Limit theorems for uniform distributions on spheres in high-dimensional euclidean spaces

1982 ◽  
Vol 19 (1) ◽  
pp. 221-228 ◽  
Author(s):  
A. J. Stam
Keyword(s):  
Normal Distribution ◽  
Uniform Distribution ◽  
Total Variation ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Geometric Probability ◽  
High Dimensional ◽  
Standard Normal Distribution ◽  
Uniform Distributions ◽  
Standard Normal

If X = (X1, · ··, Xn) has uniform distribution on the sphere or ball in ℝ with radius a, then the joint distribution of , ···, k, converges in total variation to the standard normal distribution on ℝ. Similar results hold for the inner products of independent n-vectors. Applications to geometric probability are given.

Limit theorems for uniform distributions on spheres in high-dimensional euclidean spaces

1982 ◽  
Vol 19 (01) ◽  
pp. 221-228 ◽  
Author(s):  
A. J. Stam
Keyword(s):  
Normal Distribution ◽  
Uniform Distribution ◽  
Total Variation ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Geometric Probability ◽  
High Dimensional ◽  
Standard Normal Distribution ◽  
Uniform Distributions ◽  
Standard Normal

If X = (X 1, · ··, Xn ) has uniform distribution on the sphere or ball in ℝ with radius a, then the joint distribution of , ···, k, converges in total variation to the standard normal distribution on ℝ. Similar results hold for the inner products of independent n-vectors. Applications to geometric probability are given.

scholarly journals A New Linear Regression Kalman Filter with Symmetric Samples

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2139
Author(s):  
Xiuqiong Chen ◽  
Jiayi Kang ◽  
Mina Teicher ◽  
Stephen S.-T. Yau
Keyword(s):  
Kalman Filter ◽  
Linear Regression ◽  
Particle Filter ◽  
Normal Distribution ◽  
Extended Kalman Filter ◽  
Nonlinear Filtering ◽  
Unscented Kalman Filter ◽  
Standard Normal Distribution ◽  
Leibler Divergence ◽  
Standard Normal

Nonlinear filtering is of great significance in industries. In this work, we develop a new linear regression Kalman filter for discrete nonlinear filtering problems. Under the framework of linear regression Kalman filter, the key step is minimizing the Kullback–Leibler divergence between standard normal distribution and its Dirac mixture approximation formed by symmetric samples so that we can obtain a set of samples which can capture the information of reference density. The samples representing the conditional densities evolve in a deterministic way, and therefore we need less samples compared with particle filter, as there is less variance in our method. The numerical results show that the new algorithm is more efficient compared with the widely used extended Kalman filter, unscented Kalman filter and particle filter.

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