Stein’s method of normal approximation for dynamical systems

2019 ◽  
Vol 20 (04) ◽  
pp. 2050021 ◽  
Author(s):  
Olli Hella ◽  
Juho Leppänen ◽  
Mikko Stenlund
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Central Limit ◽  
Normal Approximation ◽  
Stein’S Method ◽  
Stein's Method ◽  
Multiplicative Constant ◽  
Correlation Decay ◽  
The Central Limit Theorem

We present an adaptation of Stein’s method of normal approximation to the study of both discrete- and continuous-time dynamical systems. We obtain new correlation-decay conditions on dynamical systems for a multivariate central limit theorem augmented by a rate of convergence. We then present a scheme for checking these conditions in actual examples. The principal contribution of our paper is the method, which yields a convergence rate essentially with the same amount of work as the central limit theorem, together with a multiplicative constant that can be computed directly from the assumptions.

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).

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.

Estimating the Error of a Permutational Central Limit Theorem

1996 ◽  
Vol 10 (4) ◽  
pp. 533-541 ◽  
Author(s):  
Chern-Ching Chao ◽  
Lincheng Zhao ◽  
Wen-Qi Liang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Error Bound ◽  
Central Limit ◽  
Normal Approximation ◽  
Random Permutation ◽  
Stein’S Method ◽  
Real Numbers ◽  
Sorting Problem ◽  
Two Measures

Motivated by two measures of presortedness, number of runs and oscillation of a permutation, related to the sorting problem, we derive an error bound for normal approximation to the distribution of Here, αij's are given real numbers and π is a uniformly distributed random permutation of {l,…, n}. The derivation is based on Stein's method.

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.

On the exact order of normal approximation in multivariate renewal theory

1985 ◽  
Vol 22 (2) ◽  
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).

scholarly journals KOLMOGOROV BOUNDS FOR THE NORMAL APPROXIMATION OF THE NUMBER OF TRIANGLES IN THE ERDŐS–RÉNYI RANDOM GRAPH

2021 ◽  
pp. 1-27
Author(s):  
Adrian Röllin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Graph ◽  
Normal Approximation ◽  
Combinatorial Theory ◽  
Characteristic Functions ◽  
The Central Limit Theorem ◽  
New Variant ◽  
Kolmogorov Metric

We bound the error for the normal approximation of the number of triangles in the Erdős–Rényi random graph with respect to the Kolmogorov metric. Our bounds match the best available Wasserstein bounds obtained by Barbour et al. [(1989). A central limit theorem for decomposable random variables with applications to random graphs. Journal of Combinatorial Theory, Series B 47: 125–145], resolving a long-standing open problem. The proofs are based on a new variant of the Stein–Tikhomirov method—a combination of Stein's method and characteristic functions introduced by Tikhomirov [(1976). The rate of convergence in the central limit theorem for weakly dependent variables. Vestnik Leningradskogo Universiteta 158–159, 166].

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