On the very weak 0-1 law for random graphs with orders

Let us draw a graph R on {0, 1, . . . , n - 1} by having an edge {i, j} with probability p_\i-j\, where ∑_i pi < ∞, and let M_n = (n, <, R). For a first-order sentence ψ let a_ψⁿ be the probability of M_n |= ψ. We know that the sequence a_ψ¹, a_ψ², . . . , a_ψⁿ, . . . does not necessarily converge. But here we find a weaker substitute which we call the very weak 0-1 law. We prove that lim_{n → ∞}(a_ψⁿ - a_ψⁿ⁺¹) = 0. For this we need a theorem on the (first-order) theory of distorted sum of models.