Limit Theorems for Competitive Density Dependent Population Processes

∣ = 0 respectively. We also use Hardy notation: f(N) ≪ g(N) if f(N) = o(g(N)). • Throughout, I will use ∂i to indicate the partial derivative with respect to the i th cooordinate and D to denote the total derivative operator: if F : R → R, (DF) = (∂jFi)ij • R+ = {x ∈ R K : xi ≥ 0, i = 1, . . . ,K}. • If x,y ∈ R , we write x ≤ y if xi ≤ yi for all i, x < y if x ≤ y and xi < yi for at least one i, and x ⊳ y if xi < yi for all i. • Given a Polish space Ω, DΩ[0,∞) denotes the space of Ω-valued càdlàg functions endowed with the Skorohod topology. • We use X D −→ X to denote convergence in distribution for a sequence {X (t)} of càdlàg stochastic processes,i.e. E [ f(X ) ]