A Berry-Esséen theorem and a functional law of the iterated logarithm for weakly associated random vectors
Considered are weakly associated random variables \(\{X_ i\), \(i\geq 1\}\) from \(R^ d\) into R and this concept means that for any permutation \(\pi\) of the integers we have \[ cov[f(X_{\pi (1)},...,X_{\pi (n)}),g(X_{\pi (n+1)},...,X_{\pi (n+m)})]\geq 0 \] for all coordinatewise increasing functions f from \(R^{nd}\) into R and g from \(R^{md}\) into R. Note that we have a kind of symmetric independence. Under additional moment conditions (moments of order E \(\| X_ 1\|^{2+\epsilon}<\infty\) \((\epsilon >0\) arbitrary)) the author proves a central limit theorem for the normed sums \(S_ n=\sum^{n}_{i=1}X_ i\) when the sequence \(\{X_ i\}\) is strictly stationary, mean-zero, and weakly associated. Estimation of the remainder term is given for convex sets, sufficient for showing its convergence to zero as \(n\to \infty.\) An inequality by \textit{C. M. Newman} [Commun. Math. Phys. 74, 119-128 (1980; Zbl 0429.60096)] and a lemma on inversion of Fourier transforms by \textit{B. von Bahr} [Ark. Mat. 7, 71-88 (1967; Zbl 0221.60014)] are of importance for the proofs. Under rather strong and complicated conditions the functional law of the iterated logarithm is shown.
functional law of the iterated logarithm
scientific article; zbMATH DE number 4088614