On the influence of extremes on the rate of convergence in the central limit theorem (Q790557)

Let \(X_ 1,...,X_ n\) be i.i.d. random variables with finite variance \(\sigma^ 2\). Let \(S_ n=X_ 1+...+X_ n\) and let \(^{(k)}S_ n=S_ n-\sum^{k}_{i=1}X_{ni}\), where \(X_{ni}\) is the i-th largest among \(X_ 1,...,X_ n\) in absolute value. The author investigates the rate of convergence in the central limit theorem for the trimmed sum \(^{(k)}S_ n\). The main conclusions are as follows: (i) If \(A_ n\), \(B_ n\) and \(k=k(n)\) satisfying \(k(n)/n\to 0\) are chosen suitably, then \(\Delta_ n=\sup_{-\infty<z<\infty}| P(^{(k)}S_ n\leq A_ nz+B_ n)-\Phi(z)|\) can be made arbitrarily close to \(0(n^{-1})\), where \(\Phi\) is the standard normal distribution function. Thus the removal of extremes can substantially improve the rate of convergence in the central limit theorem. However the optimal \(A_ n\) is of a rather involved nature and is most unlikely to be known in practice. (ii) The improvements described in (i) cannot be achieved using a simple scale factor. Indeed, if \(EX_ 1=0\), if \[ \Delta_ n^{(1)} = \sup_{-\infty<z<\infty}| P(^{(k)}S_ n\leq(n-k)^{1/2}\sigma z)- \Phi(z)| \] and \[ \Delta_ n^{(2)}=\sup_{-\infty<z<\infty}| P(S_ n\leq n^{1/2}\sigma z)-\Phi(z)|, \] then \[ (\Delta_ n^{(1)}+n^{-1})/(\Delta_ n^{(2)}+n^{-1}) \] does not converge to zero. Therefore the rate of convergence of \(^{(k)}S_ n/(n- k)^{1/2}\sigma\) to N(0,1) is at least as slow as that of \(S_ n/n^{1/2}\sigma\) up to terms of order \(n^{-1}\).