Asymptotic normality of sums of random elements with values in a real separable Hilbert space

1969 ◽  
Vol 13 (3-4) ◽  
pp. 221-255 ◽  
Author(s):  
Bengt Ros�n
Keyword(s):  
Hilbert Space ◽  
Asymptotic Normality ◽  
Separable Hilbert Space ◽  
Random Elements ◽  
Real Separable Hilbert Space

Asymptotic normality of sums of Hilbert space valued random elements

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Alfredas Račkauskas
Keyword(s):  
Hilbert Space ◽  
Asymptotic Normality ◽  
Separable Hilbert Space ◽  
Linear Process ◽  
Weighted Sums ◽  
Bounded Operators ◽  
Linear Processes ◽  
Summation Methods ◽  
Random Elements

Abstract We investigate the asymptotic normality of distributions of the sequence {\sum_{k\in\mathbb{Z}}u_{n,k}X_{k}} , {n\in\mathbb{N}} , where {(X_{k},k\in\mathbb{Z})} either is a sequence of i.i.d. random elements or constitutes a linear process with i.i.d. innovations in a separable Hilbert space. The weights {(u_{n,k})} are in general a family of linear bounded operators. This model includes operator weighted sums of Hilbert space valued linear processes, operator-wise discounted sums in a Hilbert space as well some extensions of classical summation methods.

scholarly journals Operator semistable probability measures on a Hilbert space

1980 ◽  
Vol 22 (3) ◽  
pp. 397-406 ◽  
Author(s):  
R.G. Laha ◽  
V.K. Rohatgi
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Probability Measures ◽  
Real Separable Hilbert Space ◽  
Semistable Probability

A characterization of the class of operator semistable probability measures on a real separable Hilbert space is given.

Time inhomogeneous generalized Mehler semigroups and skew convolution equations

2016 ◽  
Vol 28 (2) ◽  
Author(s):  
Shun-Xiang Ouyang ◽  
Michael Röckner
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Real Separable Hilbert Space ◽  
Convolution Equations

AbstractA time inhomogeneous generalized Mehler semigroup on a real separable Hilbert space ℍ is defined through

CLT for Operator Abel Sums of Random Elements

2008 ◽  
Vol 15 (4) ◽  
pp. 785-792
Author(s):  
Tengiz Shervashidze ◽  
Vaja Tarieladze
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Random Element ◽  
Second Order ◽  
Identity Operator ◽  
Random Elements ◽  
Continuous Operators ◽  
Independent Identically Distributed

Abstract Let (ξ 𝑘)𝑘 ≥ 1 be a sequence of independent, identically distributed second order mean zero random elements in a separable Hilbert space 𝐻 and 𝐴 be an element of a certain class of linear continuous operators 𝐻 → 𝐻 such that ‖𝐴‖ < 1. Denote . We prove that if ‖𝐼 – 𝐴‖ tends to zero, where 𝐼 is the identity operator, then the normalized sum (𝐼 – 𝐴2)1/2 η 𝐴 converges in distribution to a Gaussian random element.

scholarly journals Second-order neutral stochastic evolution equations with heredity

2004 ◽  
Vol 2004 (2) ◽  
pp. 177-192 ◽  
Author(s):  
Mark A. McKibben
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Continuous Dependence ◽  
Evolution Equations ◽  
Separable Hilbert Space ◽  
Second Order ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Integro Differential Equation ◽  
Real Separable Hilbert Space

Existence, continuous dependence, and approximation results are established for a class of abstract second-order neutral stochastic evolution equations with heredity in a real separable Hilbert space. A related integro-differential equation is also mentioned, as well as an example illustrating the theory.

scholarly journals On Abstract Wiener Measure

1972 ◽  
Vol 46 ◽  
pp. 155-160 ◽  
Author(s):  
Balram S. Rajput
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Gaussian Measure ◽  
Separable Hilbert Space ◽  
Reflexive Banach Space ◽  
Wiener Measure ◽  
Real Separable Hilbert Space ◽  
Cylinder Measure

In a recent paper, Sato [6] has shown that for every Gaussian measure n on a real separable or reflexive Banach space (X, ‖ • ‖) there exists a separable closed sub-space X〵 of X such that and is the σ-extension of the canonical Gaussian cylinder measure of a real separable Hilbert space such that the norm is contiunous on and is dense in The main purpose of this note is to prove that ‖ • ‖ x〵 is measurable (and not merely continuous) on .

scholarly journals Eigencurves for linear elliptic equations

2019 ◽  
Vol 25 ◽  
pp. 45
Author(s):  
Mauricio A. Rivas ◽  
Stephen B. Robinson
Keyword(s):  
Hilbert Space ◽  
Elliptic Equations ◽  
Separable Hilbert Space ◽  
Elliptic Boundary ◽  
Bilinear Forms ◽  
Asymptotic Results ◽  
Elliptic Boundary Value ◽  
Real Separable Hilbert Space ◽  
Linear Elliptic Boundary ◽  
Two Parameter

This paper provides results forvariational eigencurvesassociated with self-adjoint linear elliptic boundary value problems. The elliptic problems are treated as a general two-parameter eigenproblem for a triple (a,b,m) of continuous symmetric bilinear forms on a real separable Hilbert spaceV.Geometric characterizationsof eigencurves associated with (a,b,m) are obtained and are based on their variational characterizations described here. Continuity, differentiability, as well as asymptotic, results for these eigencurves are proved. Finally, two-parameter Robin–Steklov eigenproblems are treated to illustrate the theory.

Convergence rates of the LIL for random fields in Hilbert spaces

2011 ◽  
Vol 61 (2) ◽  
Author(s):  
Ke-Ang Fu ◽  
Xiao-Rong Yang
Keyword(s):  
Hilbert Space ◽  
Gamma Function ◽  
Hilbert Spaces ◽  
Random Fields ◽  
Convergence Rates ◽  
Separable Hilbert Space ◽  
Partial Ordering ◽  
Covariance Operator ◽  
Dimensional Lattice ◽  
Real Separable Hilbert Space

AbstractConsidering the positive d-dimensional lattice point Z +d (d ≥ 2) with partial ordering ≤, let {X k: k ∈ Z +d} be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $$ S_n = \sum\limits_{k \leqslant n} {X_k } $$, n ∈ Z +d. Let σ i2, i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ 2. Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $$ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) $$. We show that when l ≥ 2 and b > −l/2, E[‖X‖2(log ‖X‖)d−2(log log ‖X‖)b+4] < ∞ implies $$ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} $$, where Γ(·) is the Gamma function and $$ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } $$.

scholarly journals On a Class of Measure-Dependent Stochastic Evolution Equations Driven by fBm

2007 ◽  
Vol 2007 ◽  
pp. 1-26 ◽  
Author(s):  
Eduardo Hernandez ◽  
David N. Keck ◽  
Mark A. McKibben
Keyword(s):  
Hilbert Space ◽  
Brownian Motion ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Evolution Equations ◽  
Separable Hilbert Space ◽  
Probability Measures ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Real Separable Hilbert Space

We investigate a class of abstract stochastic evolution equations driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space. We establish the existence and uniqueness of a mild solution, a continuous dependence estimate, and various convergence and approximation results. Finally, the analysis of three examples is provided to illustrate the applicability of the general theory.

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