### 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 2001 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Journal of Statistical Physics*,

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

**Central limit theorems for nonlinear hierarchical sequences of random variables.** / Wehr, Jan; Woo, Jung M.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 104, no. 3-4, pp. 777-797. https://doi.org/10.1023/A:1010384806884

}

TY - JOUR

T1 - Central limit theorems for nonlinear hierarchical sequences of random variables

AU - Wehr, Jan

AU - Woo, Jung M.

PY - 2001/8

Y1 - 2001/8

N2 - Let a random variable x0 and a function f: [a, b]k → [a, b] be given. A hierarchical sequence {xn: n = 0, 1, 2,...} of random variables is defined inductively by the relation xn = f(xn-1, 1, xn-1, 2...., xn-1 k), where {xn-1, i: i = 1, 2,..., k} is a family of independent random variables with the same distribution as xn-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.

AB - Let a random variable x0 and a function f: [a, b]k → [a, b] be given. A hierarchical sequence {xn: n = 0, 1, 2,...} of random variables is defined inductively by the relation xn = f(xn-1, 1, xn-1, 2...., xn-1 k), where {xn-1, i: i = 1, 2,..., k} is a family of independent random variables with the same distribution as xn-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.

KW - Central limit theorem

KW - Hierarchical lattices

KW - Random resistor networks

KW - Renormalization group

UR - http://www.scopus.com/inward/record.url?scp=0035537484&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035537484&partnerID=8YFLogxK

U2 - 10.1023/A:1010384806884

DO - 10.1023/A:1010384806884

M3 - Article

VL - 104

SP - 777

EP - 797

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -