“Markov chain must have a beginning”. In memory of Prof. Loo-Keng Hua. (English) Zbl 0632.60064

Chung, K. L.

J. Math. Res. Expo. 6, No. 1, 1-3 (1986).

Let W denote the class of uniformly bounded r-tuples (generalizing \(V= class\) of r-valued probability distributions), and let \(P=(p_{ij})_{1\leq i,j\leq r}\) be a finite stochastic matrix. Clearly \(P: W\to W\), and \(v\in W\) is said to have a history of length n in W, if there exists \(u\in W\) such that \(P^ nu=v\). Further v is said to be periodic with period d under P if \(vP^ d=v.\)
Proposition 1 states that there exists an integer d (period) with the following property. If \(v\in W\) and if v has an arbitrarily long history in W then v is periodic with period d. A similar (periodless) continuous time version is given in Proposition 2. The reader will find this note probably tantalizing. What happens if \(P^{-1}\) exists? Why is step (6) (there may be an omission) correct, and how does this relate to the interpretation of d as the lcm of the periods of positive recurrent states in the \(W=V\) case?
It must be added that there are several misprints and syntax error in this paper, and that the author’s style, so well-known also for his clarity, is here not easy to recognize. The note, written in honour of his late former teacher Loo-Keng Hua, was written and edited under time pressure.

Reviewer: F.Th.Bruss

MSC:

60J10

Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

stochastic matrix; periodic; positive recurrent states