### Abstract

We study the Domany-Kinzel model, which is a class of discrete time Markov processes with two parameters (p_{1}, p_{2}) ∈ [0, 1]^{2} and whose states are subsets of Z, the set of integers. When p_{1} = αβ and p_{2} = α(2β - β^{2}) with (α, β) ∈ [0, 1]^{2}, the process can be identified with the mixed site-bond oriented percolation model on a square lattice with the probabilities of open site a and of open bond β. For the attractive case, 0 ≤ p_{1} ≤ p_{2} ≤ 1, the complete convergence theorem is easily obtained. On the other hand, the case (p_{1}, p_{2}) = (1,0) realizes the rule 90 cellular automaton of Wolfram in which, starting from the Bernoulli measure with density θ, the distribution converges weakly only if θ ∈ {0, 1/2, 1}. Using our new construction of processes based on signed measures, we prove limit theorems which are also valid for nonattractive cases with (p_{1}, p_{2}) ∈ (1, 0). In particular, when p_{2} ∈ [0, 1] and p_{1} is close to 1, the complete convergence theorem is obtained as a corollary of the limit theorems.

Original language | English |
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Pages (from-to) | 933-947 |

Number of pages | 15 |

Journal | Annals of Probability |

Volume | 30 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2002 Apr 1 |

Externally published | Yes |

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### Keywords

- Complete convergence theorem
- Limit theorem
- Nonattractive process
- The Domany-Kinzel model

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*30*(2), 933-947. https://doi.org/10.1214/aop/1023481012