Distribution of the norm of a stable random vector of a hilbert space

1987 ◽  
Vol 26 (2) ◽  
pp. 114-120 ◽  
Author(s):  
V. Bentkus ◽  
D. Pap
Keyword(s):  
Hilbert Space ◽  
Random Vector ◽  
Stable Random Vector

scholarly journals U-Statistic for Multivariate Stable Distributions

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Mahdi Teimouri ◽  
Saeid Rezakhah ◽  
Adel Mohammadpour
Keyword(s):  
Asymptotic Normality ◽  
Random Vector ◽  
Spectral Measure ◽  
Stable Distributions ◽  
Tail Index ◽  
Univariate Case ◽  
Stable Random Vector

A U-statistic for the tail index of a multivariate stable random vector is given as an extension of the univariate case introduced by Fan (2006). Asymptotic normality and consistency of the proposed U-statistic for the tail index are proved theoretically. The proposed estimator is used to estimate the spectral measure. The performance of both introduced tail index and spectral measure estimators is compared with the known estimators by comprehensive simulations and real datasets.

scholarly journals An optimal linear filter for estimation of random functions in Hilbert space

2021 ◽  
Vol 62 ◽  
pp. 274-301
Author(s):  
Phil George Howlett ◽  
Anatoli Torokhti
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Random Vector ◽  
Optimal Filter ◽  
Linear Filter ◽  
Closed Linear Operator ◽  
Random Functions ◽  
Best Estimate ◽  
Covariance Operators ◽  
Optimal Linear

Let \(\boldsymbol{f}\) be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space \(H\), and let \(\boldsymbol{g}\) be an associated square-integrable, zero-mean, random vector with realizations which are not observable in a Hilbert space \(K\). We seek an optimal filter in the form of a closed linear operator \(X\) acting on the observable realizations of a proximate vector \(\boldsymbol{f}_{\epsilon} \approx \boldsymbol{f}\) that provides the best estimate \(\widehat{\boldsymbol{g}}_{\epsilon} = X\! \boldsymbol{f}_{\epsilon}\) of the vector \(\boldsymbol{g}\). We assume the required covariance operators are known. The results are illustrated with a typical example.   doi:10.1017/S1446181120000188

scholarly journals Finite-Size Scaling of Typicality-Based Estimates

2020 ◽  
Vol 75 (5) ◽  
pp. 465-473 ◽  
Author(s):  
Jürgen Schnack ◽  
Johannes Richter ◽  
Tjark Heitmann ◽  
Jonas Richter ◽  
Robin Steinigeweg
Keyword(s):  
Hilbert Space ◽  
Standard Deviation ◽  
Random Vector ◽  
Probability Distributions ◽  
Finite Size ◽  
System Size ◽  
Thermodynamic Quantities ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Full Temperature

AbstractAccording to the concept of typicality, an ensemble average can be accurately approximated by an expectation value with respect to a single pure state drawn at random from a high-dimensional Hilbert space. This random-vector approximation, or trace estimator, provides a powerful approach to, e.g. thermodynamic quantities for systems with large Hilbert-space sizes, which usually cannot be treated exactly, analytically or numerically. Here, we discuss the finite-size scaling of the accuracy of such trace estimators from two perspectives. First, we study the full probability distribution of random-vector expectation values and, second, the full temperature dependence of the standard deviation. With the help of numerical examples, we find pronounced Gaussian probability distributions and the expected decrease of the standard deviation with system size, at least above certain system-specific temperatures. Below and in particular for temperatures smaller than the excitation gap, simple rules are not available.

scholarly journals AN OPTIMAL LINEAR FILTER FOR ESTIMATION OF RANDOM FUNCTIONS IN HILBERT SPACE

2021 ◽  
pp. 1-28
Author(s):  
PHIL HOWLETT ◽  
ANATOLI TOROKHTI
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Random Vector ◽  
Optimal Filter ◽  
Linear Filter ◽  
Closed Linear Operator ◽  
Random Functions ◽  
Best Estimate ◽  
Covariance Operators ◽  
Optimal Linear

Abstract Let $\boldsymbol{f}$ be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space H, and let $\boldsymbol{g}$ be an associated square-integrable, zero-mean, random vector with realizations which are not observable in a Hilbert space K. We seek an optimal filter in the form of a closed linear operator X acting on the observable realizations of a proximate vector $\boldsymbol{f}_{\epsilon } \approx \boldsymbol{f}$ that provides the best estimate $\widehat{\boldsymbol{g}}_{\epsilon} = X \boldsymbol{f}_{\epsilon}$ of the vector $\boldsymbol{g}$ . We assume the required covariance operators are known. The results are illustrated with a typical example.

Hilbert Space

1993 ◽  
Author(s):  
J. R. Retherford
Keyword(s):  
Hilbert Space
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