Abstract
In dimension d ≥ 3, we present a general assumption under which the renewal theorem established by Spitzer (1964) for i.i.d. sequences of centered nonlattice r.v. holds true. Next we appeal to an operator-type procedure to investigate the Markov case. Such a spectral approach has been already developed by Babillot (Ann Inst Henri Poincaré, Sect B, Tome 24(4):507–569, 1988), but the weak perturbation theorem of Keller and Liverani (Ann Sc Norm Super Pisa CI Sci XXVIII(4):141–152, 1999) enables us to greatly weaken the moment conditions of Babillot (Ann Inst Henri Poincaré, Sect B, Tome 24(4):507–569, 1988). Our applications concern the v-geometrically ergodic Markov chains, the ρ-mixing Markov chains, and the iterative Lipschitz models, for which the renewal theorem of the i.i.d. case extends under the (almost) expected moment condition.
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