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Created by Knut M. Synstadfrom the Noun Project

Excursion Time

Definition (Excursion time)

Define the length of the rr-th excursion to state ii as the difference between passage times or Si(r)=Ti(r)−Ti(r−1)S_{i}^{(r)}=T_{i}^{(r)}-T_{i}^{(r-1)}for r≥1r\ge1.

Lemma (Excursion Time Lemma)

By strong markov property Si(2),Si(3),…S_{i}^{(2)},S_{i}^{(3)},\ldots are iid. With common distribution is distribution of Si(2)S_{i}^{(2)} conditional on X0=iX_{0}=i

Lemma (Excursion Time Lemma)

Assume X1=iX_{1}=i and PP recurrent, then:

  1. Si(2),Si(3),…S_{i}^{(2)},S_{i}^{(3)},\ldots are iid
  2. P(lim⁡r→∞1rTi(r)=mi)=1P\left(\lim_{r\to\infty} \frac{1}{r}T_{i}^{(r)}=m_{i}\right)=1 where mi=Ei[Ti(1)]m_{i}=E_{i}[T_{i}^{(1)}]