Statistical and Machine Learning Approaches for Network Analysis

7.4 Asymptotic Expansion via the Saddle Point Method

This section provides the tool that is used to infer asymptotic expansions. The result can be obtained by using a double saddle point approach, see, for example, Refs. [7, 16–19] for details concerning this method. Note that further coefficients of the asymptotic expansions can be calculated in the same way, but the expressions are so complicated that it does not make sense to provide them outside a computer algebra system. A maple worksheet is available on request from the author.

Lemma 7.4 [7] Let f(x, y) and g(x, y) be analytic functions locally around (x, y) = (0, 0) such that all coefficients are non-negative and that there exists M such that all indices (m₁, m₂) with m₁, m₂ ≥ M can be represented as a finite linear combination of the set with positive integers as coefficients.

Let R₁ and R₂ be compact intervals of the positive real line such that R = R₁ × R₂ is contained in the regions of convergence of f(x, y) and g(x, y). Furthermore set

Then, we have

uniformly for (m₁/k, m₂/k) S, where x₀ and y₀ are uniquely determined ...