An invariance principle for stationary \(\rho\)-mixing sequences with infinite variance
Let \(\{X_ k\), \(k\in\mathbb{Z}\}\) be a strictly stationary \(\rho\)-mixing sequence of real-valued random variables, that is, a strictly stationary sequence satisfying the condition \[ \rho(n)=\sup\{| \text{Corr}(f_ 1,f_ 2)|:\;f_ 1\in L_ 2({\mathcal F}^ 0_{-\infty}),\;f_ 2\in L_ 2({\mathcal F}^ \infty_ n)\}\downarrow 0 \qquad (n\to\infty), \] where \({\mathcal F}^ b_ a\) denotes the \(\sigma\)-algebra of events generated by \(X_ a,\dots,X_ b\). The author establishes the weak invariance principle for the sequence \(\{X_ i\}\) when \(E X_ 0=0\) and \(V(X_ 0)=\infty\); more precisely, he shows that there exists a sequence \(\{A_ n\), \(n\in N\}\) of positive numbers with \(A_ n\to\infty\) as \(n\to\infty\) such that the sequence \(\bigl\{\bigl\{{1\over {A_ n}} \sum_{j=1}^{[nt]} X_ j\;(0\leq t\leq 1)\bigr\}:\;n\geq 1\bigr\}\) converges weakly to a Wiener process if \[ H(x):=E X^ 2_ 0 I(| x_ 0|<x) \qquad\text{and}\qquad G(x):=E X^ 2_ 0 g(X_ 0)I(| X_ 0|\leq x) \] are slowly varying as \(x\to \infty\) and some additional conditions on \(\rho(n)\) are satisfied, where \(g\) is some increasing continuous function. The result improves the previous results of \textit{R. C. Bradley} [Ann. Probab. 16, No. 1, 313-332 (1988; Zbl 0643.60018)] and of the author [Chin. Ann. Math., Ser. B 10, 427-433 (1989; Zbl 0683.60023)].
scientific article; zbMATH DE number 205745