scholarly journals An Edgeworth Expansion for the Ratio of Two Functionals of Gaussian Fields and Optimal Berry–Esseen Bounds

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2223
Author(s):  
Yoon-Tae Kim ◽  
Hyun-Suk Park
Keyword(s):  
Differential Equations ◽  
Rate Of Convergence ◽  
Edgeworth Expansion ◽  
Normal Approximation ◽  
Gaussian Fields ◽  
Kolmogorov Distance ◽  
Likelihood Estimator ◽  
Infinite Dimensional ◽  
A New Technique ◽  
Exact Rate Of Convergence

This paper is concerned with the rate of convergence of the distribution of the sequence {Fn/Gn}, where Fn and Gn are each functionals of infinite-dimensional Gaussian fields. This form very frequently appears in the estimation problem of parameters occurring in Stochastic Differential Equations (SDEs) and Stochastic Partial Differential Equations (SPDEs). We develop a new technique to compute the exact rate of convergence on the Kolmogorov distance for the normal approximation of Fn/Gn. As a tool for our work, an Edgeworth expansion for the distribution of Fn/Gn, with an explicitly expressed remainder, will be developed, and this remainder term will be controlled to obtain an optimal bound. As an application, we provide an optimal Berry–Esseen bound of the Maximum Likelihood Estimator (MLE) of an unknown parameter appearing in SDEs and SPDEs.

scholarly journals On a Berry-Esseen type bound for the maximum likelihood estimator of a parameter for some stochastic partial differential equations

2004 ◽  
Vol 2004 (2) ◽  
pp. 109-122 ◽  
Author(s):  
M. N. Mishra ◽  
B. L. S. Prakasa Rao
Keyword(s):  
Maximum Likelihood ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Rate Of Convergence ◽  
Maximum Likelihood Estimator ◽  
Stochastic Partial Differential Equations ◽  
Likelihood Estimator ◽  
Partial Differential ◽  
Distribution Of The Maximum

This paper is concerned with the study of the rate of convergence of the distribution of the maximum likelihood estimator of a parameter appearing linearly in the drift coefficients of two types of stochastic partial differential equations (SPDEs).

scholarly journals Rate of convergence for traditional Pólya urns

2020 ◽  
Vol 57 (4) ◽  
pp. 1029-1044
Author(s):  
Svante Janson
Keyword(s):  
Rate Of Convergence ◽  
Dirichlet Distribution ◽  
Fixed Number ◽  
Kolmogorov Distance ◽  
Initial Composition ◽  
Exact Distributions ◽  
Polya Urn ◽  
The One ◽  
Lévy Distance ◽  
Direct Calculations

AbstractConsider a Pólya urn with balls of several colours, where balls are drawn sequentially and each drawn ball is immediately replaced together with a fixed number of balls of the same colour. It is well known that the proportions of balls of the different colours converge in distribution to a Dirichlet distribution. We show that the rate of convergence is $\Theta(1/n)$ in the minimal $L_p$ metric for any $p\in[1,\infty]$, extending a result by Goldstein and Reinert; we further show the same rate for the Lévy distance, while the rate for the Kolmogorov distance depends on the parameters, i.e. on the initial composition of the urn. The method used here differs from the one used by Goldstein and Reinert, and uses direct calculations based on the known exact distributions.

scholarly journals On Lagrange interpolation with equally spaced nodes

2000 ◽  
Vol 62 (3) ◽  
pp. 357-368 ◽  
Author(s):  
Michael Revers
Keyword(s):  
Rate Of Convergence ◽  
Lagrange Interpolation ◽  
End Points ◽  
Quantitative Version ◽  
Equidistant Nodes ◽  
Lagrange Interpolation Polynomials ◽  
Interpolation Polynomials ◽  
Exact Rate Of Convergence

A well-known result due to S.N. Bernstein is that sequence of Lagrange interpolation polynomials for |x| at equally spaced nodes in [−1, 1] diverges everywhere, except at zero and the end-points. In this paper we present a quantitative version concerning the divergence behaviour of the Lagrange interpolants for |x|3 at equidistant nodes. Furthermore, we present the exact rate of convergence for the interpolatory parabolas at the point zero.

scholarly journals State space decomposition for non-autonomous dynamical systems

2011 ◽  
Vol 141 (5) ◽  
pp. 957-974 ◽  
Author(s):  
Xiaopeng Chen ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
State Space ◽  
Decomposition Theorem ◽  
Navier Stokes ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
The Lorenz System ◽  
A Chain ◽  
Autonomous Dynamical Systems

The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

Inhomogeneous thermal conduction equations

1978 ◽  
Vol 56 (7) ◽  
pp. 928-935
Author(s):  
C. S. Lai
Keyword(s):  
Differential Equations ◽  
Thermal Conduction ◽  
Series Solution ◽  
Nonlinear Differential Equations ◽  
Three Dimensional ◽  
Incompressible Fluids ◽  
Self Similar Solution ◽  
Self Similar ◽  
A New Technique ◽  
The One

The method of self-similar solution of partial differential equations is applied to the one-, two-, and three-dimensional inhomogeneous thermal conduction equations with the thermometric conductivities χ ~ rmWn. Analytical solutions are obtained for the case that the total amount of heat is conserved. For the case that the temperature is maintained constant at r = 0, a new technique of the series solution about the point of intercept is proposed to solve the resultant nonlinear differential equations. The solutions obtained are useful in studying the thermal conduction characteristics of some incompressible fluids.

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