Convergence law for random graphs with specified degree sequence

The degree sequence of an n-vertex graph is d₀, ..., d_{n - 1}, where each d_i is the number of vertices of degree i in the graph. A random graph with degree sequence d₀, ..., d_{n - 1} is a randomly selected member of the set of graphs on {0, ..., n - 1} with that degree sequence, all choices being equally likely. Let λ ₀, λ ₁, ... be a sequence of nonnegative reals summing to 1. A class of finite graphs has degree sequences approximated by λ₀, λ₁, ... if, for every i and n, the members of the class of size n have λ_i n + o(n) vertices of degree i. Our main result is a convergence law for random graphs with degree sequences approximated by some sequence λ₀, λ₁, .... With certain conditions on the sequence λ₀, λ₁, ..., the probability of any first-order sentence on random graphs of size n converges to a limit as n grows.