For some reason my thinking is very fuzzy today, so I apologize for the following rather silly question below...

Let $T$ be an ergodic transformation of $(X,\Omega, \mathbb{P})$ and let $X$ be partitioned into $n < \infty$ disjoint sets $R_j$ of positive measure. For $x \in R_k$ define $\tau(x) := \inf \{\ell>0:T^\ell x \in R_k\}$. The Kac lemma (see, e.g. http://arxiv.org/abs/math/0505625) gives that $\int_{R_k} \tau(x) \ d\mathbb{P}(x) = 1$. Now $\int_X \tau(x) \ d\mathbb{P}(x) = \sum_k \int_{R_k} \tau(x) \ d\mathbb{P}(x) = n$, or equivalently $\mathbb{E}\tau = n$.

Can anyone provide a sanity check on the above assertion that the expected return time is just the size of the partition? I've never seen this explicitly stated as a corollary of the Kac lemma, which seems odd.


I found a 2003 paper of Choe containing your sanity check, called "A universal law of logarithm of the recurrence time". See the first few lines of section 3 on page 888. The "$K_n$" used there is essentially your $\tau$, but corresponding to a partition $P_n$ in a sequence of partitions. See definition 1.7 on page 885 for what the notation means.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.