scholarly journals A sharp lower-tail bound for Gaussian maxima with application to bootstrap methods in high dimensions

2022 ◽  
Vol 16 (1) ◽  
Author(s):  
Miles E. Lopes ◽  
Junwen Yao
Keyword(s):  
High Dimensions ◽  
Lower Tail ◽  
Bootstrap Methods ◽  
Tail Bound

scholarly journals On Tail Bounds for Random Recursive Trees

2012 ◽  
Vol 49 (2) ◽  
pp. 566-581 ◽  
Author(s):  
Götz Olaf Munsonius
Keyword(s):  
Analysis Of Algorithms ◽  
Wiener Index ◽  
Binary Search ◽  
Random Trees ◽  
Stochastic Domination ◽  
Lower Tail ◽  
Probabilistic Analysis Of Algorithms ◽  
The Laplace Transform ◽  
Recursive Trees ◽  
Tail Bound

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of random b-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

scholarly journals On Tail Bounds for Random Recursive Trees

2012 ◽  
Vol 49 (02) ◽  
pp. 566-581 ◽  
Author(s):  
Götz Olaf Munsonius
Keyword(s):  
Analysis Of Algorithms ◽  
Wiener Index ◽  
Binary Search ◽  
Random Trees ◽  
Stochastic Domination ◽  
Lower Tail ◽  
Probabilistic Analysis Of Algorithms ◽  
The Laplace Transform ◽  
Recursive Trees ◽  
Tail Bound

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of randomb-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

Preface to the special issue, CMAM 2011, no. 3.

2011 ◽  
Vol 11 (3) ◽  
pp. 272
Author(s):  
Ivan Gavrilyuk ◽  
Boris Khoromskij ◽  
Eugene Tyrtyshnikov
Keyword(s):  
Separation Of Variables ◽  
Numerical Algorithms ◽  
Multilevel Methods ◽  
High Dimensional ◽  
Special Issue ◽  
High Dimensions ◽  
Guiding Principle ◽  
Tensor Methods ◽  
Closed World ◽  
The One

Abstract In the recent years, multidimensional numerical simulations with tensor-structured data formats have been recognized as the basic concept for breaking the "curse of dimensionality". Modern applications of tensor methods include the challenging high-dimensional problems of material sciences, bio-science, stochastic modeling, signal processing, machine learning, and data mining, financial mathematics, etc. The guiding principle of the tensor methods is an approximation of multivariate functions and operators with some separation of variables to keep the computational process in a low parametric tensor-structured manifold. Tensors structures had been wildly used as models of data and discussed in the contexts of differential geometry, mechanics, algebraic geometry, data analysis etc. before tensor methods recently have penetrated into numerical computations. On the one hand, the existing tensor representation formats remained to be of a limited use in many high-dimensional problems because of lack of sufficiently reliable and fast software. On the other hand, for moderate dimensional problems (e.g. in "ab-initio" quantum chemistry) as well as for selected model problems of very high dimensions, the application of traditional canonical and Tucker formats in combination with the ideas of multilevel methods has led to the new efficient algorithms. The recent progress in tensor numerical methods is achieved with new representation formats now known as "tensor-train representations" and "hierarchical Tucker representations". Note that the formats themselves could have been picked up earlier in the literature on the modeling of quantum systems. Until 2009 they lived in a closed world of those quantum theory publications and never trespassed the territory of numerical analysis. The tremendous progress during the very recent years shows the new tensor tools in various applications and in the development of these tools and study of their approximation and algebraic properties. This special issue treats tensors as a base for efficient numerical algorithms in various modern applications and with special emphases on the new representation formats.

Sign in / Sign up

Export Citation Format

Share Document