### Abstract

Let a random variable x_{0} and a function f: [a, b]^{k} → [a, b] be given. A hierarchical sequence {x_{n}: n = 0, 1, 2,...} of random variables is defined inductively by the relation x_{n} = f(x_{n-1, 1}, x_{n-1, 2}...., x_{n-1 k}), where {x_{n-1, i}: i = 1, 2,..., k} is a family of independent random variables with the same distribution as x_{n-1}. We prove a central limit theorem for this hierarchical sequence of random variables when a function f satisfies a certain averaging condition. As a corollary under a natural assumption we prove a central limit theorem for a suitably normalized sequence of conductivities of a random resistor network on a hierarchical lattice.

Original language | English (US) |
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Pages (from-to) | 777-797 |

Number of pages | 21 |

Journal | Journal of Statistical Physics |

Volume | 104 |

Issue number | 3-4 |

DOIs | |

State | Published - Aug 1 2001 |

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### Keywords

- Central limit theorem
- Hierarchical lattices
- Random resistor networks
- Renormalization group

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*104*(3-4), 777-797. https://doi.org/10.1023/A:1010384806884