An \(L_ 1\)-convergence theorem for heterogeneous mixingale arrays with trending moments
Consider an array of pairs \(\bigl\{ X_{nt},F_{nt}:-\infty<t<\infty\), \(n \geq 1 \bigr\}\) where \(X_{nt}\) are random variables defined on a probability space \((\Omega,F,P)\) and \(F_{nt}\) are \(\sigma\)-subfields of \(F\), increasing in \(t\). The array is an \({\mathcal L}_ 1\)-mixingale in the sense that there exist constants \(\{c_{nt}\}\) and a non-negative sequence \(\{\zeta_ m\}\) converging to zero when \(n \to \infty\) such that \(\bigl\| {\mathcal E}(X_{nt} \mid F_{n,t-m}) \bigr\|_ 1\leq c_{nt}\zeta_ m\) and \(\bigl\| X_{nt}-{\mathcal E}(X_{nt}\mid F_{n,t-m}) \bigr\|_ 1 \leq c_{nt}\zeta_{m+1}\) for all \(t\), \(n\) and \(m \geq 0\). For such an array, the following theorem has been proved: Suppose \(\limsup_{n \to \infty} \sum^{k_ n}_{t=1} c_{nt}< \infty\), \(\limsup_{n \to \infty}\sum^{k_ n}_{t=1}c^ 2_{nt}=0\) and \(\{X_{nt}/C_{nt}\}\) is uniformly integrable, \(k_ n\) being an increasing, integer-valued function of \(n\) and \(k_ n \to \infty\) as \(n \to \infty\). Then \(\lim_{n \to \infty}{\mathcal E} \bigl|\sum^{k_ n}_{t=1}X_{nt} \bigr|=0\).
scientific article; zbMATH DE number 168304