Stability of quadratic forms in independent random variables in Banach spaces

1981 ◽  
pp. 314-323 ◽  
Author(s):  
J. Szulga ◽  
W. A. Woyczynski
Keyword(s):  
Banach Spaces ◽  
Quadratic Forms ◽  
Random Variables ◽  
Independent Random Variables

scholarly journals Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables

1979 ◽  
Vol 2 (2) ◽  
pp. 309-323
Author(s):  
W. J. Padgett ◽  
R. L. Taylor
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Real Numbers ◽  
Martingale Theory ◽  
Random Elements ◽  
Schauder Bases

Let{Xk}be independent random variables withEXk=0for allkand let{ank:n≥1, k≥1}be an array of real numbers. In this paper the almost sure convergence ofSn=∑k=1nankXk,n=1,2,…, to a constant is studied under various conditions on the weights{ank}and on the random variables{Xk}using martingale theory. In addition, the results are extended to weighted sums of random elements in Banach spaces which have Schauder bases. This extension provides a convergence theorem that applies to stochastic processes which may be considered as random elements in function spaces.

scholarly journals Convergence to Zero of Quadratic Forms in Independent Random Variables

1973 ◽  
Vol 1 (5) ◽  
pp. 838-848 ◽  
Author(s):  
Gary N. Griffiths ◽  
Ronald D. Platt ◽  
F. T. Wright
Keyword(s):  
Quadratic Forms ◽  
Random Variables ◽  
Independent Random Variables

RUC-systems and Besselian systems in Banach spaces

1989 ◽  
Vol 106 (1) ◽  
pp. 163-168 ◽  
Author(s):  
D. J. H. Garling ◽  
N. Tomczak-Jaegermann
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Span ◽  
Random Variables ◽  
Biorthogonal System ◽  
Independent Random Variables ◽  
Closed Linear Span ◽  
Image Position

Let (rj) be a Rademacher sequence of random variables – that is, a sequence of independent random variables, with , for each j. A biorthogonal system in a Banach space X is called an RUC-system[l] if for every x in [ej] (the closed linear span of the vectors ej), the seriesconverges for almost every ω. A basis which, together with its coefficient functionals, forms an RUC-system is called an RUC-basis. A biorthogonal system is an RLTC-svstem if and only if there exists 1 ≤ K < ∞ such thatfor each x in [ej]: the RUC-constant of the system is the smallest constant K satisfying (1) (see [1], proposition 1.1).

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