Long Memory of Max-Stable Time Series as Phase Transition: Asymptotic Behaviour of Tail Dependence Estimators

In this paper, we consider a simple estimator for tail dependence coefficients of a max-stable time series and show its asymptotic normality under a mild condition. The novelty of our result is that this condition does not involve mixing properties that are common in the literature. More importantly, our condition is linked to the transition between long and short range dependence (LRD/SRD) for max-stable time series. This is based on a recently proposed notion of LRD in the sense of indicators of excursion sets which is meaningfully defined for infinite-variance time series. In particular, we show that asymptotic normality with standard rate of convergence and a function of the sum of tail coefficients as asymptotic variance holds if and only if the max-stable time series is SRD.


Introduction
In extreme value analysis, one is typically interested in events which are extreme in the sense that a high threshold u is exceeded, i.e. events of the form X 0 > u where X 0 is part of a time series (X t ) t∈Z .Although lim u→∞ P(X 0 > u) = 0, one studies the behaviour as u → ∞ because the limiting probability might be positive, provided that it exists.Assuming that the limit indeed exists, χ h denotes the limiting probability that, conditional on an extreme event at time t = 0, another extreme event occurs within h ∈ Z time steps.If χ h > 0, the random variables X 0 and X h are called asymptotically dependent, while they are called asymptotically independent if χ h = 0.For a longer discussion on modelling asymptotic dependence using tail dependence coefficients such as χ h let us refer to Chapter 9.5 in [1].
If (X t ) t∈Z is a max-stable time series, it is well-known that χ h always exists and is of the form χ h = 2−θ h , h ∈ Z, where θ h ∈ [1,2] denotes the corresponding pairwise extremal coefficient.The latter coefficient was first defined in [11] and is an important characteristic quantity in extreme value statistics.In the maxstable case, it has been shown that θ h = 2, or, equivalently, χ h = 0 if and only if X 0 and X h are fully independent.Thus, components of a max-stable vector are either asymptotically dependent or fully independent and the pairwise extremal coefficients can be interpreted as a measure for the strength of dependence.Hence, the pairwise extremal coefficient has gained a lot of attention and has been thoroughly studied in the literature, e.g. in [3], [4] and [10].
In this paper, we will consider the case that (X t ) t∈Z is a stationary maxstable time series with α-Fréchet, i.e. heavy-tailed, marginal distributions and focus on the estimation of limits of conditional exceedance probabilities of the form P(X t1 > u, . . ., X tN > u | X 0 > u) as u → ∞ where 0 < t 1 < . . .< t N for some N ∈ N.This is well-defined as the existence of the limit χ T : = lim u→∞ P(X t > u for all t ∈ T | X 0 > u) ( = lim u→∞ P(X t > u for all t ∈ T ) P(X 0 > u) is ensured for any set T = {0, t 1 , . . ., t N } by Lemma 11 in the appendix.
In practice, estimation has to be based on a single realization of the time series up to time n, i.e. on dependent observations X 1 , . . ., X n .A straight-forward estimator can be constructed by replacing the desired probabilities in (1) with relative frequencies, i.e. a natural estimator of χ T is given by χ T,n := where Note that we need u = u n to depend on n with u n → ∞ in order to get a good approximation of the limiting quantity χ T .Furthermore, our estimator uses observations up until time point n + t N rather than just n.Asymptotically, the use of t N more observations has no influence but eases notation.Unfortunately, the larger u n gets, the less data can be effectively used for our estimation since we do not count those data that are less than u n .Thus, the effective size of the sample that we use is only To make sure that the effective sample size grows to ∞ as n → ∞, a natural assumption is lim n→∞ nP( The main result of our paper is that we prove asymptotic normality of our estimator (3) in Theorem 6.The novelty of this result is that Theorem 6 characterizes the dependence of the underlying time series in terms of the weak assumption which is equivalent to the finiteness of the asymptotic variance of the estimator.This assumption is weaker than common assumptions in terms of strong mixing properties involving α-mixing coefficients.
More importantly, our condition ∞ t=1 (2 − θ t ) < ∞ opens a connection to the phenomenon of long range dependence (LRD).While this phenomenon has seen increasing relevance in other fields, there are only few results relating to LRD of max-stable time series.Clearly, this is the case because LRD was previously investigated mainly for finite-variance processes in terms of auto-covariance functions.However, there have been recent investigations into LRD for infinitevariance processes.For example, [7] defines LRD in terms of the covariance of indicators of excursion sets, cf.Definition 1.The advantage of this definition is that it allows to investigate LRD for infinite-variance processes because the covariance of indicators is always well-defined as the indicator functions are bounded.Furthermore, [8] have recently shown that max-stable time series are LRD in the sense of Definition 1 iff In addition, our condition marks the point of a phase transition.Recently, Chapter 9 of [9] describes the phenomenon of LRD as a phase transition after which the convergence rate in a limit theorem changes magnitude.In this paper, we show that the limiting variance of our estimator's denominator undergoes such a phase transition as X becomes LRD.Hence, our paper takes the first steps to investigate LRD in an extreme value statistics setting.Furthermore, we round off our paper by discussing ways to extend our results into the realm of LRD even further.
Our paper is structured as follows.Section 2 reviews well-known facts about max-stable time series that we need in order to prove our main result.Section 3 investigates the limiting behavior of Var p bn (u n ) and Var P T,bn (u n ) for arbitrary integer sequences b n → ∞ under the SRD assumption ∞ t=1 (2 − θ t ) < ∞.In Section 4, we prove Theorem 5 that shows the asymptotic normality of p bn (u n ) and P T,bn (u n ) in the SRD-case.This result is then used to prove our main result.Section 5 proves the aforementioned phase transition and discusses conjectures on extending Theorem 6 to the LRD case.Finally, auxillary lemmas and the proof of Theorem 5 can be found in Appendix A.

Max-Stable Time Series
Let us review properties of max-stable time series and their extremal coefficients.These properties are crucial to prove asymptotic normality of the estimators p bn (u n ) and P T,bn (u n ) which is a key step to prove our main result, Theorem 6.
Let X = (X t ) t∈Z be a stationary max-stable time series with α-Fréchet margins.Then, by the spectral representation according to [6], one can write where Γ 1 < Γ 2 < . . .are the arrival times of a unit rate Poisson process on (0, ∞) and, independently of this Poisson process, W (1) , W (2) , . . .are independent copies of some stochastic process W = (W t ) t∈Z satisfying E(W α t ) = 1 for all t ∈ Z.A well-known consequence of this is that for any finite index set T ⊂ Z, one obtains It can be seen that the joint c.d.f. of (X t ) t∈T can be used to define a measure . By construction, the exponent measure µ T satisfies µ T (A) < ∞ for any measurable set A ⊂ E T bounded away from 0 ∈ R |T | and is homogeneous of order −α, i.e. µ T (cA) = c −α µ T (A) for all c > 0 and the same sets A as before.Using this definition, Equation (6) implies that or, more generally, lim for any measurable set A ⊂ E T bounded away from 0 ∈ R |T | and satisfying µ(∂A) = 0.In particular, this also ensures the existence of the limit where we used that µ {0} ([1, ∞)) = lim u→∞ u α P(X 0 > u) = 1 according to the first part of Lemma 11.
A dependence measure that is closely related to χ T is the extremal coefficient In the case that T = {0, h} with h = 0, we have that Here, we denoted θ T = θ h for T = {0, h}.For more general finite index sets T , we obtain Let us note that the last equality follows from Lemma 11 with A = {0} and B = T \ {0}.
Finally, let us review the notion of LRD that was discussed in the introduction.This notion is defined in terms of the covariance of indicators of excursion sets.

Definition 1 ([7]). A real-valued stationary stochastic process
for any finite measure µ on R. Otherwise, i.e. if there exists a finite measure µ such that the integral in inequality (7) is infinite, X is long range dependent.For stochastic processes in discrete time, the integral T dt should be replaced by the summation t∈T .
Notice that this definition is defined even for infinite-variance processes.Hence, it is also applicable to the max-stable time series we are investigating in this paper.In fact, due to [8] it is known that a max-stable time series is LRD in the sense of Definition 1 iff Furthermore, it is worth pointing out that this definition is invariant under strictly monotonic transformations.This is desirable in an extreme value setting because it means that this notion of LRD does not depend on whether we use α-Fréchet, Gumbel or Weibull margins.

Variance behavior
Let us consider the asymptotic variance behavior of p n (u n ) and P T,n (u n ).For our purposes it is suitable to replace n by a more flexible sequence b n that fulfills b n → ∞ as n → ∞.We will denote the corresponding estimators by p bn (u n ) and P T,bn (u n ), respectively.For the estimator p bn (u n ), we will see that the typical rate of convergence b n p(u n ) holds if and only if the time series is short range dependent in the sense of Definition 1, i.e.
Let (X t ) t∈Z be a max-stable stationary time series with α-Fréchet margins and extremal coefficients (θ t ) t∈Z .Then, for any real sequence u n → ∞ and any integer sequence b n → ∞, we have In particular, the variance is finite if and only if For the second term, we notice that each summand is bounded by due to the the second part of Lemma 12. From the first part of Lemma 12, we know that u −α n /p(u n ) is bounded and by assumption we know that (2 − θ t ), t ∈ Z, is summable.Therefore, we can apply dominated convergence and obtain where we used that P( Second Case: 0 as the max-stable time series (X t ) t∈Z is positively associated.Thus, for any fixed k ∈ N and sufficiently large n, we have Using the same calculations as above (here dominated convergence does not need to be applied as the sum consists of a fixed number of summands), we obtain As this holds for any k ∈ N, we finally get Next, let us consider the numerator of our estimator (3).To do so, denote where (X t ) t∈Z is a stationary max-stable time series and C T is defined in (5).These quantities always exist as they can be expressed in terms of the exponent measure of the underlying time series.
Proposition 3. Let (X t ) t∈Z be a max-stable stationary time series with α-Fréchet margins and pairwise extremal coefficients (θ t ) t∈Z .Assume that ∞ t=1 (2− θ t ) < ∞ and choose a real sequence u n → ∞ as well as an integer sequence b n → ∞.Then, we have where κ t,T is defined as in (9).
Proof.We have that For fixed k ∈ N, the sum of the first and the second term converges to (details similar to the proof of Proposition 2).From Lemma 12 it follows that that we can bound the third term by K • ∞ t=k+1 (2 − θ t ) for some constant K > 0. As this bound goes to zero as k → ∞, taking the limit superior as k → ∞ gives the desired result.
This follows from χ T = χ {0}, T \{0} and Lemma 12(b) which implies that for some constant K > 0. The claim now follows from Lemma 11.(c) From combining Prop. 2 and 3 with Markov's inequality we get that and, consequently,

Asymptotic normality
As we've concluded in the last section, the properly normalized numerator and denominator of our estimator (3) are weakly consistent under SRD.In this section, we want to show that they are also asymptotically normally distributed.Furthermore, we use this to prove that our ratio estimator is asymptocially normally distributed too.Theorem 5. Let (X t ) t∈Z be a max-stable stationary time series with α-Fréchet margins and pairwise extremal coefficients θ t .Assume that as n → ∞, where σ 2 0 and σ 2 T are the limiting variances as defined in Propositions 2 and 3.
Proof.For legibility this proof has been moved to Appendix A.
Finally, let us combine the results from Theorem 5 to show that our quotient estimator χ T,n is asymptotically normal.

Theorem 6. Under the assumptions of Theorem 5 it holds that
as n → ∞ for some σ 2 > 0.
Remark 7. The novelty of this result is the weak assumption ∞ t=1 (2−θ t ) < ∞ on the dependence structure of the time series.Typically, asymptotic normality is proven under the assumption of strong mixing properties of the time series.For example, [5] proves asymptotic normality of a more general version of our estimator (3) under the assumption that X is, among other assumptions, strongly mixing with mixing rate (α t ) t∈Z that decays sufficiently fast such that where m n , r n → ∞ with m n /n → 0 and r n /m n → 0 as n → ∞.Compared to this, our assumption is much simpler and also weaker as these mixing assumptions in particular imply that the limiting variance σ 2 0 in Theorem 5 is finite.
Proof.First, let us show the following multivariate CLT: For where Σ is a covariance matrix.In order to prove (11), it suffices to show z We can prove this using the same large/small block technique that we used to prove Theorem 5. Hence, the calculations are the same as in Appendix A. Note that the variance of the small blocks are negligible again because Var(X + Y ) ≤ Var(X) + Var(Y ) + 2 Var(X) Var(Y ).Furthermore, the Lipschitz constant of the functions in the analogue of (20) will be bound by |z 1 | + |z 2 |.This does not affect the remainder of the proof.
Thus, we can consider iid.copies of Now, we have to verify Lindeberg's condition for all ε > 0. Similar to our proof in Appendix A, Chebychev's inequality yields that for some constant K > 0.
By the same assumption on the block length b n,1 that we used in Appendix A, where Then, CLT (12) follows from Slutsky's theorem and CLT (11).Finally, from Remark 4(b) we know that This completes our proof.

Long Range Dependence as Phase Transition
We have seen in the previous sections that our main results, Theorems 5 and 6, hold if Interestingly, this exact condition is indicative of a max-stable time series being long or short range dependent in the sense of Definition 1.More precisely, [8,Theorem 4.3] shows that a max-stable time series is LRD in the sense of Definition 1 iff ∞ t=1 (2 − θ t ) = ∞.In this section, we show that the limiting behavior of p bn (u n ) undergoes a phase transition as the underlying time series becomes LRD.Moreover, we close this section with an additional discussion of conjectures on how to extend our results even further.
Proposition 2 shows that the asymptotic variance of the numerator is infinite if ∞ t=1 (2 − θ t ) = ∞ and the typical normalization np(u n ) is used.This begs the question whether a different normalization of the denominator can yield a finite limiting variance in the case where ∞ t=1 (2 − θ t ) < ∞.And this is indeed the case.Proposition 8. Let (X t ) t∈Z be a max-stable stationary time series with α-Fréchet margins and pairwise extremal coefficients (θ t ) t∈Z .Furthermore, assume that 2 − θ t = Ct −δ , t ∈ Z, for some constants C > 0 and δ ∈ (0, 1).Then, for any real sequence u n → ∞ and any integer sequence b n → ∞, we have Proof.Since δ < 1, the first and second term converge to zero as n → ∞.Due to Lemma 12, we know for t > 0 that Clearly, the convergence rate of the variance of P T,bn (u n ) is bounded by the convergence rate of the variance of p bn (u n ).Hence, there exists a constant Thus, the properly normalized variance of P T,bn (u n ) is bounded by a finite constant, possibly zero.
As for our ratio estimator χ T,bn , we conjecture that its limiting variance uses the same rate of convergence as the denominator.This could possibly shown with a Delta-method argument if assumptions are appropriately chosen such that remainder terms of the resulting Taylor expansion are negligible.Conjecture 10.Let (X t ) t∈Z be a max-stable stationary time series with α-Fréchet margins and pairwise extremal coefficients (θ t ) t∈Z .Furthermore, assume that 2 − θ t = Ct −δ , t ∈ Z, for some constants C > 0 and δ ∈ (0, 1).Then, under appropriate assumptions on the asymptotic covariance of p n (u n ) and P T,n (u n ) there exists a constant K > 0 such that for any real sequence u n → ∞ and any integer sequence b n → ∞, we have Finally, in light of Conjecture 10, we conjecture that if 2 − θ t = Ct −δ , t ∈ Z, for some constants C > 0 and δ ∈ (0, 1), we conjecture an LRD-analogue of our main result, Theorem 6, i.e. the limiting distribution of By applying the Taylor expansion of the function x → exp(−x) around 0 for each exponential term, we finally obtain Now, the assertion follows from the first part of the lemma.
The following lemma will give us two crucial auxiliary results to prove asymptotic normality.The first result bounds the covariance of indicators.The second result gives a bound on the covariance's convergence rate.
Lemma 12. Let (X t ) t∈Z be a max-stable stationary time series with α-Fréchet margins and extremal coefficients θ T , T ⊂ Z finite, where we denote the pairwise extremal coefficients by θ h := θ {0,h} , h > 0, for short.Furthermore, let A, B ⊂ Z be finite and disjoint.Then, the following bounds hold.
Proof.We begin to prove the first part.Equation (14) yields Now, we notice that, for arbitrary sets A 1 , A 2 , A 3 ⊂ Z, we have that, by Equation ( 15) in [10],

Applying this bound iteratively to sets of the type
respectively we can decompose the sets C ∩ A and C ∩ B into singletons and, using that θ {t} = 1 for all t ∈ Z, we eventually obtain Thus, we obtain The assertion follows from the fact that, for each (s, t) ∈ A × B, there exist 2 |A|−1 subsets of A that contain s and 2 |B|−1 subsets of B that contain t.
Next, let us prove the second part.Plugging in Equation ( 14) and the expression for χ A,B from the second part of Lemma 11, we obtain Then, applying analogous calculations as in (17) to both parts of the maximum yields where we used θ C∩A + θ C∩B − θ C = θ C∩A + θ C∩B − θ C∩(A∪B) ≥ 0 by inequality (15) and θ M ≤ |M | for all finite sets M ⊂ Z.The assertion follows by employing Equation ( 16) analogously to the proof of the first part of the lemma.This proof follows the same large/small block argument as given in the proof of Theorem 3.2 in [5].We prove only the limit theorem for P T,bn (u n ).The result for p bn (u n ) follows analogously.Let us start by denoting C := C T , p 0 := Using these block sizes we can define blocks for i = 1, . . ., k n .For any index sex B ⊂ N 0 we write S n (B) = t∈B Y n,t .
Next, let us show that the small blocks J n,i , i = 1, . . ., k n , are asymptotically negligible, i.e.Cov S n (J n,i ), S n (J n,j ) =: P 1 + P 2 .
Thus, let us show that both P As the index set T is bounded and the block distance b n,1 − b n,2 between two subsequent small blocks tends to infinity as n → ∞, the index set s+T and t+T in the sum above are all disjoint for sufficiently large n.Hence, we can apply the first part of Lemma 11.Counting the pairwise distances between elements of the blocks J n,i and J n,j we get that there is a constant K > 0 such that Consequently, we have proven (18) and the limiting distribution of S n and kn i=1 S n ( I n,i ) are equal if the limit exists.Now, let us consider iid.copies S n ( I n,l ), l = 1, . . ., k n , of S n ( I n,1 ).Using a telescoping series we get for any t ∈ R that Since we have assumed that b 2 n,1 /(np(u n )) → 0 as n → ∞, the RHS converges to zero as n → ∞.This completes the proof.

1 and P 2 1 = b n,2 b n, 1 j=1
converge to zero as n → ∞.By Proposition 3 and b n,2 /b n,1 → 0 as n → ∞, it follows that thatP Var b n,2 p(u n ) P T,bn,2 (u n ) p(u n ) Cov S n (J n,i ), S n (J n,j ) Cov 1 u −1 n X s+T ∈ C , 1 u −1 n X t+T ∈ C

eeeSS+ Cov sin t l− 1 s=1S+ Cov sin t l− 1 s=1S 1 s=1S
itSn( I n,l ) − E kn l=1 e it Sn( I n,l ) itSn( In,s) e itSn( I n,l ) − e it Sn( I n,l ) itSn( In,s) e itSn( I n,l ) − e it Sn( I n,l ) n ( I n,s ) , exp itS n (I n,l ) n ( I n,s ) , cos tS n ( I n,l ) + Cov cos t l−1 s=1 S n ( I n,s ) , sin tS n ( I n,l ) n ( I n,s ) , cos tS n ( I n,l ) n ( I n,s ) , sin tS n ( I n,l )(19)Next, let us bound each of those covariances.To do so, let us rewriteCov cos t l−n ( I n,s ) , cos tS n ( I n,l )as n → ∞.First, notice that the limiting variance is correct by Proposition 3 sinceVar kn l=1 S n (I n,l ) = k n Var S n (I n,1 ) = Var b n,1 p(u n ) P T,bn (u n ) p(u n ) .Thus, we need to check the Lindeberg condition of the CLT for triangular arrays of iid.mean-zero random variables, i.e. we need to verifyk n E S n (I n,1 ) 2 1 | S n (I n,1 )| > ε → 0, (n → ∞),for any ε > 0. Using Chebyshev's inequality and the fact that | S n (I n,1 )| ≤ b n,1 / np(u n ), we get k n E S n (I n,1 ) 2 1 | S n (I n,1 )| > ε ≤ b 2 n,1 np(u n ) k n Var S n (I n,1 ) ε 2 .
1, . . ., n.Now, we want to consider k n large blocks of size b n,1 .To this end, choose b n,1 ∈ N such that k n = n/b n,1 is a positive integer and b n,1 → ∞ as n → ∞.Moreover, for technical reasons that will become clear in the last part of this proof, we assume that b 2 n,1 /(np(u n )) → 0 as n → ∞.Note that this can always be achieved by letting b n,1 grow sufficiently slowly.In addition to b n,1 we choose small block sizes b n,2