On \(\rho\)-mixing except on small sets
Let \(\{X_ k\}_{k\in {\mathbb{Z}}}\) be a strictly stationary sequence on (\(\Omega\),\({\mathcal F},P)\) and \({\mathcal F}^ L_ J=\sigma (X_ j:\) \(J\leq j\leq L\}\), -\(\infty \leq J\leq L\leq \infty\). With \(\tau\) denoting one of the usual mixing coefficients \(\phi\) and \(\rho\), the condition of ``\(\tau\)-mixing except on small sets'' (ex.s.s.) for \(\{X_ k\}\) amounts to the existence of a sequence of events \({\mathcal D}_ n\in {\mathcal F}^ 0_{-\infty}\) such that \(\lim_{n\to \infty}P({\mathcal D}_ n)=1,\lim_{n\to \infty}\tau ({\mathcal F}^ 0_{-\infty},{\mathcal F}^{\infty}_ n;P(\cdot | {\mathcal D}_ n))=0.\) It is shown that the absolute regularity property coincides with ``\(\phi\)-mixing ex.s.s.'' while strong mixing for \(\{X_ k\}\) coincides with ``\(\rho\)-mixing ex.s.s.''. The conditions with ``exception on small sets'' were introduced in the theory of limit theorems by \textit{R. C. Bradley} and \textit{M. Peligrad} [Stochastic Processes Appl. 22, 271-289 (1986; Zbl 0609.60048)].
scientific article; zbMATH DE number 4213137