A generalization of a lemma of Kac

Abstract

Denote by $\tau (k,.):A \rightarrow {\mathbb {N}}$ the k-th return function for a set A with positive measure and a transformation $T:\Omega \rightarrow \Omega$ that preserves $\mu$. Kac’s lemma asserts that $\int _A \tau (1,.) = \mu (\Omega ) - \mu (E^{*})$. Where $E^{*}$ is the set of points that never enter A. We generalize this formula for arbitrary time of return:

$$\begin{aligned} \int \limits _A \tau (k,.) = k\big (\mu (\Omega ) - \mu (E^{*})\big ). \end{aligned}$$

We also prove that the distributions of $\tau (1,.)$ and $\tau (k,.)-\tau (k-1,.)$ are equal for arbitrary k.