*Mathew Penrose*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506263
- eISBN:
- 9780191707858
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506263.003.0009
- Subject:
- Mathematics, Probability / Statistics

This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) ...
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This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) arguments are described for estimating the number of connected sets in the lattice, and elements of (lattice) percolation theory are described. A multiparameter ergodic theorem is given, and the basic theory of continuum percolation is described. Some of the theory of Poisson point processes are recalled, including the superposition, thinning, and scaling theorems.Less

This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) arguments are described for estimating the number of connected sets in the lattice, and elements of (lattice) percolation theory are described. A multiparameter ergodic theorem is given, and the basic theory of continuum percolation is described. Some of the theory of Poisson point processes are recalled, including the superposition, thinning, and scaling theorems.

*James Davidson*

- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.003.0013
- Subject:
- Economics and Finance, Econometrics

A number of probabilistic approaches to the concept of dependence in stochastic sequences are contrasted. The fundamental idea is a shift transformation. The notions of stationarity and (serial) ...
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A number of probabilistic approaches to the concept of dependence in stochastic sequences are contrasted. The fundamental idea is a shift transformation. The notions of stationarity and (serial) independence are defined, with examples. Next, invariant events and the idea of ergodicity enter, and the ergodic theorem is proved. The final sections deal with the notions of regularity and mixing.Less

A number of probabilistic approaches to the concept of dependence in stochastic sequences are contrasted. The fundamental idea is a shift transformation. The notions of stationarity and (serial) independence are defined, with examples. Next, invariant events and the idea of ergodicity enter, and the ergodic theorem is proved. The final sections deal with the notions of regularity and mixing.