On the convergence of weighted sums of \(L_q\)-mixingale arrays (Q1964615)

A sequence of random variables \(\{U_n, n\geq 1\}\) is said to converge completely to a constant \(a\) if \(\sum_{n\geq 1} P(|U_n-a|\geq \varepsilon)<\infty\) for each \(\varepsilon>0\). Generalizing McLeish's concept of \(L_2\)-mixingale sequences the author introduces \(L_q\)-mixingale sequences and proves complete convergence, \(L_q\)-convergence and convergence in probability of weighted sums of triangular arrays of \(L_q\)-mixingales. An essential point in proving the results consists in applying a Bernstein-type inequality for bounded martingale difference sequences.