Central limit theory for Peaks-over-Threshold partial sums of long memory linear time series

Abstract

Over the last 30 years, extensive work has been devoted to developing central limit theory for partial sums of subordinated long memory linear time series. A much less studied problem, motivated by questions that are ubiquitous in extreme value theory, is the asymptotic behavior of such partial sums when the subordination mechanism has a threshold depending on sample size, so as to focus on the right tail of the time series. This article substantially extends longstanding asymptotic techniques by allowing the subordination mechanism to depend on the sample size in this way and to grow at a polynomial rate, while permitting the innovation process to have infinite variance. The cornerstone of our theoretical approach is a tailored reduction principle, which enables the use of classical results on partial sums of long memory linear processes. In this way we obtain asymptotic theory for certain Peaks-over-Threshold estimators with deterministic or random thresholds. Applications cover both heavy- and light-tailed regimes, yielding unexpected results which, to the best of our knowledge, are new to the literature. A simulation study illustrates the relevance of our findings in finite samples.

Figures (4)

• Figure 1: Illustration of the convergence of pre-asymptotic scaling factors. Top: Values of $s_{n,d,\alpha}$ in \ref{['eq:scale-partialsums']} for $n\in \{10^3, 10^4, 10^5, 10^6\}$, $d=0.1$ and $\alpha=1.9$. Bottom: Values of $\alpha \mathbf{P}[X>u]/ (u f_{X}(u))$ for the unit $\alpha$-stable distribution along the levels $u=q_X(\tau)$ for $\tau\in [0.99,0.999]$, with $\alpha=1.9$.
• Figure 2: Histograms of $N=10{,}000$ realizations of the centered and rescaled exceedance counts in \ref{['eqn:PoT_rescaled']}. Top panels: raw realizations, middle panels: realizations rescaled by the parameter $\widehat{s}$ (of a stable distribution fitted to the partial sums), bottom panels: realizations rescaled by the parameter $\widetilde{s}$ (of a stable distribution fitted directly to these raw realizations). The length of the time series $(X_t)$, generated with $a_j=j^{-(1-d)}$ and symmetric unit $\alpha$-stable innovations (for $d=0.1$ and $\alpha=1.9$), is $n=10^k$ with $k=3$ (left panels), $5$ (middle panels), $7$ (right panels), with $u_n=1-n^{-1/5} = 1.38$, $2.67$, $3.74$.
• Figure 3: Histograms of $N=10{,}000$ realizations of the centered and rescaled Hill estimator in \ref{['eqn:Hill_rescaled']}. Top panels: raw realizations, middle panels: realizations rescaled by the parameter $\widehat{s}$ (of a stable distribution fitted to the partial sums), bottom panels: realizations rescaled by the parameter $\widetilde{s}$ (of a stable distribution fitted directly to these raw realizations). The length of the time series $(X_t)$, generated with $a_j=j^{-(1-d)}$ and symmetric unit $\alpha$-stable innovations (for $d=0.1$ and $\alpha=1.9$), is $n=10^k$ with $k=3$ (left panels), $5$ (middle panels), $7$ (right panels), with $u_n=1-n^{-1/5} = 1.38$, $2.67$, $3.74$.
• Figure 4: General position of the curves representing the functions $\gamma_1$ (blue dotted line), $\gamma_2$ (orange dotted line) and $\gamma_3$ (green dotted line) in the cases $\alpha(1-d)(1-2d)> 1$ (left panel) and $\alpha(1-d)(1-2d)\leq 1$ (right panel). The purple curve represents the function $\gamma_1\wedge \gamma_2 \wedge \gamma_3$.

Theorems & Definitions (49)

• Remark 2.1: On Assumption \ref{['asu:coef']}
• Remark 2.2: On Assumption \ref{['asu:G_n']}
• Remark 2.3: On Assumption \ref{['asu:f']}(i)
• Remark 2.4: On Assumption \ref{['asu:f']}(ii)
• Theorem 3.1: Central limit theorems for partial sums
• Remark 3.2: On the limiting distribution in Theorem \ref{['thm:clt_partsum']}
• Theorem 3.3: $L^r-$moment bound
• Remark 3.4: On the existence of $\gamma$ and $r$ in Theorem \ref{['thm:approx']}
• Theorem 3.5: Optimal $L^r-$moment bound
• Corollary 3.6: Central limit theorem