Estimate of the bounded Lipschitz metric for sums of weakly dependent random variables
Let \(X_ 1,X_ 2,..\). be a sequence of weakly dependent random variables (m-dependent or having the \(\phi\)-mixing property or that of absolute regularity with the coefficient \(\beta)\). Denote \[ S_ n=\sum^{n}_{j=1}X_ j,\quad B^ 2_ n=E S^ 2_ n,\quad Y_ j=X_ j/B_ n,\quad j=1,...,n, \] \[ Z_ n=S_ n/B_ n,\quad L_{n,r}=\sum^{n}_{j=1}E | Y_ j|^ r,\quad \bar L_{n,r}=\sum^{n}_{j=1}E | Y_ j|^ rI\{| Y_ j| \leq t\} \] with positive t and \(I\{\) \(A\}\) as an indicator of the event A, Ḻ\({}_{n,r}=L_ n-\bar L_{n,r}\). Let BL(R) be the class of bounded Lipschitz functions h with the usual norm \[ \| h\| =\sup \{| h(x)|:\quad x\in R\}+\inf \{c:\quad | h(x)-h(y)| \leq c| x-y|,\quad x,y\in R\}. \] The estimations from above are obtained for the supremum w.r.t. \(h\in BL(R)\) of the ratio \(| E h(Z_ n)-E h(N)| /\| h\|\) in terms of n, \(L_ r\), Ḻ\({}_{n,r}\), \(\bar L_{n,r}\) and m, \(\phi\) or \(\beta\), respectively (depending on the case considered), where N is a random variable with standard normal distribution. For example, in the case of m-dependency (m\(\geq 0)\) and finiteness of E \(| X_ j|^ r\), \(2<r\leq 3\), \(j=1,...,n\), the estimation has the form \(C(m+1)^{r-1}L_{n,r}\) with an absolute constant C.
scientific article; zbMATH DE number 4140921