An invariance principle for dependent random variables (Q5905373)

A sequence of random variables, \(X_ 1,X_ 2,\ldots,\) is \(\phi\)-mixing if as \(m\to\infty \sup\{| P[A\mid B]-P[B]|\}\to0\), where the supremum is taken over all sets \(B\) in \(\sigma(X_ i:i\geq n+m)\), nontrivial sets \(A\) in \(\sigma(X_ i:i\leq n)\), and \(n\geq 1.\) Assume that \(EX_ n=0\) and \(EX^ 2_ n<+\infty\) for each \(n\). Define \(S_ n\) to be the sum of the \(X_ i\) for \(1\leq i\leq n\), and define \(s^ 2_ n=E S^ 2_ n\) (assumed \(\to+\infty\) always). For a sequence of positive integers \(k_ n\), \(n\geq0\), increasing sufficiently slowly, define a random process on [0,1] by \(W_ n(t)=S_{m_ n(t)}/s_ n\), where \(m_ n(t)=\max\{i\geq0:k_ i\leq tk_ n\}\). This paper presents some sufficient and necessary conditions for the weak convergence of \(W_ n\) to a standard Brownian process on \(D[0,1].\) It provides an invariance principle for \(\phi\)-mixing sequences under Lindeberg's condition which improves, for example, the results of \textit{D. L. McLeish} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 32, 165-178 (1975; Zbl 0288.60034)] and \textit{M. Peligrad} [Ann. Probab. 13, 1304-1313 (1985; Zbl 0597.60018)]. This paper does not require \(\lim s^ 2_ n/n=\sigma^ 2>0\) nor uniform integrability of the \(X^ 2_ i\). A related paper is that of \textit{H. Dehling}, \textit{M. Denker} and \textit{W. Philipp} [ibid. 14, 1359-1370 (1986; Zbl 0605.60027)].