scholarly journals Rates of convergence in the central limit theorem for empirical processes

1986 ◽  
pp. 73-109 ◽  
Author(s):  
Pascal Massart
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Rates Of Convergence ◽  
Empirical Processes ◽  
The Central Limit Theorem

scholarly journals Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

2021 ◽  
Author(s):  
Felix Herold ◽  
Daniel Hug ◽  
Christoph Thäle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Hyperbolic Space ◽  
Central Limit ◽  
Hausdorff Measure ◽  
Rates Of Convergence ◽  
Limit Problem ◽  
Dimensional Hausdorff Measure ◽  
The Central Limit Theorem ◽  
Order Quantities

AbstractPoisson processes in the space of $$(d-1)$$ ( d - 1 ) -dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature $$-1$$ - 1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $$d\ge 2$$ d ≥ 2 , it is shown that in case (ii) the central limit theorem holds for $$d\in \{2,3\}$$ d ∈ { 2 , 3 } and fails if $$d\ge 4$$ d ≥ 4 and $$k=d-1$$ k = d - 1 or if $$d\ge 7$$ d ≥ 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

On the exact order of normal approximation in multivariate renewal theory

1985 ◽  
Vol 22 (02) ◽  
pp. 280-287 ◽  
Author(s):  
Ştefan P. Niculescu ◽  
Edward Omey
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Normal Approximation ◽  
Renewal Theory ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Exact Order ◽  
First Passage ◽  
The Central Limit Theorem

Equivalence of rates of convergence in the central limit theorem for the vector of maximum sums and the corresponding first-passage variables is established. A similar result for the vector of partial sums and the corresponding renewal variables is also given. The results extend to several dimensions the bivariate results of Ahmad (1981).

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