Extreme values statistics for Markov chains via the (pseudo-) regenerative method

This paper is devoted to the study of specific statistical methods for extremal events in the markovian setup, based on the regenerative method and the Nummelin technique. Exploiting ideas developed in Rootzén (Adv Appl Probab 20:371–390, 1988), the principle underlying our methodology consists of first generating a random number l of approximate pseudo-renewal times τ ₁, τ ₂, ..., τ _l for a sample path X ₁, ..., X _n drawn from a Harris chain X with state space E, from the parameters of a minorization condition fulfilled by its transition kernel, and then computing submaxima over the approximate cycles thus obtained: $\max_{1+\tau_1\leq i \leq \tau_2}f(X_i),\ldots ,\max_{1+\tau_{l-1}\leq i \leq \tau_l}f(X_i)This paper is devoted to the study of specific statistical methods for extremal events in the markovian setup, based on the regenerative method and the Nummelin technique. Exploiting ideas developed in Rootzén (Adv Appl Probab 20:371–390, 1988), the principle underlying our methodology consists of first generating a random number l of approximate pseudo-renewal times τ ₁, τ ₂, ..., τ _l for a sample path X ₁, ..., X _n drawn from a Harris chain X with state space E, from the parameters of a minorization condition fulfilled by its transition kernel, and then computing submaxima over the approximate cycles thus obtained: max_{1+t1 ￡ i ￡ t2}f(X_i),  ?,  max_{1+tl-1 ￡ i ￡ tl}f(X_i)\max_{1+\tau_1\leq i \leq \tau_2}f(X_i),\ldots ,\max_{1+\tau_{l-1}\leq i \leq \tau_l}f(X_i) for any measurable function f:E→ℝ. Estimators of tail features of the sample maximum max_{1 ≤ i ≤ n} f(X _i ) are then constructed by applying standard statistical methods, tailored for the i.i.d. setting, to the submaxima as if they were independent and identically distributed. In particular, the asymptotic properties of extensions of popular inference procedures based on the conditional maximum likelihood theory, such as Hill’s method for the index of regular variation, are thoroughly investigated. Using the same approach, we also consider the problem of estimating the extremal index of the sequence {f(X _n )} _{n ∈ ℕ} under suitable assumptions. Eventually, practical issues related to the application of the methodology we propose are discussed and preliminary simulation results are displayed.