Introduction The aim of this note1 is to give a self-contained treatment of weak convergence of martingales and integrable additive functionals in general Harris recurrent Markov processes in continuous time. If a Harris process X = (Xt)to has a recurrent atom, then necessary and sufficient conditions for weak convergence of martingales associated to X have two components: first, either ergodicity of X or - in case of null recurrence - regular variation at infinity of tails of lifecycle length distributions (life cycles are excursions of the process between suitably defined successive visits to the atom) second, an integrability condition (with respect to invariant mesure) on the predictable quadratic variation. The norming functions are expressed in terms of tails of the lifecycle length distribution they vary regularly at infinity with some index 0 a 1. Limit processes are either Brownian motion (case a = 1), or Brownian motion subject to independent time change by a Mittag-Leffler process (the process inverse to a stable increasing process) of index 0 a 1. No other weak limits under linear scaling of time and suitable norming can occur. Brownian .motion in the limit does not characterize ergodicity of the process X, but arises also in a null recurrent case on the frontier to ergodicity. For general Harris processes, recurrent atoms and thus i.i.d life cycles for the process X do not exist. So we consider instead of X a family of Harris proces- ses (X m ) m where Xm for large m is very close to X, and where trajectories of Xm have from time to time flats of independent exponential length over which Nummelin splitting can be applied. In this way we get for every one of the processes Xm a recurrent atom, i.i.d life cycles and thus limit^theorems as above, for martingales and integrable additive functionals of X171. These limit theorems depend on m only through a set of constants which converge to a limit as m tends to infinity. In this way, we can deduce the desired limit theorem for martingales and Jntegrable additive functionals of X from the family of limit theorems for (X m ) m . Of course, since life cycles for Xm have been introduced artificially and are different at each stage m, we need an int- rinsic representation of the norming function for X-martingales: this intrinsic norming function is given in terms of regular variation at 0 of resolvants of X. This is a new look on a partially very old topic. A first famous paper on convergence of integrable additive functionals is by Darling and Kac in 1957 ([D-K 57], re-exposed in the book by Bingham, Goldie and Teugels [B-G-T 87]): they prove that under a 'Darling-Kac condition', norming functions for additive functionals of X are necessarily regularly varying, and limit laws (for 1 Received by the editor Dec. 20, 2000, rev. Aug. 30, 2001, accepted Sept. 4, 2001. 1
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