  It is the purpose of this paper to investigate the issue
  of estimating the regularity index $\beta>0$ of a discrete
  heavy-tailed r.v. $S$, \textit{i.e.} a r.v. $S$ valued
  in $\mathbb{N}^*$ such that $\mathbb{P}(S>n)=L(n)\cdot n^{-\beta}$
  for all $n\geq 1$, where $L:\mathbb{R}^*_+\to \mathbb{R}_+$
  is a slowly varying function. As a first go, we consider the
  situation where inference is based on independent copies
  $S_1,\; \ldots,\; S_n$ of the generic variable $S$. Just
  like the popular Hill estimator in the continuous
  heavy-tail situation, the estimator $\widehat{\beta}$ we
  propose can be derived by means of a suitable
  reformulation of the regularly varying condition,
  replacing $S$'s survivor function by its empirical
  counterpart. Under mild assumptions, a non-asymptotic
  bound for the deviation between $\widehat{\beta}$ and
  $\beta$ is established, as well as limit results
  (consistency and asymptotic normality). Beyond the
  i.i.d. case, the inference method proposed is extended
  to the estimation of the regularity index of a
  regenerative $\beta$-null recurrent Markov chain. Since
  the parameter $\beta$ can be then viewed as the tail
  index of the (regularly varying) distribution of the
  return time of the chain $X$ to any (pseudo-)
  regenerative set, in this case, the estimator is
  constructed from the successive regeneration times.
  Because the durations between consecutive regeneration
  times are asymptotically independent, we can prove that
  the consistency of the estimator promoted is preserved.
  In addition to the theoretical analysis carried out,
  simulation results provide empirical evidence of the
  relevance of the inference technique proposed.
