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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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