Dependent versions of a central limit theorem for the squared length of a sample mean (Q1347177)

Consider the sample means \(\{\overline{X}_ n\}\) of a suitably- normalized \(p\)-dimensional square-integrable martingale difference sequence \(\{X_ i, {\mathcal F}_ i, i\geq 1\}\), where \(p = p_ n\) is a function of \(n\). Suppose that the \(2\delta\)-th moments of all components of all the \(X_ i\)'s are uniformly bounded for some \(\delta \geq 4\), and that, for each \(l \geq 1\) and all \(a\), \(b\) in \(\{1,2,\dots,p\}\), \[ \sum^ \infty_{i= 1} \sum^ \infty_{j = 1} \sum^ \infty_{k = 1} | \text{cum} (X_{ia}, X_{ja}, X_{kb}, X_{l b}| = o(n/p)\qquad \text{ as } n \to \infty. \] Assume that, for some integer \(k\) and real sequence \(\{\varphi_ l, l \geq 0\}\), \((E \| E(X_ i X_ i' ({\mathcal F}_{i-j}) - I_ p \|^ 4)^{1/4} \leq p \varphi_ 0\) or \(p \varphi_{j-k_ p}\) according as \(j \leq k_ p\) or \(j > k_ p\), where \(I_ p\) is the \(p \times p\) identity matrix and \(l^ 2 \varphi_ l \to 0\) as \(l \to \infty\). If \(p^ 3/n \to 0\), then it is shown that \((n \| X_ n \|^ 2 - p) / \sqrt {2p}\) converges in law to the standard normal distribution. The same conclusion is shown to hold under the weaker hypothesis \(p^ 2/n \to 0\) if \(k = 0\) and \(\varphi_ l = O(l^{-1-\varepsilon})\) for some \(\varepsilon > 0\). These results extend work of \textit{S. Portnoy} [Ann. Stat. 16, No. 1, 356- 366 (1988; Zbl 0637.62026)] in the i.i.d. case. An application to a test of serial correlation is presented.