An expansion for self-interacting random walks

We derive a perturbation expansion for general interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the true (weakly) self-avoiding walk and loop-erased random walk. We use the expansion to prove a law of large numbers and central limit theorem for two models: (i) A directed version of once-reinforced random walk on Zd for sufficiently small reinforcement parameters. This model is such that if the reinforcement parameter is set to zero, then the resulting random walk has independent increments with a non-zero drift; and (ii) Excited random walk in dimension d > 8 when the excitement parameter is sufficiently small.