Statistical inference for heavy tailed series with extremal independence

Abstract

We consider stationary time series $\{X_{j},j\in \mathbb {Z}\}$ whose finite dimensional distributions are regularly varying with extremal independence. We assume that for each h ≥ 1, conditionally on X₀ to exceed a threshold tending to infinity, the conditional distribution of X_h suitably normalized converges weakly to a non degenerate distribution. We consider in this paper the estimation of the normalization and of the limiting distribution.

Appendix: Convergence in \(\ell ^{\infty }\)

Theorem A.1

Giné and Nickl (2016, Theorem 3.7.23) Let$\{\mathbb {Z}_{n},n \in \mathbb {N}\}$, be a sequence of processes with values in$\ell ^{\infty }(\mathcal {F})$indexed by a semi-metric space${\mathcal {F}}$. Then the following statements are equivalent.

• (i) The finite dimensional distributions of the processes$\mathbb {Z}_{n}$converge in law and there exists a pseudometricρon$\mathcal {F}$such that$(\mathcal {F}, \rho )$is totally bounded and for all𝜖 > 0,

  $$ \lim_{\delta\to0} \limsup_{n\to\infty} \mathbb P^{*}\left( \sup_{\rho(f,g)<\delta} |\mathbb{Z}_{n}(f)-\mathbb{Z}_{n}(g)| > \epsilon\right) = 0 . $$

  (A.1)

• (ii) There exists a process$\mathbb {Z}$whose law is a tight Borel probability measure on$\ell ^{\infty }(\mathcal {F})$and such that$\mathbb {Z}_{n} \overset {{w}}{\Longrightarrow } \mathbb {Z}$in$\ell ^{\infty }(\mathcal {F})$.

Moreover, if (i) holds, then the process$\mathbb {Z}$ in (ii) has a version with bounded uniformly continuous paths for ρ.

The following result provides a sufficient condition for (A.1) above. Let {Z_{n, i}, 1 ≤ i ≤ m_n}, n ≥ 1, be a triangular array of rowwise i.i.d. processes by a class $\mathcal {F}$. Define the random pseudometric d_n on $\mathcal {F}$ by

$$ \begin{array}{@{}rcl@{}} {d_{n}^{2}}(f,g) = \sum\limits_{i=1}^{m_{n}} \{Z_{n,i}(f)-Z_{n,i}(g)\}^{2} , \ \ f,g\in \mathcal{F} . \end{array} $$

Let $N(\epsilon ,\mathcal {F},d_{n})$ be the minimum number of balls with radius 𝜖 in the pseudometric d_n needed to cover $\mathcal {F}$. Let $\mathbb {Z}_{n}$ be the empirical process defined by

$$ \begin{array}{@{}rcl@{}} \mathbb{Z}_{n}(f) = \sum\limits_{i=1}^{m_{n}} \{Z_{n,i}(f)-\mathbb E[Z_{n,i}(f)]\} , \ \ f \in \mathcal{F} \end{array} $$

Define $\|H\|_{\mathcal {F}} = \sup _{f\in \mathcal {F}} |H(f)|$ for a functional H on $\mathcal {F}$.

Theorem A.2 (Adapted from van der Vaart and Wellner 1996, Theorem 2.11.1)

Assume that the stochastic processes$\{{Z}_{n,i}(f),f\in \mathcal {F}\}$, $i=1,\dots ,m_{n}$, n ≥ 1, are separable and that the pseudometric space$\mathcal {F}$is totally bounded. Assume moreover that for all ζ > 0,

$$ \lim\limits_{n\to\infty} {m_{n}} \mathbb E[\|Z_{n,1}\|^{2}_{\mathcal{F}}\mathbbm{1}{\left\{\|Z_{n,1}\|_{\mathcal{F}}>\zeta\right\}}] = 0 . $$

(2)

Assume that for every sequence {δ_n} which decreases to zero,

$$ \lim\limits_{n\to\infty} \sup_{f,g\in\mathcal{F}\atop \rho(f,g)\leq\delta_{n}} {\mathbb E[d_{n}^{2}}(f,g)] = 0 , $$

(3)

Assume finally that there exists a measurable majorant$N^{*}(\epsilon ,\mathcal {F},d_{n})$ of $N(\epsilon ,\mathcal {F},d_{n})$such that for every sequence {δ_n} which decreases to zero,

$$ {\int}_{0}^{\delta_{n}} \sqrt{\log N^{*}(\epsilon,\mathcal{F},d_{n})} \mathrm{d} \epsilon \stackrel{\mathbb P}{\longrightarrow} 0 . $$

(4)

Then$\mathbb {Z}_{n}$is asymptotically ρ-equicontinuous, i.e. Eq. A.1holds.

Condition (4) holds if $\mathcal {F}$ is linearly ordered. We now provide a sufficient condition for (4) when the class $\mathcal {F}$ is approximable by subclasses with finite VC-dimension which possibly increase at a certain rate. We consider a triangular array of independent random elements $\mathbb {X}_{n,i}$, 1 ≤ i ≤ m_n, in a measurable space $(\mathsf {E},\mathcal {E})$ and assume that $Z_{n,i} = v_{n}^{-1/2}f(\mathbb {X}_{n,i})$. We consider a class $\widehat {\mathcal {G}}$ of functions on E and the random semi-metric d_n on $\widehat {\mathcal {G}}$ defined by

$$ \begin{array}{@{}rcl@{}} {d_{n}^{2}}(f,g) = v_{n}^{-1}\sum\limits_{i=1}^{m_{n}} \{f(\mathbb{X}_{n,i}) - g(\mathbb{X}_{n,i})\}^{2} . \end{array} $$

The following result formalizes ideas which can be found in the proof of the result in Drees and Rootzén (2010, Example 4.4).

Lemma A.3

Let$\{\widehat {\mathcal {G}}_{k},k\geq 1\}$be an a non-decreasing sequence of subclasses of$\widehat {\mathcal {G}}$. Assume that:

• (i) The envelope function Gof$\widehat {\mathcal {G}}$is measurable.

• (ii) There exists a constant$\text {cst}_{\mathcal {G}}$such that for every$k\in \mathbb {N}^{*}$, $\widehat {\mathcal {G}}_{k}$is VC-subgraph class with index$\text {VC}(\widehat {\mathcal {G}}_{k})$not greater than$\text {cst}_{\mathcal {G}} k$.

• (iii) For every k ≥ 1, there exists a measurable functionG_ksuch that for all$f\in \widehat {\mathcal {G}}$, there exists$f_{k}\in \widehat {\mathcal {G}}_{k}$such that |f − f_k|≤ G_k.

• (iv) There exists$\theta \in (0,\infty )$such that

  $$ \frac{4}{v_{n}}\sum\limits_{i=1}^{m_{n}} G^{2}(\mathbb{X}_{n,i}) \overset{\mathbb{P}}{\longrightarrow} \theta . $$

  (5)

• (v) There exists xς ∈ (0, 1) such that

  $$ \frac{1}{v_{n}}\sum\limits_{i=1}^{m_{n}} {G_{k}^{2}}(\mathbb{X}_{n,i}) = O_{P}(k^{-1/\varsigma}) . $$

  (6)

Then Eq. 4holds.

Proof

Define the (random) probability measure Q_n on E by

$$ \begin{array}{@{}rcl@{}} Q_{n} = \frac1{m_{n}} \sum\limits_{i=1}^{m_{n}} \delta_{\mathbb{X}_{n,i}} . \end{array} $$

Define the L²(Q_n) distance on $\widehat {\mathcal {G}}$ by

$$ \begin{array}{@{}rcl@{}} d_{L^{2}(Q_{n})}^{2} (f,g) = m_{n}^{-1}\sum\limits_{i=1}^{m_{n}} \{f(\mathbb{X}_{n,i})-g(\mathbb{X}_{n,i})\}^{2} . \end{array} $$

For 𝜖 > 0, define

$$ \begin{array}{@{}rcl@{}} K_{n}(\epsilon) & =& \min\left\{k\in\mathbb{N}: \frac{4}{v_{n}}\sum\limits_{i=1}^{m_{n}} {G_{k}^{2}}(\mathbb{X}_{n,i}) < \frac{\epsilon^{2}}2 \right\} . \end{array} $$

Then, for $f,g\in \widehat {\mathcal {G}}$ and k > K_n(𝜖), we have by Assumption (iii) of our lemma,

$$ \begin{array}{@{}rcl@{}} {d_{n}^{2}}(f,g) & \leq& 2{d_{n}^{2}}(f_{k},g_{k}) + \frac{4}{v_{n}}\sum\limits_{i=1}^{m_{n}} {G_{k}^{2}}(\mathbb{X}_{n,i}) \leq \frac{2m_{n}}{v_{n}} d_{L^{2}(Q_{n})}^{2}(f_{k},g_{k}) + \frac{\epsilon^{2}}2 . \end{array} $$

This bound implies that

$$ N(\widehat{\mathcal{G}},d_{n},\epsilon) \leq N\left( \mathcal{G}_{K_{n}(\epsilon)},d_{L^{2}(Q_{n})},\epsilon \left( \frac{v_{n}}{4m_{n}}\right)^{1/2}\right) + 1 . $$

(A.7)

Set

$$ \begin{array}{@{}rcl@{}} {\zeta_{n}^{2}} = \frac{4m_{n}}{v_{n}} Q_{n}(G^{2}) = \frac{4}{v_{n}}\sum\limits_{i=1}^{m_{n}} G^{2}(\mathbb{X}_{n,i}) \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} J_{k}(\epsilon) = \text{VC}(\widehat{\mathcal{G}}_{k})(16\mathrm{e})^{\text{VC}(\widehat{\mathcal{G}}_{k})} \epsilon^{-(2(\text{VC}(\widehat{\mathcal{G}}_{k})-1))} . \end{array} $$

Since $\widehat {\mathcal {G}}_{k}\subset \widehat {\mathcal {G}}$, the envelope function of $\widehat {\mathcal {G}}_{k}$ is smaller than G. Thus, by Theorem 2.6.7 in van der Vaart and Wellner (1996) we obtain for each k

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!N\left( \widehat{\mathcal{G}}_{k},d_{L^{2}(Q_{n})},\epsilon \!\left( \frac{v_{n}}{4m_{n}}\right)^{1/2}\right) \!& \leq& \text{cst} J_{k}\left( \epsilon\sqrt{\frac{v_{n}}{4m_{n} Q_{n}(G^{2})}}\right) \\ & \leq &\text{cst} J_{k}(\epsilon \zeta_{n}^{-1/2}) \leq \text{cst} J_{k}(\epsilon) (\zeta_{n}\vee1)^{\text{VC}(\widehat{\mathcal{G}}_{k})} \!. \end{array} $$

(A.8)

Combining (A.7) and (A.8) yields

$$ \begin{array}{@{}rcl@{}} \log N(\widehat{\mathcal{G}},d_{n},\epsilon) \leq \text{cst} + \log J_{K_{n}(\epsilon)}(\epsilon)+\text{VC}(\widehat{\mathcal{G}}_{K_{n}(\epsilon)}) \log(\zeta_{n}\vee1) . \end{array} $$

By Eq. 5, $\log (\zeta _{n}\vee 1)= O_{P}(1)$, thus we need to prove that for all ζ > 0,

$$ \begin{array}{@{}rcl@{}} \lim\limits_{\delta\to0} \limsup\limits_{n\to\infty} \mathbb P\left( {\int}_{0}^{\delta}\sqrt{\log J_{K_{n}(\epsilon)}(\epsilon)} \mathrm{d} \epsilon > \zeta\right) = 0 , \end{array} $$

(A.9a)

$$ \begin{array}{@{}rcl@{}} \lim\limits_{\delta\to0} \limsup\limits_{n\to\infty} \mathbb P\left( {\int}_{0}^{\delta}\sqrt{\text{VC}(\widehat{\mathcal{G}}_{K_{n}(\epsilon)})} \mathrm{d} \epsilon > \zeta\right) = 0 . \end{array} $$

(A.9b)

By assumption (6), K_n(𝜖) = O_P(𝜖^− 2ς). Thus, for ξ ∈ (0, 1), A₀ can be chosen such that K_n(𝜖) ≤ A₀𝜖^− 2ς with probability greater than 1 − ξ. Since ς ∈ (0, 1) and $\text {VC}(\widehat {\mathcal {G}}_{k})=O(k)$ by Assumption (ii), this yields, with probability tending to 1,

$$ \begin{array}{@{}rcl@{}} {\int}_{0}^{\delta}\sqrt{\text{VC}(\widehat{\mathcal{G}}_{K_{n}(\epsilon)})} \mathrm{d} \epsilon \leq \text{cst} {\int}_{0}^{\delta} \epsilon^{-\varsigma} \mathrm{d} \epsilon = O(\delta^{1-\varsigma}) . \end{array} $$

Similarly, with probability tending to 1, we have

$$ \begin{array}{@{}rcl@{}} {\int}_{0}^{\delta} \sqrt{\log J_{K_{n}(\epsilon)}(\epsilon)} \mathrm{d}\epsilon = O(\delta^{1-\varsigma}) . \end{array} $$

This proves (A.9a,b). □