scholarly journals Stein’s method of exchangeable pairs in multivariate functional approximations

2021 ◽  
Vol 26 (none) ◽  
Author(s):  
Christian Döbler ◽  
Mikołaj J. Kasprzak
Keyword(s):  
Stein’S Method ◽  
Stein's Method ◽  
Exchangeable Pairs

Two new proofs of the Erdös–Kac Theorem, with bound on the rate of convergence, by Stein's method for distributional approximations

2009 ◽  
Vol 147 (1) ◽  
pp. 95-114 ◽  
Author(s):  
ADAM J. HARPER
Keyword(s):  
Rate Of Convergence ◽  
Normal Approximation ◽  
Poisson Approximation ◽  
Stein’S Method ◽  
Stein's Method ◽  
Quantitative Form ◽  
Approximation Argument ◽  
Exchangeable Pairs ◽  
Distributional Approximations

AbstractIn this paper, we apply Stein's method for distributional approximations to prove a quantitative form of the Erdös–Kac Theorem. We obtain our best bound on the rate of convergence, on the order of log log log n (log log n)−1/2, by making an intermediate Poisson approximation; we believe that this approach is simpler and more probabilistic than others, and we also obtain an explicit numerical value for the constant implicit in the bound. Different ways of applying Stein's method to prove the Erdös–Kac Theorem are discussed, including a Normal approximation argument via exchangeable pairs, where the suitability of a Poisson approximation naturally suggests itself.

scholarly journals Step Size in Stein's Method of Exchangeable Pairs

2009 ◽  
Vol 18 (6) ◽  
pp. 979-1017
Author(s):  
NATHAN ROSS
Keyword(s):  
Normal Distribution ◽  
Convergence Rates ◽  
Normal Approximation ◽  
Poisson Approximation ◽  
Stein’S Method ◽  
Stein's Method ◽  
Step Size ◽  
Rigorous Results ◽  
Exchangeable Pairs ◽  
Natural Way

Stein's method of exchangeable pairs is examined through five examples in relation to Poisson and normal distribution approximation. In particular, in the case where the exchangeable pair is constructed from a reversible Markov chain, we analyse how modifying the step size of the chain in a natural way affects the error term in the approximation acquired through Stein's method. It has been noted for the normal approximation that smaller step sizes may yield better bounds, and we obtain the first rigorous results that verify this intuition. For the examples associated to the normal distribution, the bound on the error is expressed in terms of the spectrum of the underlying chain, a characteristic of the chain related to convergence rates. The Poisson approximation using exchangeable pairs is less studied than the normal, but in the examples presented here the same principles hold.

scholarly journals Exponential bounds via Stein’s method and exchangeable pairs

ScienceAsia ◽  
2018 ◽  
Vol 44 (4) ◽  
pp. 277
Author(s):  
Patcharee Sumritnorrapong ◽  
Kritsana Neammanee ◽  
Jiraphan Suntornchost
Keyword(s):  
Stein’S Method ◽  
Stein's Method ◽  
Exchangeable Pairs ◽  
Exponential Bounds

On the rate of Poisson process approximation to a Bernoulli process

1997 ◽  
Vol 34 (4) ◽  
pp. 898-907 ◽  
Author(s):  
Aihua Xia
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Discrete Time ◽  
Wasserstein Distance ◽  
Stein’S Method ◽  
Upper Bounds ◽  
Stein's Method ◽  
Bernoulli Process ◽  
Logarithmic Factor ◽  
Process Approximation

This note gives the rate for a Wasserstein distance between the distribution of a Bernoulli process on discrete time and that of a Poisson process, using Stein's method and Palm theory. The result here highlights the possibility that the logarithmic factor involved in the upper bounds established by Barbour and Brown (1992) and Barbour et al. (1995) may be superfluous in the true Wasserstein distance between the distributions of a point process and a Poisson process.

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