Peak-over-threshold estimators for spectral tail processes: random vs deterministic thresholds

Abstract

The extreme value dependence of regularly varying stationary time series can be described by the spectral tail process. Drees et al. (Extremes 18(3), 369–402, 2015) proposed estimators of the marginal distributions of this process based on exceedances over high deterministic thresholds and analyzed their asymptotic behavior. In practice, however, versions of the estimators are applied which use exceedances over random thresholds like intermediate order statistics. We prove that these modified estimators have the same limit distributions. This finding is corroborated in a simulation study, but the version using order statistics performs a bit better for finite samples.

Appendices

Appendix A: Tables

Table 1 Approximate theoretical quantiles $F^{\leftarrow }(\beta )$, for β = 90% or 95% (with estimated standard deviations in parentheses). In the copula model $F^{\leftarrow }(\beta )$ is completely determined by the marginal t₄ distribution

Table 2 Probabilities P{Θ₁ > 1} and P{Θ₁ > 1/2} in each model

Table 3 Estimated p_β(x) and e_β(x) for x ∈{1,1/2} (with estimated standard deviations in parentheses)

Appendix B: Stochastic recurrence equations

Consider the stochastic recurrence equation

$$ X_{t}=C_{t}X_{t-1}+D_{t},\quad t\in\mathbb{Z}, $$

(B.1)

where (C_t, D_t), $t\in \mathbb {Z}$, is a sequence of iid $[0,\infty )^{2}$-valued random variables. It is well known that there exists a unique strictly stationary causal solution, provided $E[\log C_{1}] <0$ and $E[\log ^{+} D_{1}] <\infty$ (Basrak et al. 2002, Cor. 2.2). In addition, assume that the distribution of C₁ is not concentrated on a lattice and that there exists α > 0 such that $E[C_{1}^{\alpha }] =1$, $E[ C_{1}^{\alpha } \log ^{+} C_{1} ] < \infty$ and $E[ D_{1}^{\alpha }] < \infty$. Then the time series is regularly varying with index α (Basrak et al. 2002, Rem. 2.5, Cor. 2.6). Its tail process is denoted by $(Y_{t})_{t\in \mathbb {Z}}$.

Drees et al. (2015) have shown that condition (B) holds for suitably chosen (logarithmically increasing) r_n, provided $(\log n)^{2}/n=o(v_{n})$ and $v_{n}=o(1/\log n)$, and that a milder version of condition (C) is satisfied. Here, we will show that our strengthened condition (C) is fulfilled, too, if we assume in addition that $r_{n}^{1+2\delta }v_{n}=O(1)$.

Let ${\varPi }_{i,j}:={\prod }_{l=i}^{j}C_{l}$ and $V_{i,j}:={\sum }_{l=i}^{j}{\varPi }_{l+1,j}D_{l}$. Iterating (B.1) yields $X_{k}={\prod }_{j+1,k}X_{j}+V_{j+1,k}$. Define u_{n, ε} = (1 − ε)u_n and v_{n, ε} := P{X₀ > u_{n, ε}}.

Consider $g:\mathbb {R}\to \mathbb {R}$ with $0\leqslant g(x)\leqslant ax^{\tau }+b$ for some $a,b\geqslant 0$, τ ∈ (0, α) and all $x\in \mathbb {R}$. Using the Potter bounds, one can show that the random variables , are uniformly integrable. Hence, for sufficiently large u,

(B.2)

Under the above conditions, one has $\rho :=E[C_{1}^{\xi }]<1$ for any ξ ∈ (0, α). Thus, by the generalized Markov inequality and the independence of the random variables C_l,

$$ P\{\varPi_{j+1,k}>u_{n,\varepsilon}/(2t)\} \leqslant \rho^{k-j}(2t/u_{n,\varepsilon})^{\xi}. $$

(B.3)

Inequality B.2, $V_{1,k}\leqslant X_{k}$ and the Potter bounds imply

for all $k\in \mathbb {N}$ and sufficiently large n. Moreover, it was shown in (Drees et al. 2015, Example A.3) that there exists a constant c > 0 such that

$$ \begin{array}{@{}rcl@{}} P\{\min\{X_{0},X_{k}\}>u\}& \leqslant& cP\{X_{0}>u\}(P\{X_{0}>u\}+\rho^{k}), \end{array} $$

(B.6)

$$ \begin{array}{@{}rcl@{}} P\{\varPi_{1,k}X_{0}>u/2,X_{0}>u\} & \leqslant& 2^{\xi+1} \rho^{k} E[Y_{0}^{\xi}] P\{X_{0}>u\} \end{array} $$

(B.7)

for all $k\in \mathbb {N}$ and all u sufficiently large.

By independence of (V_{j+ 1, k}, π_{j+ 1, k}) and (X₀, X_j), one has

where in the last step we have used Inequalities B.3, B.5 and B.6. Using (a + b)^ξ ≤ 2^ξ(a^ξ + b^ξ) for all a, b > 0, Inequalities B.4, B.7, $V_{1,j}\leqslant X_{j}$ and the independence of X₀ and V_{1, j}, we can bound the last expected value as follows:

To sum up, we have shown that

$$ \begin{array}{@{}rcl@{}} P\!\!\!\!\!&&\{\min\{X_{0},X_{j},X_{k}\}>u_{n,\varepsilon}\}\\ &&\leqslant c2^{1+\alpha}v_{n,\varepsilon}^{2}(v_{n,\varepsilon}+\rho^{j})+\rho^{k-j}2^{2\xi+1}E[Y_{0}^{\xi}]v_{n,\varepsilon} ((2^{\alpha}+1)\rho^{j}+2^{\alpha}v_{n,\varepsilon}). \end{array} $$

This yields

$$ P(\min\{X_{j},X_{k}\}>u_{n,\varepsilon}\ |\ X_{0}>u_{n,\varepsilon})\leqslant C(v_{n,\varepsilon}^{2}+v_{n,\varepsilon}(\rho^{j}+\rho^{k-j})+\rho^{k})=:\tilde s_{n}(j,k) $$

for a suitable constant C > 0. Now, note that $\tilde s_{n}(j,k)\to C\rho ^{k}=:\tilde s_{\infty }(j,k)$ for all $j\leqslant k$, and

$$ \begin{array}{@{}rcl@{}} \sum\limits_{1\leqslant j\leqslant k\leqslant r_{n}}\tilde s_{n}(j,k)&=&C(\frac{r_{n}(r_{n}+1)}{2}v_{n,\varepsilon}^{2}+v_{n,\varepsilon}\sum\limits_{l=0}^{r_{n}-1}(r_{n}-l)\rho^{l}\\ &&+v_{n,\varepsilon}\sum\limits_{j=1}^{r_{n}}(r_{n}-j+1)\rho^{j}+ \sum\limits_{k=1}^{r_{n}}k\rho^{k})\\ &\to & C \sum\limits_{k=1}^{\infty} k\rho^{k}= \sum\limits_{1\leqslant j\leqslant k<\infty}\tilde s_{\infty}(j,k)<\infty, \end{array} $$

because r_nv_{n, ε} → 0. Thus, condition (2.10) is fulfilled.

Next, we verify condition (2.9) which is equivalent to

$$ \begin{array}{@{}rcl@{}} \lim_{L\to\infty}\limsup_{n\to\infty} \sum\limits_{j=L+1}^{r_{n}}E[\psi(u_{n,\varepsilon}^{-1}X_{0})\psi(u_{n,\varepsilon}^{-1}X_{k})\ |\ X_{0}>u_{n,\varepsilon}]=0 \end{array} $$

(B.8)

for . This can be done by direct calculations, but here we give a more elegant proof using general results for Markov processes under the additional assumptions that the time series is aperiodic and irreducible. This is e.g. true if (C₁, D₁) is absolutely continuous; see Buraczewski et al. (2016), Proposition 2.2.1 and Lemma 2.2.2. According to Lemma 4.3 of Kulik et al. (2019), convergence (B.8) holds when conditions (i)–(vi) in Assumption 2.1 of this paper are fulfilled. Since $(X_{t})_{t\in \mathbb {Z}}$ is a regularly varying Markov chain, condition (i) and (ii) are trivial. The arguments given in subsection 5.2 of Mikosch and Wintenberger (2013) show that the Lyapunov drift condition (iii) holds with V (x) = 1 + |x|^p for any p ∈ (0, α). The small set condition (iv) follows from subsection 2.2 of this paper in combination with Theorem 9.4.10 and Corollary 14.1.6 of Douc et al. (2018). With the above choice of V, condition (v) is obvious. Using Ineq. B.2 and the Potter bounds one may conclude, for all s > 0,

for some η with arbitrarily small modulus (η is positive when s > 1 and negative for s ∈ (0, 1)) and sufficiently large C > 0, $n\in \mathbb {N}$. Thus

such that condition (vi) is also satisfied. Hence, condition (2.9) is fulfilled.

It remains to prove Eq. 2.11. Since to all p > 0 there exists c_p > 0 such that , it suffices to show that, for some $p,\tilde p>0$,

(B.9)

is bounded. By induction, one can conclude from the drift condition that to all p ∈ (0, α) there exist β ∈ (0, 1) and b > 0 such that $E[{X_{k}^{p}}\mid X_{0}=y]\leqslant \beta ^{k} y^{p}+b/(1-\beta )$ (Douc et al. 2018, Prop. 14.1.8). Hence

for sufficiently large n, by Ineq. B.2, provided $p+\tilde p<\alpha$. Choose p ∈ (α(1 + δ)/(1 + 2δ), α) and $\tilde p>0$ sufficiently small such that $p+\tilde p<\alpha$. Then Eq. B.9 can be bounded by a multiple of ${\sum }_{k=1}^{r_{n}} \beta ^{k/(1+\delta )}+r_{n} u_{n,\varepsilon }^{-p/(1+\delta )}$. By the regular variation of X₀ with index α and the choice of p, one has $u_{n,\varepsilon }^{-p/(1+\delta )}=o(v_{n,\varepsilon }^{1/(1+2\delta )})$. Thus, term B.9 is bounded if $r_{n}^{1+2\delta }v_{n,\varepsilon }$ is bounded.

Appendix C: Proofs

Lemma C.1

If condition (2.10) from (C) holds, then

Proof

Let v_{n, ε} = P{X₀ > (1 − ε)u_n}. By regular variation and stationarity of $(X_{t})_{t\in \mathbb {Z}}$

□

Taking up the notation of Drees and Rootzén (2010), we consider the empirical process $\tilde {Z}_{n}$ defined by

$$ \tilde{Z}_{n}(\psi):=(nv_{n})^{-1/2}\sum\limits_{i=1}^{n}(\psi(X_{n,i})-E[\psi(X_{n,i})]), $$

where ψ is one of the functions $\phi _{0,s},\phi _{1,s},\phi _{2,x,s}^{t}$ or $\phi _{3,y,s}^{t}$ (defined in Eqs. 2.5–2.8). The asymptotic normality of the Hill estimator and our main Theorem 2.1 can be derived from the following result about the process convergence of $\tilde Z_{n}$.

Proposition C.2

Let $(X_{t})_{t\in \mathbb {Z}}$ be a stationary, regularly varying process with index α > 0. Suppose that the conditions (A(x₀)), (B) and (C) are fulfilled for some $x_{0}\geqslant 0$. Then, for all $y_{0}\in [x_{0},\infty )\cap (0,\infty )$, the sequence of processes $(\tilde {Z}_{n}(\psi ))_{\psi \in \varPhi }$ with index set

$$ \varPhi:=\!\left\{\phi_{0,s},\phi_{1,s},\phi_{2,x,s}^{t},\phi_{3,y,s}^{t}\ |\ s\in[1-\varepsilon,1\!+\varepsilon],x\geqslant x_{0},y\geqslant y_{0}, |t|\in\{1,\dots,\tilde{t}\}\right\} $$

converges weakly in $l^{\infty }(\varPhi )$ to a centered Gaussian process Z with covariance function given by

$$ \begin{array}{@{}rcl@{}} \text{cov}\left( Z\left( \psi_{1}\right),Z\left( \psi_{2}\right)\right)&=& E\left[\psi_{1}\left( \bar{Y}_{0}\right)\psi_{2}\left( \bar{Y}_{0}\right)\right]+ \sum\limits_{k=1}^{\infty}\left( E\left[\psi_{1}\left( \bar{Y}_{0}\right)\psi_{2}\left( \bar{Y}_{k}\right)\right]\right.\\ &&\left. + E\left[\psi_{1}\left( \bar{Y}_{k}\right)\psi_{2}\left( \bar{Y}_{0}\right)\right]\right)\\ &=& \sum\limits_{k=-\infty}^{\infty}E\left[\psi_{1}\left( \bar{Y}_{0}\right)\psi_{2}\left( \bar{Y}_{k}\right)\right] \end{array} $$

(C.1)

for ψ₁, ψ₂ ∈Φ, where

Proof

Weak convergence of the finite-dimensional distributions of $(\tilde {Z}_{n}(\psi ))_{\psi \in \varPhi }$ can be established as in the proof of Proposition B.1 of Drees et al. (2015). Note that here the threshold (1 − ε)u_n is used instead of u_n, while the components of X_{n, i} are standardized with u_n. Moreover, we standardize the process using v_n = P{X₀ > u_n} instead of P{X₀ > (1 − ε)u_n} = (1 − ε)^−αv_n(1 + o(1)). Therefore, we obtain as limiting covariance function

$$ \begin{array}{@{}rcl@{}} \text{cov} \!\!\!&&\left( Z\left( \psi_{1}\right),Z\left( \psi_{2}\right)\right)= (1-\varepsilon)^{-\alpha}\left( \vphantom{{\sum}_{k=1}^{\infty}}E\left[\psi_{1}\left( \left( 1-\varepsilon\right)\bar{Y}_{0}\right)\psi_{2}\left( \left( 1-\varepsilon\right)\bar{Y}_{0}\right)\right]\right. \\ && \left. +{\sum}_{k=1}^{\infty}\left( E\!\left[\psi_{1}\!\left( (1\! -\!\varepsilon)\bar{Y}_{0}\right)\psi_{2}\!\left( \left( 1 - \varepsilon\right)\bar{Y}_{k}\right)\right] \!+ \! E\left[\psi_{1}\!\left( \left( 1\! - \!\varepsilon\right)\bar{Y}_{k}\right)\psi_{2}\left( \left( 1\! -\! \varepsilon\right)\bar{Y}_{0}\right)\right]\right)\right). \end{array} $$

Now recall that Y_k = Y₀Θ_k for a Pareto random variable Y₀ independent of the spectral process. Since P{(1 − ε)Y₀ > 1} = (1 − ε)^α, Y₀ has the same distribution as (1 − ε)Y₀ conditionally on {(1 − ε)Y₀ > 1}, and $\psi _{i}(y_{-\tilde t},\ldots , y_{\tilde t})$ vanishes if $y_{0}\leqslant 1$, one has

$$ \begin{array}{@{}rcl@{}} E[\psi_{1}{\kern-.5pt}({\kern-.5pt}(1\! -\! \varepsilon)\bar Y_{0})\psi_{2}({\kern-.5pt}(1\! -\! \varepsilon)\bar Y_{k}){\kern-.5pt}]\! & =&\! E{\kern-.5pt}[\psi_{1}({\kern-.5pt}(1\! -\! \varepsilon)Y_{0}({\kern-.5pt}\varTheta_{t}{\kern-.5pt})_{|t|\leqslant \tilde t})\psi_{2}({\kern-.5pt}(1\! -\! \varepsilon)Y_{0}(\varTheta_{k+t})_{|t|\leqslant \tilde t})]\\ & = &\! (1-\varepsilon)^{\alpha} E[\psi_{1}(Y_{0}(\varTheta_{t})_{|t|\leqslant \tilde t})\psi_{2}(Y_{0}(\varTheta_{k+t})_{|t|\leqslant \tilde t})]\\ &= &\! (1-\varepsilon)^{\alpha} E[\psi_{1}(\bar Y_{0})\psi_{2}(\bar Y_{k})]. \end{array} $$

Now, the asserted representation (C.1) is obvious. The second representation can be similarly concluded from the equation

$$ E\left[ \sum\limits_{i=1}^{r_{n}} \psi_{1}\left( X_{n,i}\right)\sum\limits_{j=1}^{r_{n}} \psi_{2}\left( X_{n,j}\right)\right] = r_{n}\sum\limits_{k=1-r_{n}}^{r_{n}-1} \left( 1-\frac{|k|}{r_{n}}\right) E\left[\psi_{1}\left( X_{n,0}\right)\psi_{2}\left( X_{n,k}\right)\right]. $$

To prove asymptotic equicontinuity of the processes (and thus their weak convergence), we apply Theorem 2.10 of Drees and Rootzén (2010). To this end, we must verify the conditions (D1), (D2’), (D3), (D5) and (D6) of this paper. Except for condition (D6), all assumptions of the theorem can be established by similar arguments as in the proof of Proposition B.1 of Drees et al. (2015).

It remains to prove that the following condition holds:(D6)

$$ \lim_{\delta\downarrow 0} \limsup_{n\to\infty}P^{*}\left\{ {\int}_{0}^{\delta} \sqrt{ \log N\left( \varepsilon_{0},\varPhi,d_{n}\right)} d\varepsilon_{0} > \tau \right\} = 0 \quad \forall \tau>0. $$

Here P^∗ denotes he outer probability, and the (random) covering number N(ε₀, Φ, d_n) is the minimum number of balls with radius ε₀ w.r.t.

$$ d_{n}\left( \psi_{1},\psi_{2}\right) := \left( \frac 1{nv_{n}} \sum\limits_{j=1}^{m_{n}} \left( \sum\limits_{i=1}^{r_{n}}\psi_{1}\left( T_{n,j,i}^{*}\right)-\psi_{2}\left( T_{n,j,i}^{*}\right)\right)^{2}\right)^{1/2}, $$

needed to cover Φ, $T_{n,j}^{*}=(T_{n,j,i}^{*})_{1\le i\le r_{n}}$ denote iid copies of $(X_{n,i})_{1\le i\le r_{n}}$ (defined in Eq. 2.4) and m_n := ⌊n/r_n⌋.

Note that this condition can be verified separately for the sets of functions {ϕ_{0, s}∣s ∈ [1 − ε, 1 + ε]}, {ϕ_{1, s}∣s ∈ [1 − ε, 1 + ε]}, ${\varPhi _{2}^{t}}:=\{\phi _{2,x,s}^{t} | x\in [x_{0},\infty ),s\in [1-\varepsilon ,1+\varepsilon ]\}$ and ${\varPhi _{3}^{t}} := \{\phi _{3,y,s}^{t}\mid y\in [y_{0},\infty ),s\in [1-\varepsilon ,1+\varepsilon ]\}$ for $|t|\in \{1,\ldots ,\tilde t\}$. For the former two function classes, (D6) easily follows from the linear order of the functions in s; see (Drees and Rootzén 2010, Example 3.8) for details. It is much more challenging to verify (D6) for the remaining two families. We give details of the proof only for the class ${\varPhi _{3}^{t}}$, but the arguments readily carry over to ${\varPhi _{2}^{t}}$.

Since this most crucial part of the proof is rather involved, we first give a brief outline. In a first step, we use VC theory to bound the covering number of the family of functions $f_{3,y,s}^{(r,t)}:([0,\infty )^{(2\tilde {t}+1)})^{r}\to \mathbb {R}$,

with $z_{i}=(z_{i,-\tilde {t}},\dots ,z_{i,\tilde {t}})$ for fixed r (see Ineq. C.2). Then we show that by choosing r equal to the (random) minimum number (denoted by $R_{n,\varepsilon _{0}}$) such that the maximal possible contribution of clusters of length larger than r to the distance d_n is less than ε₀/2, one can bound the covering number $N(\varepsilon _{0},{\varPhi _{3}^{t}},d_{n})$ by some function of $R_{n,\varepsilon _{0}}$ (cf. Ineq. C.3). Finally, Condition (D6) follows if one shows that, with large probability, $R_{n,\varepsilon _{0}}$ grows only polynomially in $\varepsilon _{0}^{-1}$.

First note that the function $\phi _{3,y,s}^{t}$ does not vanish on the set $V_{y,s}^{t}:=$$\{(x_{-\tilde {t}},\dots ,x_{\tilde {t}})\in [0,\infty )^{2\tilde {t}+1}\ |\ x_{0}/x_{-t}>y,\ x_{-t}>0,\ x_{0}>s\}$. We now show that the subgraphs $M_{y,s}^{(r,t)}:=\{(\lambda ,z_{1},\dots ,z_{r})\in \mathbb {R}\times ([0,\infty )^{(2\tilde {t}+1)})^{r}\ |\$$\lambda <f_{3,y,s}^{(r,t)}(z_{1},\dots ,z_{r})\}$ of $f_{3,y,s}^{(r,t)}$ form a VC class. To this end, consider an arbitrary set $A=\{(\lambda ^{(l)},x_{1}^{(l)},\dots ,x_{r}^{(l)})\ |\ 1\leqslant l\leqslant m\}\subset \mathbb {R}\times [0,\infty )^{r(2\tilde {t}+1)}$ of m points. For $1\leqslant i\leqslant r$, $1\leqslant l\leqslant m$, define lines $\{(x_{i,0}^{(l)}/x_{i,-t}^{(l)},s)\ |\ s\in [1-\varepsilon ,1+\varepsilon ]\}$ and $\{(y,x_{i,0}^{(l)})\ |\ y\in [y_{0},\infty )\}$, that divide the set $[y_{0},\infty )\times [1-\varepsilon ,1+\varepsilon ]$ into at most (mr + 1)² rectangles. If $(y,s),(\tilde {y},\tilde {s})\in [y_{0},\infty )\times [1-\varepsilon ,1+\varepsilon ]$ belong to the same rectangle then the symmetric difference $V_{y,s}^{t}\vartriangle V_{\tilde {y},\tilde {s}}^{t}$ does not contain any of the points $x_{i}^{(l)}$, $1\leqslant i\leqslant r$, $1\leqslant l\leqslant m$. Hence, the equality $f_{3,y,s}^{(r,t)}(x_{1}^{(l)},\dots ,x_{r}^{(l)})=f_{3,\tilde {y},\tilde {s}}^{(r,t)}(x_{1}^{(l)},\dots ,x_{r}^{(l)})$ holds for all $1\leqslant l\leqslant m$, and the intersections $A\cap M_{y,s}^{t}$ and $A\cap M_{\tilde {y},\tilde {s}}^{t}$ are identical. Thus, $(M_{y,s}^{t})_{y\in [y_{0},\infty ),s\in [1-\varepsilon ,1+\varepsilon ]}$ can pick at most (mr + 1)² different subset of A. If $m>4\log r$ and r is sufficiently large then $m-2\log _{2} m>3 \log r>\log _{2}(4r^{2})$ which implies $2^{m}>4m^{2}r^{2}\geqslant (mr+1)^{2}$. Hence, the family of subgraphs $(M_{y,s}^{t})_{y\in [y_{0},\infty ),s\in [1-\varepsilon ,1+\varepsilon ]}$ cannot shatter A, which shows that the VC-index of ${\mathcal{F}}_{3}^{(r,t)}:=\{f_{3,y,s}^{(r,t)}\ |\ y\in [y_{0},\infty ),s\in [1-\varepsilon ,1+\varepsilon ]\}$ is less than $4\log r$ if r is sufficiently large. By Theorem 2.6.7 of van der Vaart and Wellner (1996), we have

$$ N\left( \delta\left( \int {G_{r}^{2}}\ \mathrm{d}Q\right)^{1/2},\mathcal{F}_{3}^{(r,t)},L_{2}(Q)\right)\leqslant K_{1}r^{16}\delta^{-K_{2}\log r} $$

(C.2)

for all small δ > 0, all probability measures Q on $([0,\infty )^{(2\tilde {t}+1)})^{r}$ such that $\int \limits {G_{r}^{2}}\ \mathrm {d}Q>0$, and suitable universal constants K₁, K₂ > 0 with $G_{r}=f_{3,y_{0},1-\varepsilon }^{(r,t)}$ denoting the envelope function of ${\mathcal{F}}_{3}^{(r,t)}$.

In the second step we show that the terms pertaining to blocks with more than r non-vanishing summands do not contribute too much to $d_{n}(\phi _{3,y,s}^{t},\phi _{3,\tilde y,\tilde s}^{t})$, if we let r tend to infinity in a suitable way.

Denote the number of independent blocks with at most r non-zero entries by with {z_i,0 > 1 − ε}, $z\in ([0,\infty )^{(2\tilde {t}+1)})^{r_{n}}$. For these blocks, define vectors $\tilde T_{n,j}^{*}$ of length r which contain all non-zero values of $T_{n,j}^{*}$, augmented by $r-H(T_{n,j}^{*})$ zeros. Let

with ε_T the Dirac measure with mass 1 in T. We can bound the squared distance between $\phi _{3,y,s}^{t}$ and $\phi _{3,\tilde y,\tilde s}^{t}$ as follows:

for all $r\in \mathbb {N}$. In particular,

$$ {d_{n}^{2}}\left( \phi_{3,y,s}^{t},\phi_{3,\tilde y,\tilde s}^{t}\right)\leqslant \frac{N_{n,R_{n,\varepsilon_{0}}}}{nv_{n}}\int \left( f_{3,y,s}^{(R_{n,\varepsilon_{0}},t)}-f_{3,\tilde y,\tilde s}^{(R_{n,\varepsilon_{0}},t)}\right)^{2}\ \mathrm{d}Q_{n,R_{n,\varepsilon_{0}}}+\frac{{\varepsilon_{0}^{2}}}{2} $$

with

If ${\int \limits } (f_{3,y,s}^{(R_{n,\varepsilon _{0}},t)}-f_{3,\tilde y,\tilde s}^{(R_{n,\varepsilon _{0}},t)})^{2}\ \mathrm {d}Q_{n,R_{n,\varepsilon _{0}}}\leqslant nv_{n}{\varepsilon _{0}^{2}}/(2N_{n,R_{n,\varepsilon _{0}}})=:{\varepsilon _{1}^{2}}$ then ${d_{n}^{2}}(\phi _{3,y,s}^{t},\phi _{3,\tilde y,\tilde s}^{t}){\leqslant \varepsilon _{0}^{2}}$; that is, for vectors $(y,s),(\tilde y,\tilde s)$ such that $f_{3,y,s}^{(R_{n,\varepsilon _{0}},t)}$ and $f_{3,\tilde y,\tilde s}^{(R_{n,\varepsilon _{0}},t)}$ belong to some ε₁-ball w.r.t. HCode $L^{2}(Q_{n,R_{n,\varepsilon _{0}}})$, the corresponding functions $\phi _{3,y,s}^{t}$ and $\phi _{3,\tilde y,\tilde s}^{t}$ belong to the same ε₀-ball w.r.t. d_n. This implies $N(\varepsilon _{0},{\varPhi _{3}^{t}},d_{n})\leqslant N(\varepsilon _{1},{\mathcal{F}}_{3}^{(R_{n,\varepsilon _{0}},t)},L_{2}(Q_{n,R_{n,\varepsilon _{0}}}))$. Note that

Using Ineq. C.2, we conclude that ${\varPhi _{3}^{t}}$ can be covered by

balls with radius ε₀ w.r.t. d_n. Since by Chebyshev’s inequality and regular variation

we conclude

$$ N(\varepsilon_{0},{\varPhi_{3}^{t}},d_{n})\leqslant K_{1}R_{n,\varepsilon_{0}}^{16}\left( \frac{\varepsilon_{0}y_{0}^{\alpha}}{2R_{n,\varepsilon_{0}}}\right)^{-K_{2}\log R_{n,\varepsilon_{0}}} $$

(C.3)

with probability tending to 1.

It remains to show that $R_{n,\varepsilon _{0}}$ does not increase too fast as ε₀ tends to 0. To this end, we decompose the unit interval into intervals (2^−(l+ 1), 2^−l], $l\in \mathbb {N}_{0}$. Check that by Markov’s inequality and Lemma C.1

for some constant K₃ depending on y₀, and M > 2K₃/η. Hence $R_{n,\varepsilon _{0}}\leqslant M\varepsilon _{0}^{-3}$ with probability greater than 1 − η, so that by Ineq. C.3

$$ {\int}_{0}^{\delta}(\log N(\varepsilon_{0},\varPhi_{3},d_{n}))^{1/2}\ \mathrm{d}\varepsilon_{0}\leqslant {\int}_{0}^{\delta}(K_{4}+K_{5}|\log \varepsilon_{0}|+K_{6}\log^{2}\varepsilon_{0})^{1/2}\ \mathrm{d}\varepsilon_{0} $$

for suitable constants K₄, K₅, K₆ > 0. Now condition (D6) is obvious. □

Lemma C.3

If the conditions (2.1), A(x₀), (B), (C) and Eq. 2.14 hold, then

$$ \begin{array}{@{}rcl@{}} (nv_{n})^{1/2}(\hat{\alpha}_{n,\hat{u}_{n}}-\alpha)& \rightsquigarrow \alpha Z(\phi_{1,1})-\alpha^{2}Z(\phi_{0,1}) \end{array} $$

where Z is the same centered Gaussian process as in Proposition C.2.

Proof

For s ∈ [1 − ε, 1 + ε], define

$$ \alpha_{n,s}:=\frac{1}{E[\log(X_{0}/(su_{n}))\ |\ X_{0}>su_{n}]}\quad\text{and}\quad \tilde{\alpha}_{n,s}:=\frac{{\sum}_{i=1}^{n}\phi_{1,s}(X_{n,i})}{{\sum}_{i=1}^{n}\phi_{0,s}(X_{n,i})} $$

(C.4)

and processes $V_{n}(s):=(nv_{n})^{1/2}(\tilde {\alpha }_{n,s}-\alpha _{n,s})$ and V (s) := s^α(αZ(ϕ_{1, s}) − α²Z(ϕ_{0, s})). Note that, by Eq. 2.1, S_n → 1 in probability, and so $\hat {\alpha }_{n,\hat {u}_{n}}=\tilde {\alpha }_{n,S_{n}}$ with probability tending to 1. In view of Eq. 2.14, $(nv_{n})^{1/2}(\hat {\alpha }_{n,\hat {u}_{n}}-\alpha )=V_{n}(S_{n})+o_{P}(1)$.

By similar arguments as in proof of Lemma 4.4 of Drees et al. (2015), one can conclude from Proposition C.2 that $V_{n}\rightsquigarrow V$ (w.r.t. the supremum norm) and that V has continuous sample paths almost surely. Using Slutsky’s lemma, we obtain $(V_{n},S_{n})\rightsquigarrow (V,1)$, and by Skorohod’s theorem, there are versions for which the convergence holds almost surely. It follows that

$$ |V_{n}(S_{n})-V(1)|\leqslant \sup_{s\in[1-\varepsilon,1+\varepsilon]}|V_{n}(s)-V(s)|+|V(S_{n})-V(1)|\to 0 $$

almost surely, from which the assertion is obvious. □

Proof of Theorem 2.1

The assertion follows from arguments along the line of reasoning used in the proof of Theorem 4.5 in Drees et al. (2015) with similar modifications as employed in the proof of Lemma C.3. □

Proof of Lemma 2.2

Fix an arbitrary δ ∈ (0, ε) and choose some $a_{\delta }^{+}\in (0,1-(1+\delta )^{-\alpha })$ and $a_{\delta }^{-}\in (0,(1-\delta )^{-\alpha }-1)$. Then, by regular variation of $F^{\leftarrow }$, we have

$$ \begin{array}{@{}rcl@{}} (1+\delta)F^{\leftarrow}(1-k_{n}/n)&> F^{\leftarrow}(1-(1-a_{\delta}^{+})k_{n}/n),\\ (1-\delta)F^{\leftarrow}(1-k_{n}/n)&< F^{\leftarrow}(1-(1+a_{\delta}^{-})k_{n}/n) \end{array} $$

for sufficiently large n, and hence $\bar F((1+\delta )F^{\leftarrow }(1-k_{n}/n))<(1-a_{\delta }^{+})k_{n}/n$ and $\bar F((1-\delta )F^{\leftarrow }(1-k_{n}/n))>(1+a_{\delta }^{-})k_{n}/n$. Let $u_{n}=F^{\leftarrow }(1-k_{n}/n)$ so that v_n = (1 + o(1))k_n/n by the regular variation of $\bar F$ (Bingham et al. 1987, Th. 1.5.12). The proof of Prop. C.2 shows that $\tilde {Z}_{n}(\phi _{1,s})$ converge weakly to a normal distribution, because ϕ_{1, s} is almost surely continuous w.r.t. HCode ${\mathcal{L}}(\bar {Y}_{0})$. Thus,

Analogously, one obtains

$$ \begin{array}{@{}rcl@{}} P\left\{\frac{X_{n-k_{n}:n}}{F^{\leftarrow}(1-k_{n}/n)}>1-\delta\right\}\geqslant P\left\{\tilde{Z}_{n}(\phi_{1,1-\delta})>-a_{\delta}^{-}k_{n}^{1/2}\right\}\to 1. \end{array} $$

Now let δ tend to 0 to conclude the assertion. □

Proof of Theorem 2.3

First note that, by Corollary 3.9 of Drees and Rootzén (2010), the asymptotic behavior of the processes $\tilde Z_{n}$ is not changed if one replaces n with r_nm_n. One may easily conclude that, up to terms of the order o_P((nv_n)^− 1/2), the estimators $\hat F_{n,su_{n}}^{(f,\varTheta _{t})}$, $\hat F_{n,su_{n}}^{(b,\varTheta _{t})}$ and $\tilde \alpha _{n,s}$ (defined in Eq. C.4) do not change either. Hence, w.l.o.g. we assume that n = m_nr_n.

The crucial observation to establish consistency of the bootstrap version is that the bootstrap processes

$$ Z_{n}^{*}(\psi) := (nv_{n})^{-1/2} \sum\limits_{j=1}^{m_{n}} \xi_{j} \sum\limits_{i\in I_{j}} (\psi(X_{n,i})-E\psi (X_{n,i})) $$

converge to the same limit Z as Z_n, both unconditionally and conditionally given X_n,1,…, X_{n, n}; see Drees (2015), Corollary 2.7. Define

$$ \tilde{\alpha}_{n,s}^{*}:=\frac{{\sum}_{j=1}^{m_{n}}(1+\xi_{j}){\sum}_{i\in I_{j}}\phi_{1,s}(X_{n,i})}{{\sum}_{j=1}^{m_{n}}(1+\xi_{j}){\sum}_{i\in I_{j}}\phi_{0,s}(X_{n,i})}, $$

and $V_{n}(s):=(nv_{n})^{1/2}(\tilde {\alpha }_{n,s}^{*}-\tilde {\alpha }_{n,s})$. By similar calculations as in the proof of Theorem 3.3 of Davis et al. (2018), one obtains $V_{n}\rightsquigarrow V$ with V (s) := s^α(αZ(ϕ_{1, s}) − α²Z(ϕ_{0, s})) denoting the limit process in Lemma C.3. Since V has a.s. continuous sample paths, $S_{n}=\hat {u}_{n}/u_{n}\to 1$ in probability and $\hat \alpha _{n,\hat u_{n}}^{*}=\tilde {\alpha }_{n,S_{n}}^{*}$ and $\hat \alpha _{n,\hat u_{n}}=\tilde {\alpha }_{n,S_{n}}$ with probability tending to 1, the convergence $(nv_{n})^{1/2}(\hat {\alpha }_{n,\hat u_{n}}^{*}-\hat {\alpha }_{n,\hat {u}_{n}})=V_{n}(S_{n})+o_{P}(1)\rightsquigarrow V(1)$ follows readily.

Now one may argue as in the proof of Theorem 3.3 of Davis et al. (2018) to verify the assertion. To this end, one must replace u_n with $\hat u_{n}=S_{n}u_{n}$ everywhere. For example, equation (6.10) of Davis et al. (2018) now becomes

where the last term is of stochastic order $r_{n}m_{n}^{1/2}v_{n}=o((nv_{n})^{1/2})$. Thus

(C.5)

Since, by the law of large numbers, for all fixed s ∈ [1 − ε, 1 + ε] and S_n → 1 in probability, a standard argument shows that the right hand side of Eq. C.5 equals nv_n(1 + o_P(1)), and the proof can be concluded as in Davis et al. (2018). □