Uniform Stochastic Convergence

2021 ◽  
pp. 471-492
Author(s):  
James Davidson
Keyword(s):  
Uniform Convergence ◽  
Metric Spaces ◽  
Main Issue ◽  
Stochastic Convergence ◽  
Convergence Conditions ◽  
Laws Of Large Numbers ◽  
Random Sequences ◽  
Large Numbers ◽  
Stochastic Equicontinuity

This chapter concerns random sequences of functions on metric spaces. The main issue is the distinction between convergence at all points of the space (pointwise) and uniform convergence, where limit points are also taken into account. The role of the stochastic equicontinuity property is highlighted. Generic uniform convergence conditions are given and linked to the question of uniform laws of large numbers.

Convergence of elements in random normed spaces

1975 ◽  
Vol 12 (1) ◽  
pp. 31-47 ◽  
Author(s):  
Robert Lee Taylor
Keyword(s):  
Normed Linear Space ◽  
Metric Spaces ◽  
Normed Spaces ◽  
Linear Topology ◽  
Laws Of Large Numbers ◽  
Convergence In Measure ◽  
Large Numbers ◽  
Random Elements ◽  
Random Normed Spaces ◽  
Necessary And Sufficient

For a random normed space of mappings into a separable normed linear space, convergence of identically distributed elements in i the random norm (norm distribution) is shown to be equivalent to convergence in measure in the weak linear topology. Convergence in measure in each coordinate of a Schauder basis is also shown to be a necessary and sufficient condition for convergence in the random norm topology. These results have laws of large numbers for random elements in separable normed linear spaces as almost immediate corollaries and illustrate some of the recently obtained laws of large numbers for random elements. Similar results are also given for elements which need not have the same norm distributions, and the results are extended to linear metric spaces. Finally, applications of the results to stochastic processes are considered.

Stochastic Convergence

2021 ◽  
pp. 400-417
Author(s):  
James Davidson
Keyword(s):  
Almost Sure Convergence ◽  
Convergence In Probability ◽  
Sufficient Condition ◽  
Stochastic Convergence ◽  
Cantelli Lemma ◽  
Laws Of Large Numbers ◽  
Large Numbers

The modes of convergence introduced in Chapter 12 are studied in detail. Conditions for almost‐sure convergence are derived via the Borel–Cantelli lemma. Convergence in probability is contrasted, and then a number of results for convergence of transformed series are given. Convergence in LP‐norm is introduced as a sufficient condition for convergence in probability. Examples are given, and the chapter concludes with a preliminary look at the laws of large numbers.

Localization of Intracisternal a Particle Antigens in Neoplastic Cells and Preimplantation Mouse Embryos

1980 ◽  
Vol 38 ◽  
pp. 696-697
Author(s):  
Thomas T.F. Huang ◽  
Patricia G. Calarco
Keyword(s):  
Endoplasmic Reticulum ◽  
Normal Development ◽  
Mouse Embryos ◽  
Neoplastic Cells ◽  
Large Numbers ◽  
Preimplantation Mouse Embryos ◽  
Preimplantation Mouse ◽  
Intracisternal A Particle ◽  
Normal Feature

The stage specific appearance of a retravirus, termed the Intracisternal A particle (IAP) is a normal feature of early preimplantation development. To date, all feral and laboratory strains of Mus musculus and even Asian species such as Mus cervicolor and Mus pahari express the particles during the 2-8 cell stages. IAP form by budding into the endoplasmic reticulum and appear singly or as groups of donut-shaped particles within the cisternae (fig. 1). IAP are also produced in large numbers in several neoplastic cells such as certain plasmacytomas and rhabdomyosarcomas. The role of IAP, either in normal development or in neoplastic behavior, is unknown.

scholarly journals Characterizing Highly Benefited Patients in Randomized Clinical Trials

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
Vivek Charu ◽  
Paul B. Rosenberg ◽  
Lon S. Schneider ◽  
Lea T. Drye ◽  
Lisa Rein ◽  
...  
Keyword(s):  
Randomized Clinical Trials ◽  
Controlled Trial ◽  
New Method ◽  
Usual Model ◽  
Randomized Controlled ◽  
Working Model ◽  
Large Numbers ◽  
High Benefit ◽  
Two Stages

AbstractPhysicians and patients may choose a certain treatment only if it is predicted to have a large effect for the profile of that patient. We consider randomized controlled trials in which the clinical goal is to identify as many patients as possible that can highly benefit from the treatment. This is challenging with large numbers of covariate profiles, first, because the theoretical, exact method is not feasible, and, second, because usual model-based methods typically give incorrect results. Better, more recent methods use a two-stage approach, where a first stage estimates a working model to produce a scalar predictor of the treatment effect for each covariate profile; and a second stage estimates empirically a high-benefit group based on the first-stage predictor. The problem with these methods is that each of the two stages is usually agnostic about the role of the other one in addressing the clinical goal. We propose a method that characterizes highly benefited patients by linking model estimation directly to the particular clinical goal. It is shown that the new method has the following two key properties in comparison with existing approaches: first, the meaning of the solution with regard to the clinical goal is the same, and second, the value of the solution is the best that can be achieved when using the working model as a predictor, even if that model is incorrect. In the Citalopram for Agitation in Alzheimer’s Disease (CitAD) randomized controlled trial, the new method identifies substantially larger groups of highly benefited patients, many of whom are missed by the standard method.

On the volume of parallel bodies: a probabilistic derivation of the Steiner formula

1995 ◽  
Vol 27 (1) ◽  
pp. 97-101 ◽  
Author(s):  
Richard A. Vitale
Keyword(s):  
Convex Bodies ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Steiner Formula ◽  
Random Convex

We give a proof of the Steiner formula based on the theory of random convex bodies. In particular, we make use of laws of large numbers for both random volumes and random convex bodies themselves.

Beta-Blocker Attenuates Cardiotoxicity Related Anthracycline Usage

2021 ◽  
Vol 104 (8) ◽  
pp. 1389-1392
Keyword(s):  
Side Effects ◽  
Cancer Therapy ◽  
Mortality Rate ◽  
Cancer Patients ◽  
Beta Blocker ◽  
The Other ◽  
Large Numbers ◽  
Therapy Effectiveness ◽  
Chemotherapy Agents

To summarize the recent trials and studies of the role of beta-blocker on the treatment for cancer patients treated with anthracycline to decrease morbidity and mortality rate. Good management of cancer will result in large numbers of cancer survivors. On the other hand, cancer therapy also has side effects, one of which is cardiotoxicity. Cardiotoxicity could reduce therapy effectiveness, hence, increase disease progression and mortality rate. Anthracyclines is one of the chemotherapy agents with cardiotoxicity as a side effect. Beta-blocker has the ability to reduce cardiotoxicity due to anthracyclines usage. Keywords: Beta-blocker; Cardiotoxicity; Anthracyclines

BUSINESS SOCIAL RESPONSIBILITY: THEORETICAL OVERVIEW

2020 ◽  
Vol 73 (1) ◽  
pp. 196-203
Author(s):  
T.B. Tauyekelova ◽  
◽  
G.O. Abdikerova ◽  
Keyword(s):  
Corporate Social Responsibility ◽  
Social Responsibility ◽  
Main Issue ◽  
Scientific Theories ◽  
Factors Influencing ◽  
The Social ◽  
Corporate Social ◽  
Theoretical Overview

The main issue discussed in the article is the social responsibility of business. The article provides various definitions of the category of social responsibility. The concept of "social responsibility of business" is a multilevel and complex category. Responsibility includes ethical categories such as morality, duty and charity.The article examines the theoretical aspects of scientific approaches to corporate social responsibility, analyzes classical and modern scientific theories and concepts. The factors influencing the growing importance of corporate social responsibility in society, issues related to the role of business in the formation of a voluntary society are considered.

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