### 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 |
---|---|

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

**Limit theorems for the nonattractive Domany-Kinzel model.** / Katori, Makoto; Konno, Norio; Tanemura, Hideki.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 30, no. 2, pp. 933-947. https://doi.org/10.1214/aop/1023481012

}

TY - JOUR

T1 - Limit theorems for the nonattractive Domany-Kinzel model

AU - Katori, Makoto

AU - Konno, Norio

AU - Tanemura, Hideki

PY - 2002/4/1

Y1 - 2002/4/1

N2 - We study the Domany-Kinzel model, which is a class of discrete time Markov processes with two parameters (p1, p2) ∈ [0, 1]2 and whose states are subsets of Z, the set of integers. When p1 = αβ and p2 = α(2β - β2) 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 ≤ p1 ≤ p2 ≤ 1, the complete convergence theorem is easily obtained. On the other hand, the case (p1, p2) = (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 (p1, p2) ∈ (1, 0). In particular, when p2 ∈ [0, 1] and p1 is close to 1, the complete convergence theorem is obtained as a corollary of the limit theorems.

AB - We study the Domany-Kinzel model, which is a class of discrete time Markov processes with two parameters (p1, p2) ∈ [0, 1]2 and whose states are subsets of Z, the set of integers. When p1 = αβ and p2 = α(2β - β2) 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 ≤ p1 ≤ p2 ≤ 1, the complete convergence theorem is easily obtained. On the other hand, the case (p1, p2) = (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 (p1, p2) ∈ (1, 0). In particular, when p2 ∈ [0, 1] and p1 is close to 1, the complete convergence theorem is obtained as a corollary of the limit theorems.

KW - Complete convergence theorem

KW - Limit theorem

KW - Nonattractive process

KW - The Domany-Kinzel model

UR - http://www.scopus.com/inward/record.url?scp=0036018199&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036018199&partnerID=8YFLogxK

U2 - 10.1214/aop/1023481012

DO - 10.1214/aop/1023481012

M3 - Article

AN - SCOPUS:0036018199

VL - 30

SP - 933

EP - 947

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 2

ER -