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On strong embeddings by Stein’s method. (English) Zbl 1338.60064
Summary: Strong embeddings, that is, couplings between a partial sum process of a sequence of random variables and a Brownian motion, have found numerous applications in probability and statistics. We extend Chatterjee’s novel use of Stein’s method for $$\{-1,+1\}$$ valued variables to a general class of discrete distributions, and provide $$\log n$$ rates for the coupling of partial sums of independent variables to a Brownian motion, and results for coupling sums of suitably standardized exchangeable variables to a Brownian bridge.
##### MSC:
 60F05 Central limit and other weak theorems 60F17 Functional limit theorems; invariance principles 60G09 Exchangeability for stochastic processes
##### Keywords:
strong approximation; coupling; zero bias; exchangeability
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