Weak law of large numbers for arrays of random variables (Q1962177)

Let \(\{X_{ni}\), \(u_n\leq i\leq \nu_n\}_{\nu\geq 1}\) be an array of random variables on \((\Omega,{\mathcal F},P)\), where \(u_n,\nu_n\in \mathbb{Z}\cup \{-\infty, \infty\}\), \(u_n\leq \nu_n\), \(n\geq 1\). Set \({\mathcal F}_{nj}= \sigma \{X_{ni}\), \(u_n\leq i\leq j\}\) and \({\mathcal F}_{n,u_n-1}= \{\emptyset, \Omega\}\), \(n\geq 1\). Let \((k_n)_{n\geq 1}\) be a sequence of positive integers such that \(\lim_{n\to \infty} k_n= \infty\). Assume that \(\sup_{n\geq 1} \frac{1}{k_n} \sum_{i=u_n}^{\nu_n} E|X_{ni} |^r< \infty\) and \(\frac{1}{k_n} \sum_{i=u_n}^{\nu_n} E|X_{ni} |^r I(|X_{ni} |^r>a)\to 0\) as \(a\to \infty\) uniformly in \(n\), where \(r>0\) is arbitrarily given. The author's main result is that \(\sum_{i=u_n}^{\nu_n} (X_{ni}-a_{ni})/ k_n^{1/r}\to 0\) in \(L^r\) as \(n\to \infty\), where \(a_{ni}= 0\) if \(0< r< 1\) and \(a_{ni}= E(X_{ni}\mid {\mathcal F}_{n,i-1})\) if \(1\leq r< 2\). This extends a result of \textit{A. Gut} [Stat. Probab. Lett. 14, No. 1, 49--52 (1992; Zbl 0769.60034)]. A weak law of large numbers for moving average processes is obtained taking \(u_n= -\infty\), \(\nu_n= \infty\), and \(k_n= n\), \(n\geq 1\).