On a global form of the central limit theorem for m-dependent random variables. (Russian) Zbl 0577.60024

Shergin, V. V.

Teor. Veroyatn. Mat. Stat. 29, 122-128 (1983).

Let \(X_ 1,X_ 2,..\). be a sequence of m-dependent random variables with \(E(X_ i)=0\) and \(E(| X|^ k)<\infty\) for some \(k\geq 2\). Let \(F_ n\) be the distribution function of \(Y_ n:=X_ 1+...+X_ n/\sigma^ 2_ n\) with \(\sigma^ 2_ n=E(X_ 1+...+X_ n)^ 2\). Under a certain boundedness condition for the second moments of \(X_ i\) it is shown that for \(0\leq l\leq k-1\) \[ \int^{+\infty}_{-\infty}| x|^ l| F_ n(x)-\Phi (x)| dx\leq C\sigma_ n^{-1}\cdot \sum^{n}\quad_{i=1}E| X_ i|^ k, \] where C is a certain constant and \(k_ i\) is 3 if \(| X_ i| \leq | \sigma_ n|\), and \(\max (2,l+1)\) otherwise. From this an estimate for the deviation of \(E| Y_ n|^ l\) from \(\int | x|^ ld\Phi (x)\) is obtained.

Reviewer: C.Baldauf

MSC:

60F05

Central limit and other weak theorems

Keywords:

central limit theorem; sequence of m-dependent random variables