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.

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.

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.

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).

Multiple comparisons and sums of dissociated random variables

1985 ◽  
Vol 17 (1) ◽  
pp. 147-162 ◽  
Author(s):  
A. D. Barbour ◽  
G. K. Eagleson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Multiple Comparisons ◽  
The Central Limit Theorem ◽  
Normally Distributed

Sufficient conditions for a sum of dissociated random variables to be approximately normally distributed are derived. These results generalize the central limit theorem for U-statistics and provide conditions which can be verified in a number of applications. The method of proof is that due to Stein (1970).

scholarly journals Error estimate in the central limit theorem

2008 ◽  
Vol 48 ◽  
Author(s):  
Aurelija Kasparavičiūtė ◽  
Kazimieras Padvelskis
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Error Estimate ◽  
Central Limit ◽  
Normal Approximation ◽  
The Central Limit Theorem ◽  
Independent Identically Distributed

In this paper, we determined, independent identically distributed random variable’s {Xk, k = 1,2,...} centered and normalized sum’s Sn = \sumn k=1 Xk distribution’s Fn(x) = P(Zn < x) exact error estimate in case of the normal approximation with one Cebyšova’s asymptotic expansion’s term.

The central limit theorem for m-dependent variables

1955 ◽  
Vol 51 (1) ◽  
pp. 92-95 ◽  
Author(s):  
P. H. Diananda
Keyword(s):  
Standard Deviation ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Sufficient Conditions ◽  
The Central Limit Theorem ◽  
Lindeberg Condition ◽  
Dependent Variables ◽  
Basic Ideas

In a previous paper (4) central limit theorems were obtained for sequences of m-dependent random variables (r.v.'s) asymptotically stationary to second order, the sufficient conditions being akin to the Lindeberg condition (3). In this paper similar theorems are obtained for sequences of m-dependent r.v.'s with bounded variances and with the property that for large n, where s′n is the standard deviation of the nth partial sum of the sequence. The same basic ideas as in (4) are used, but the proofs have been simplified. The results of this paper are examined in relation to earlier ones of Hoeffding and Robbins(5) and of the author (4). The cases of identically distributed r.v.'s and of vector r.v.'s are mentioned.

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