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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 Xh suitably normalized converges weakly to a non degenerate distribution. We consider in this paper the estimation of the normalization and of the limiting distribution.

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Acknowledgements

The research of Clemonell Bilayi-Biakana and Rafal Kulik was supported by the NSERC grant 210532-170699-2001. The research of Philippe Soulier was partially supported by the LABEX MME-DII.

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Correspondence to Rafał Kulik.

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

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.

  1. (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)
  2. (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 {Zn, i, 1 ≤ imn}, n ≥ 1, be a triangular array of rowwise i.i.d. processes by a class \(\mathcal {F}\). Define the random pseudometric dn 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 dn 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 {δn} 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 {δn} 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 ≤ imn, 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 dn 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:

  1. (i)

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

  2. (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\(\text {cst}_{\mathcal {G}} k\).

  3. (iii)

    For every k ≥ 1, there exists a measurable functionGksuch that for all\(f\in \widehat {\mathcal {G}}\), there exists\(f_{k}\in \widehat {\mathcal {G}}_{k}\)such that |ffk|≤ Gk.

  4. (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)
  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 Qn 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(Qn) 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 > Kn(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), Kn(𝜖) = OP(𝜖− 2ς). Thus, for ξ ∈ (0, 1), A0 can be chosen such that Kn(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). □

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Bilayi-Biakana, C., Kulik, R. & Soulier, P. Statistical inference for heavy tailed series with extremal independence. Extremes 23, 1–33 (2020). https://doi.org/10.1007/s10687-019-00365-z

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Keywords

  • Heavy tailed time series
  • Limiting conditional distribution
  • Statistical inference

AMS 2000 Subject Classifications

  • 62G32