Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain
(Xn,n ≥
0) with invariant distribution μ. We shall investigate the
long time behaviour of some functionals of the chain, in particular the additive
functional \hbox{$S_{n}=\sum_{i=1}^{n}f(X_{i})$} for a possibly non square integrable function
f. To this
end we shall link ergodic properties of the chain to mixing properties, extending known
results in the continuous time case. We will then use existing results of convergence to
stable distributions, obtained in [M. Denker and A. Jakubowski, Stat. Probab.
Lett. 8 (1989) 477–483; M. Tyran-Kaminska, Stochastic
Process. Appl. 120 (2010) 1629–1650; D. Krizmanic, Ph.D. thesis
(2010); B. Basrak, D. Krizmanic and J. Segers, Ann. Probab. 40
(2012) 2008–2033] for stationary mixing sequences. Contrary to the usual
L^2 framework
studied in [P. Cattiaux, D. Chafai and A. Guillin, ALEA, Lat. Am. J. Probab. Math.
Stat. 9 (2012) 337–382], where weak forms of ergodicity are
sufficient to ensure the validity of the Central Limit Theorem, we will need here strong
ergodic properties: the existence of a spectral gap, hyperboundedness (or
hypercontractivity). These properties are also discussed. Finally we give explicit
examples.