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.

翻译：本文旨在研究离散重尾随机变量$S$的正则性指数$\beta>0$的估计问题，即取值于$\mathbb{N}^*$且满足$\mathbb{P}(S>n)=L(n)\cdot n^{-\beta}$（对所有$n\geq 1$成立）的随机变量$S$，其中$L:\mathbb{R}^*_+\to \mathbb{R}_+$为缓变函数。首先考虑基于通用变量$S$的独立复制样本$S_1,\; \ldots,\; S_n$进行推断的情形。与连续重尾情形中经典的Hill估计量类似，我们提出的估计量$\widehat{\beta}$可通过适当重构正则变化条件得到，即将$S$的生存函数替换为其经验对应函数。在温和假设下，建立了$\widehat{\beta}$与$\beta$之间偏差的非渐近界，并获得了极限结果（相合性与渐近正态性）。超越独立同分布情形，所提出的推断方法可扩展用于估计再生$\beta$-零常返马尔可夫链的正则性指数。由于此时参数$\beta$可视为链$X$到任意（伪）再生集的返回时间（服从正则变化分布）的尾指数，故该情形下的估计量是基于连续再生时间构造的。鉴于相继再生时间间隔具有渐近独立性，我们可以证明所提估计量的相合性得以保持。除理论分析外，仿真结果也为所提推断技术的适用性提供了实证依据。