The Nagaev-Guivarc’h method via the Keller-Liverani theorem

Abstract

Abstract. -The Nagaev-Guivarc'h method, via the perturbation operator theorem of Keller and Liverani, has been exploited in recent papers to establish limit theorems for unbounded functionals of strongly ergodic Markov chains. The main difficulty of this approach is to prove Taylor expansions for the dominating eigenvalue of the Fourier kernels. The paper outlines this method and extends it by stating a multidimensional local limit theorem, a one-dimensional Berry-Esseen theorem, a first-order Edgeworth expansion, and a multidimensional Berry-Esseen type theorem in the sense of the Prohorov metric. When applied to the exponentially L 2 -convergent Markov chains, to the v-geometrically ergodic Markov chains and to the iterative Lipschitz models, the three first above cited limit theorems hold under moment conditions similar, or close (up to ε > 0), to those of the i.i.d. case. Résumé

Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

“…The required characteristic expansion is obtained in some cases using classical perturbation theory as in [AD01b], but other tools are also required in other cases: weak perturbation theory [KL99,GL06,HP08] and interpolation spaces [BL76]. Finally, Appendix A describes another application of our techniques, to the speed in the central limit theorem.…”

“…A very similar result has been proved in [GL06], but with slightly stronger assumptions that will not be satisfied in the forthcoming application to Gibbs-Markov maps (in particular, [GL06] requires (3.16) below to hold for 0 ≤ i < j ≤ N , instead of 1 ≤ i < j ≤ N ). Let us also mention [HP08] for related results.…”

A Renewal Theorem for Strongly Ergodic Markov Chains in Dimension d ≥ 3 and Centered Case

“…This moment condition is not only clearly stronger than the previous one, but actually it is not fulfilled in general when ξ is unbounded. A typical example is presented in [17,Section 3] for the usual linear autoregressive model and ξ(x) = x. Similar improvements concerning the moment conditions are obtained in Sections 4.2 and 4.3 for the two others (above cited) Markov models.…”

“…In this Markov context, we first present a general assumption providing an operator-type formula for the term E f (X n ) e i t,Sn of Hypothesis R(m). Next, we briefly compare the spectral method developed in [1] with that presented in [17].…”

“…This new approach, in which the KellerLiverani perturbation theorem [10,18,19] plays a central role, is fully described in [17] and applied to prove some usual refinements of the central limit theorem (CLT). It is used in [11] to establish a one-dimensional (non-centered) Markov renewal theorem.…”