  We study limit theorems for partial sums of instantaneous functions of a
homogeneous Markov chain on a general state space. The summands are
heavy-tailed and the limits are stable distributions. The conditions imposed on
the transition operator $P$ of the Markov chain ensure that the limit is the
same as if the summands were independent. Such a~scheme admits a physical
interpretation, as given in Jara et al. (Ann. Appl. Probab., 19 (2009),
2270--2300).
  We considerably extend the results of Jara et al., (ibid.) and Cattiaux and
Manou-Abi (ESAIM Probab. Stat., 18 (2014), 468--486). We show that the theory
holds under the assumption of operator uniform integrability in $L^2$ of $P$ (a
notion introduced by Wu (J. Funct. Anal., 172 (2000), 301--376)) plus the
$L^2$-spectral gap property. If we strengthen the uniform integrability in
$L^2$ to the hyperboundedness, then the $L^2$-spectral gap property can be
relaxed to the strong mixing at geometric rate (in practice: to geometric
ergodicity).
  We provide an example of a Markov chain on a countable space that is
uniformly integrable in $L^2$ (and admits an $L^2$-spectral gap), while it is
not hyperbounded. Moreover, we show by example that hyperboundedness is still a
weaker property than $\phi$-mixing, what enlarges the range of models of
interest.
  What makes our assumptions working is a new, efficient version of the
Principle of Conditioning that operates with conditional characteristic
functions rather than predictable characteristics.
