It is proved that a kernel, doubly Markovian operator T is asymptotically periodic if and only if its deterministic \(\sigma \)-field \(\varSigma _d(T)\) (equivalently \(\varSigma _d(T^*)\)) is finite. It follows that kernel doubly Markovian operator T is asymptotically periodic if and only if \(T^*\) is asymptotically periodic.

中文翻译：

---

核双马尔可夫算子的渐近周期

事实证明，一个内核，双马尔可夫操作牛逼是渐近周期的当且仅当它的确定性\（\西格玛\） -场\（\ varSigma _D（T）\） （等同\（\ varSigma _D（T ^ *） \））是有限的。因此，当且仅当\（T ^ * \）是渐近周期性时，核双马氏算子T才是渐近周期性的。