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In this appendix, we recall the definition of and two limit theorems for the so-called density dependent Markov chains [48], [49]. First, we fix some notation. The set of integers (resp. real numbers) is denoted by Z (resp. R) The space of d-dimensional vectors with integer (resp. real) components is denoted by Z d (resp. R d ) The absolute value of a scalar b is denoted by |b|. The euclidean norm of a vector z is denoted by z. The transpose of a vector z (resp. a matrix G) is denoted by z T (resp. G T ) Consider a one-parameter family of continuous time Markov chains {Z (n) (t), t ? 0}, where {Z (n) (t)} has state space S (n) ? Z d and transition rate matrix Q (n) = [q (n) (Z, Z )] Z,Z ?S (n), 2002. ,