On Convergence of Series of Random Elements via Maximal Moment Relations with Applications to Martingale Convergence and to Convergence of Series with p-Orthogonal Summands

2001 ◽  
Vol 8 (2) ◽  
pp. 377-388
Author(s):  
Andrew Rosalsky ◽  
Andrei I. Volodin
Keyword(s):  
Banach Space ◽  
Convergent Series ◽  
Difference Sequence ◽  
Martingale Difference ◽  
Nonlinear Anal ◽  
Convergence Of Series ◽  
Special Cases ◽  
Random Elements ◽  
Maximal Moment ◽  
Martingale Convergence

Abstract The rate of convergence for an almost surely convergent series of Banach space valued random elements is studied in this paper. As special cases of the main result, known results are obtained for a sequence of independent random elements in a Rademacher type p Banach space, and new results are obtained for a martingale difference sequence of random elements in a martingale type p Banach space and for a p-orthogonal sequence of random elements in a Rademacher type p Banach space. The current work generalizes, simplifies, and unifies some of the recent results of Nam and Rosalsky [Teor. Īmovīr. ta Mat. Statist. 52: 120–131, 1995] and Rosalsky and Rosenblatt [Bull. Inst. Math. Acad. Sinica 11: 185–208, 1983, Nonlinear Anal. 30: 4237–4248, 1997].

On Convergence of Series of Random Elements via Maximal Moment Relations with Applications to Martingale Convergence and to Convergence of Series with 𝑝-Orthogonal Summands. Correction

2003 ◽  
Vol 10 (4) ◽  
pp. 799-802
Author(s):  
Andrew Rosalsky ◽  
Andrei I. Volodin
Keyword(s):  
Acta Math ◽  
Convergence Of Series ◽  
Corrected Version ◽  
Random Elements ◽  
Maximal Moment ◽  
Martingale Convergence

Abstract A result by Móricz, Su, and Taylor from Acta Math. Hungar. 65(1994), 1–16, was misstated in the authors' paper in Georgian Math. J. 8(2001), No. 2, 377–388, where due to this misstatement the invalid formulation and proof of a corollary is given. In this correction note, the needed result is correctly stated and a corrected version of the invalid corollary is proved.

On the rate of convergence of series of banach space valued random elements

1997 ◽  
Vol 30 (7) ◽  
pp. 4237-4248 ◽  
Author(s):  
Andrew Rosalsky ◽  
Joseph Rosenblatt
Keyword(s):  
Banach Space ◽  
Rate Of Convergence ◽  
Convergence Of Series ◽  
Random Elements

Some mean convergence theorems for weighted sums of Banach space valued random elements

Stochastics ◽  
2021 ◽  
pp. 1-19
Author(s):  
Pingyan Chen ◽  
Manuel Ordóñez Cabrera ◽  
Andrew Rosalsky ◽  
Andrei Volodin
Keyword(s):  
Banach Space ◽  
Weighted Sums ◽  
Convergence Theorems ◽  
Mean Convergence ◽  
Random Elements

On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements

2002 ◽  
Vol 47 (3) ◽  
pp. 533-547 ◽  
Author(s):  
Tien-Chung Hu ◽  
Tien-Chung Hu ◽  
Deli Li ◽  
Deli Li ◽  
Andrew Rosalsky ◽  
...  
Keyword(s):  
Banach Space ◽  
Complete Convergence ◽  
Weighted Sums ◽  
Random Elements

scholarly journals On Domain of Nörlund Sequences

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 268 ◽  
Author(s):  
Kuddusi Kayaduman ◽  
Fevzi Yaşar
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Sequence Space ◽  
Convergent Series ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Nörlund Matrix ◽  
Inclusion Relations ◽  
The Matrix ◽  
New Space

In 1978, the domain of the Nörlund matrix on the classical sequence spaces lp and l∞ was introduced by Wang, where 1 ≤ p < ∞. Tuğ and Başar studied the matrix domain of Nörlund mean on the sequence spaces f0 and f in 2016. Additionally, Tuğ defined and investigated a new sequence space as the domain of the Nörlund matrix on the space of bounded variation sequences in 2017. In this article, we defined new space and and examined the domain of the Nörlund mean on the bs and cs, which are bounded and convergent series, respectively. We also examined their inclusion relations. We defined the norms over them and investigated whether these new spaces provide conditions of Banach space. Finally, we determined their α­, β­, γ­duals, and characterized their matrix transformations on this space and into this space.

Limit theorem for the maximum of dependent Gaussian random elements in a Banach space

1997 ◽  
Vol 49 (7) ◽  
pp. 1129-1133 ◽  
Author(s):  
V. A. Koval’ ◽  
R. Schwabe
Keyword(s):  
Banach Space ◽  
Limit Theorem ◽  
Random Elements

On the Rate of Complete Convergence for Weighted Sums of Arrays of Banach Space Valued Random Elements

2003 ◽  
Vol 47 (3) ◽  
pp. 455-468 ◽  
Author(s):  
T. C. Hu ◽  
D. Li ◽  
A. Rosalsky ◽  
A. I. Volodin
Keyword(s):  
Banach Space ◽  
Complete Convergence ◽  
Weighted Sums ◽  
Random Elements

scholarly journals The space $D$ in several variables: random variables and higher moments

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Random Variables ◽  
Higher Moments ◽  
Multilinear Form ◽  
Standard Case ◽  
Random Functions ◽  
Several Variables ◽  
Random Elements ◽  
Point Evaluations

We study the Banach space $D([0,1]^m)$ of functions of several variables that are (in a certain sense) right-continuous with left limits, and extend several results previously known for the standard case $m=1$. We give, for example, a description of the dual space, and we show that a bounded multilinear form always is measurable with respect to the $\sigma$-field generated by the point evaluations. These results are used to study random functions in the space. (I.e., random elements of the space.) In particular, we give results on existence of moments (in different senses) of such random functions, and we give an application to the Zolotarev distance between two such random functions.

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