The abscissa of convergence of the Laplace transform

1992 ◽  
Vol 29 (2) ◽  
pp. 353-362 ◽  
Author(s):  
Peter Hall ◽  
Jozef L. Teugels ◽  
Ann Vanmarcke
Keyword(s):  
Weak Convergence ◽  
Convergence Rate ◽  
Laplace Transform ◽  
Sample Size ◽  
Convergence Result ◽  
Optimal Convergence Rate ◽  
Optimal Convergence ◽  
Abscissa Of Convergence ◽  
The Laplace Transform ◽  
The Mean

Assume that we want to estimate – σ, the abscissa of convergence of the Laplace transform. We show that no non-parametric estimator of σ can converge at a faster rate than (log n)–1, where n is the sample size. An optimal convergence rate is achieved by an estimator of the form where xn = O(log n) and is the mean of the sample values overshooting xn. Under further parametric restrictions this (log n)–1 phenomenon is also illustrated by a weak convergence result.

The abscissa of convergence of the Laplace transform

1992 ◽  
Vol 29 (02) ◽  
pp. 353-362 ◽  
Author(s):  
Peter Hall ◽  
Jozef L. Teugels ◽  
Ann Vanmarcke
Keyword(s):  
Weak Convergence ◽  
Convergence Rate ◽  
Laplace Transform ◽  
Sample Size ◽  
Convergence Result ◽  
Optimal Convergence Rate ◽  
Optimal Convergence ◽  
Abscissa Of Convergence ◽  
The Laplace Transform ◽  
The Mean

Assume that we want to estimate – σ, the abscissa of convergence of the Laplace transform. We show that no non-parametric estimator of σ can converge at a faster rate than (log n)–1, where n is the sample size. An optimal convergence rate is achieved by an estimator of the form where xn = O(log n) and is the mean of the sample values overshooting xn. Under further parametric restrictions this (log n)–1 phenomenon is also illustrated by a weak convergence result.

The solution of theoretical seismic problems by best rational function approximations for Laplace transform inversion

1975 ◽  
Vol 65 (4) ◽  
pp. 927-935
Author(s):  
I. M. Longman ◽  
T. Beer
Keyword(s):  
Laplace Transform ◽  
Rational Function ◽  
High Accuracy ◽  
Time Variable ◽  
Inversion Method ◽  
The Laplace Transform ◽  
Previously Treated ◽  
The Mean ◽  
Function Approximations ◽  
Laplace Transform Inversion

Abstract In a recent paper, the first author has developed a method of computation of “best” rational function approximations ḡn(p) to a given function f̄(p) of the Laplace transform operator p. These approximations are best in the sense that analytic inversion of ḡn(p) gives a function gn(t) of the time variable t, which approximates the (generally unknown) inverse f(t) of f̄(p in a minimum least-squares manner. Only f̄(p) but not f(t) is required to be known in order to carry out this process. n is the “order” of the approximation, and it can be shown that as n tends to infinity gn(t) tends to f(t) in the mean. Under suitable conditions on f(t) the convergence is extremely rapid, and quite low values of n (four or five, say) are sufficient to give high accuracy for all t ≧ 0. For seismological applications, we use geometrical optics to subtract out of f(t) its discontinuities, and bring it to a form in which the above inversion method is very rapidly convergent. This modification is of course carried out (suitably transformed) on f̄(p), and the discontinuities are restored to f(t) after the inversion. An application is given to an example previously treated by the first author by a different method, and it is a certain vindication of the present method that an error in the previously given solution is brought to light. The paper also presents a new analytical method for handling the Bessel function integrals that occur in theoretical seismic problems related to layered media.

CERTAIN ISOMETRIES RELATED TO THE BILATERAL LAPLACE TRANSFORM

2006 ◽  
Vol 11 (3) ◽  
pp. 331-346 ◽  
Author(s):  
S. B. Yakubovich
Keyword(s):  
Hilbert Space ◽  
Entire Function ◽  
Laplace Transform ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Inversion Formulas ◽  
Mean Convergence ◽  
The Laplace Transform ◽  
The Mean

We study certain isometries between Hilbert spaces, which are generated by the bilateral Laplace transform In particular, we construct an analog of the Bargmann‐Fock type reproducing kernel Hilbert space related to this transformation. It is shown that under some integra‐bility conditions on $ the Laplace transform FF(z), z = x + iy is an entire function belonging to this space. The corresponding isometrical identities, representations of norms, analogs of the Paley‐Wiener and Plancherel's theorems are established. As an application this approach drives us to a different type of real inversion formulas for the bilateral Laplace transform in the mean convergence sense.

Reachability of Optimal Convergence Rate Estimates for High-Order Numerical Convex Optimization Methods

2019 ◽  
Vol 99 (1) ◽  
pp. 91-94
Author(s):  
A. V. Gasnikov ◽  
E. A. Gorbunov ◽  
D. A. Kovalev ◽  
A. A. M. Mokhammed ◽  
E. O. Chernousova
Keyword(s):  
Convex Optimization ◽  
Convergence Rate ◽  
Optimization Methods ◽  
High Order ◽  
Optimal Convergence Rate ◽  
Optimal Convergence

Trimmed estimators for large dimensional sparse covariance matrices

2018 ◽  
Vol 08 (01) ◽  
pp. 1950003
Author(s):  
Guangren Yang ◽  
Xia Cui
Keyword(s):  
Convergence Rate ◽  
Covariance Matrix ◽  
Covariance Matrices ◽  
Optimal Convergence Rate ◽  
Sample Covariance Matrices ◽  
Optimal Convergence ◽  
Sample Covariance ◽  
Starting Point ◽  
Pairwise Product

In this paper, we will propose two new estimators for sparse covariance matrix. Our starting point is to make the estimator of each element of covariance matrix more robust. More precisely, we will trim the observations for each pairwise product of components of population as a first step. Then we form the sample covariance matrices based on the trimmed data. Finally, we apply the thresholding to the derived sample covariance matrices. These two new estimators will be shown to achieve the optimal convergence rate.

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