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

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

Hua, Loo-Keng