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A Central Limit Theorem for Punctuated Equilibrium

Current evolutionary biology models usually assume that a phenotype undergoes gradual change. This is in stark contrast to biological intuition, which indicates that change can also be punctuated-the phenotype can jump. Such a jump could especially occur at speciation, i.e. dramatic change occurs that drives the species apart. Here we derive a Central Limit Theorem for punctuated equilibrium. We show that, if adaptation is fast, for weak convergence to normality to hold, the variability in the occurrence of change has to disappear with time.