

We show that if a suitable type of simplex inR ⁿ is randomly rotated and its vertices projected onto a fixed subspace, they are as a point set affine-equivalent to a Gaussian sample in that subspace. Consequently, affine-invariant statistics behave the same for both mechanisms. In particular, the facet behavior for the convex hull is the same, as observed by Affentranger and Schneider; other results of theirs are translated into new results for the convex hulls of Gaussian samples. We show conversely that the conditions on the vertices of the simplex are necessary for this equivalence. Similar results hold for randomorthogonal transformations. Yuliy Baryshnikov was supported in part by the Alexander von Humboldt-Stiftung. Richard Vitale was supported in part by ONR Grant N00014-90-J-1641 and NSF Grant DMS-9002665.     

混合样本下的经验似然估计

研究强平稳φ混合随机变量序列均值的经验似然估计问题,利用拉格朗日乘子以及一些重要概率不等式讨论均值有限且方差不等于零的强平稳φ混合序列,并给出其总体均值和M-泛函统计推断以及置信区间.     

The mean value of the squared path-difference distance for rooted phylogenetic trees

The path-difference metric is one of the oldest distances for the comparison of fully resolved phylogenetic trees, but its statistical properties are still quite unknown. In this paper we compute the mean value of the square of the path-difference metric between two fully resolved rooted phylogenetic trees with n leaves, under the uniform distribution. This complements previous work by Steel and Penny, who computed this mean value for fully resolved unrooted phylogenetic trees.     

On convex hull of Gaussian samples

Let X _i = {X _i (t), t ∈ T} be i.i.d. copies of a centered Gaussian process X = {X(t), t ∈ T} with values in\( {\mathbb{R}^d} \) defined on a separable metric space T. It is supposed that X is bounded. We consider the asymptotic behavior of convex hulls

$ {W_n} = {\text{conv}}\left\{ {{X_1}(t), \ldots, {X_n}(t),\,\,t \in T} \right\} $

and show that, with probability 1,

$ \mathop {{\lim }}\limits_{n \to \infty } \frac{1}{{\sqrt {{2\ln n}} }}{W_n} = W $

(in the sense of Hausdorff distance), where the limit shape W is defined by the covariance structure of X: W = conv{K _t , t ∈ T}, Kt being the concentration ellipsoid of X(t). We also study the asymptotic behavior of the mathematical expectations E f(W _n ), where f is an homogeneous functional.