Analysis of the Jante’s Law Process and Proof of Conjecture: Proof of Theorem 1

Table of Links

Abstract and Introduction

Preliminaries

Reduction to the case of uniform geometry

All original points are eventually removed, a. s.

Proof of Theorem 1

Coupling Y (⋅) and Z(⋅)

Acknowledgements and References

Appendix

5 Proof of Theorem 1

Without loss of generality, we assume that the initial state Z(0) is deterministic. This is harmless since if for each deterministic choice of Z(0) the limit z∞ exists a.s. and has an absolutely continuous distribution, then if instead Z(0) is random, z∞ still exists a.s. and its distribution is a mixture of absolutely continuous distributions, which is necessarily absolutely continuous.

7 i.e., a set defined by a number of polynomial inequalities and equalities; in our case, a.s. these will be just inequalities.