A Sufficient Condition for Function Space Controllability of a Linear Neutral System

1978 ◽  
Vol 16 (3) ◽  
pp. 429-435 ◽  
Author(s):  
Hernan Rivera Rodas ◽  
C. E. Langenhop
Keyword(s):  
Function Space ◽  
Neutral System ◽  
Sufficient Condition ◽  
Function Space Controllability

Point processes, regular variation and weak convergence

1986 ◽  
Vol 18 (01) ◽  
pp. 66-138 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Weak Convergence ◽  
Function Space ◽  
Regular Variation ◽  
Point Processes ◽  
Sufficient Condition ◽  
Space Setting

A method is reviewed for proving weak convergence in a function-space setting when regular variation is a sufficient condition. Point processes and weak convergence techniques involving continuity arguments play a central role. The method is dimensionless and holds computations to a minimum. Many applications of the methods to processes derived from sums and maxima are given.

scholarly journals A NECESSARY AND SUFFICIENT CONDITION FOR CERTAIN MARTINGALE INEQUALITIES IN BANACH FUNCTION SPACES

2007 ◽  
Vol 49 (3) ◽  
pp. 431-447 ◽  
Author(s):  
MASATO KIKUCHI
Keyword(s):  
Function Space ◽  
Probability Space ◽  
Banach Function Space ◽  
Sufficient Conditions ◽  
Function Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Function Spaces ◽  
Martingale Inequalities ◽  
Necessary And Sufficient

AbstractLet X be a Banach function space over a nonatomic probability space. We investigate certain martingale inequalities in X that generalize those studied by A. M. Garsia. We give necessary and sufficient conditions on X for the inequalities to be valid.

scholarly journals On certain martingale inequalities for maximal functions and mean oscillations

2011 ◽  
Vol 109 (2) ◽  
pp. 309 ◽  
Author(s):  
Masato Kikuchi ◽  
Yasuhiro Kinoshita
Keyword(s):  
Function Space ◽  
Probability Space ◽  
Banach Function Space ◽  
Maximal Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Martingale Inequalities ◽  
Integrable Martingale ◽  
Necessary And Sufficient

Let $X$ be a Banach function space over a nonatomic probability space. For a uniformly integrable martingale $f=(f_n)$ with respect to a filtration ${\mathcal F}=({\mathcal F}_n)$, let $Mf =\sup_n |f_n|$ and $\theta_{\mathcal F}f=\sup_n E[|f_{\infty}- f_{n-1}| \mid{\mathcal F}_n]$. We give a necessary and sufficient condition on $X$ for the inequality $\parallel \theta_{\mathcal F}f \parallel_X \leq C\parallel Mf\parallel_X$ to hold.

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