On necessary and sufficient conditions for the self-normalized central limit theorem

2018 ◽  
Vol 61 (10) ◽  
pp. 1741-1748 ◽  
Author(s):  
Qiman Shao
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Self ◽  
Necessary And Sufficient

Bounded normal approximation in simulations of highly reliable Markovian systems

1999 ◽  
Vol 36 (4) ◽  
pp. 974-986 ◽  
Author(s):  
Bruno Tuffin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Relative Error ◽  
Normal Approximation ◽  
Sufficient Conditions ◽  
The Central Limit Theorem ◽  
Normal Law ◽  
Markovian Systems ◽  
Necessary And Sufficient

In this paper, we give necessary and sufficient conditions to ensure the validity of confidence intervals, based on the central limit theorem, in simulations of highly reliable Markovian systems. We resort to simulations because of the frequently huge state space in practical systems. So far the literature has focused on the property of bounded relative error. In this paper we focus on ‘bounded normal approximation’ which asserts that the approximation of the normal law, suggested by the central limit theorem, does not deteriorate as the reliability of the system increases. Here we see that the set of systems with bounded normal approximation is (strictly) included in the set of systems with bounded relative error.

A note on the central limit theorem for doubly stochastic Poisson processes

1976 ◽  
Vol 13 (04) ◽  
pp. 809-813
Author(s):  
Holger Rootzén
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Poisson Processes ◽  
Doubly Stochastic ◽  
The Central Limit Theorem ◽  
Doubly Stochastic Poisson Process ◽  
Necessary And Sufficient

In this note, necessary and sufficient conditions for the central limit theorem for the number of events in a doubly stochastic Poisson process are given.

scholarly journals On the rate of convergence of moments in the central limit theorem

1978 ◽  
Vol 25 (2) ◽  
pp. 250-256 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Expansions ◽  
Sufficient Conditions ◽  
Rates Of Convergence ◽  
The Third ◽  
The Central Limit Theorem ◽  
Convergence Of Moments ◽  
Necessary And Sufficient

AbstractAn early extension of Lindeberg's central limit theorem was Bernstein's (1939) discovery of necessary and sufficient conditions for the convergence of moments in the central limit theorem. Von Bahr (1965) made a study of some asymptotic expansions in the central limit theorem, and obtained rates of convergence for moments. However, his results do not in general imply that the moments converge. Some better rates have been obtained by Bhattacharya and Rao for moments between the second and third. In this paper we give improved rates of convergence for absolute moments between the third and fourth.

A note on the central limit theorem for doubly stochastic Poisson processes

1976 ◽  
Vol 13 (4) ◽  
pp. 809-813 ◽  
Author(s):  
Holger Rootzén
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Poisson Processes ◽  
Doubly Stochastic ◽  
The Central Limit Theorem ◽  
Doubly Stochastic Poisson Process ◽  
Necessary And Sufficient

In this note, necessary and sufficient conditions for the central limit theorem for the number of events in a doubly stochastic Poisson process are given.

Bounded normal approximation in simulations of highly reliable Markovian systems

1999 ◽  
Vol 36 (04) ◽  
pp. 974-986 ◽  
Author(s):  
Bruno Tuffin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Relative Error ◽  
Normal Approximation ◽  
Sufficient Conditions ◽  
The Central Limit Theorem ◽  
Normal Law ◽  
Markovian Systems ◽  
Necessary And Sufficient

In this paper, we give necessary and sufficient conditions to ensure the validity of confidence intervals, based on the central limit theorem, in simulations of highly reliable Markovian systems. We resort to simulations because of the frequently huge state space in practical systems. So far the literature has focused on the property of bounded relative error. In this paper we focus on ‘bounded normal approximation’ which asserts that the approximation of the normal law, suggested by the central limit theorem, does not deteriorate as the reliability of the system increases. Here we see that the set of systems with bounded normal approximation is (strictly) included in the set of systems with bounded relative error.

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