Sur un processus de vie et de mort de particules sur [0,1]. (On a birth and death process of particles on [0,1]). (French) Zbl 0624.60095

Robert, Philippe

Ann. Inst. Henri Poincaré, Probab. Stat. 23, 225-235 (1987).

At each unit of time, \(n=1,2,...\), a particle is created at random on [0,1]. With probability p, it stays on [0,1], otherwise this particle together with the first particle to its right, if any, are annihilated.
The author studies the asymptotic distribution of the particles as \(n\to \infty\). If \(p\geq\), the particles become uniformly dense in [0,1] with density (2p-1), so that for large n the expected number of particles in [a,b] is approximately n(b-a)(2p-1).
If \(0<p<\), the process converges to a stationary state characterized as follows: the number of particles in an interval [a,b], \(a>0\), is finite with probability one, and has expectation \(\alpha\) log(b/a), for \(\alpha >0\) constant; 0 is an accumulation point of particles, and if the point process of particles has a change of time-scale, \(t\to \log (1/t)\), \(0<t\leq 1\), it becomes a stationary point process on [0,\(\infty)\).

Reviewer: D.P.Kennedy

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60G57	Random measures
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)

Keywords:

asymptotic distribution; accumulation point of particles; point process of particles; change of time-scale; stationary point process