Orthogonal Decompositions and Functional Limit Theorems for Random Graph Statistics (Memoirs of the American Mathematical Society) by Svante Janson (PDF)

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This book develops a method to obtain limit theorems for various functionals of random graphs. The method is based on a certain orthogonal decomposition. Janson’s results include limit theorems for the two standard random graph models, $G_{n,p}$ and $G_{n,m}$, as well as functional limit theorems for the evolution of a random graph and results on the maximum of a function during the evolution. Janson obtains both normal and nonnormal limits, and the method provides an explanation for the appearance of nonnormal limits. Applications to subgraph counts and to vertex degrees are presented as examples.

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