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A functional Hungarian construction for sums of independent random variables. (English) Zbl 1021.60027
The authors develop a Hungarian construction for the partial sum process of independent non-identically distributed random variables. The process is indexed by functions $$f$$ defined on $$[0,1]$$ and belonging to some class $$\mathcal H$$. The main result is as follows: Given a sequence $$X_1,\dots,X_n$$ of independent mean zero random variables satisfying a generalized Sakhanenko condition, suppose $$N_1,\dots,N_n$$ are independent mean zero Gaussian r.v. with $$\text{Var }N_i=\text{Var }X_i$$ for all $$i$$ and defined on $$(\Omega,{\mathcal F},P)$$. Then a sequence $$\widetilde X_1,\dots,\widetilde X_n$$ of independent r.v. can be constructed on $$\Omega$$ such that $$\widetilde{X}_i\overset {d}= X_i$$ for all $$i$$ and $\sup_{f\in{\mathcal H}} P\left(\left|\sum_{i=1}^n f(i/n)(\widetilde X_i - N_i)\right|> \lambda_n^{-1}x \log^2 n\right)\leq c_1\exp\{-c_2 x\}$ for $$x\geq 0$$, where $$\mathcal H$$ is the Hölder ball of smoothness 1/2 while $$\lambda_n$$ appears in generalized Sakhanenko condition. The main reason for a construction with the supremum outside the probability is that otherwise a substantial loss of the classical $$n^{-1/2}\log n$$ rate of approximation occurs. This strong approximation result may be useful e.g. for proving asymptotic equivalence of statistical experiments.

##### MSC:
 60F17 Functional limit theorems; invariance principles 60F15 Strong limit theorems 62G07 Density estimation
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