Long Range Dependence for Stable Random Processes

We investigate long and short memory in $\alpha$-stable moving averages and max-stable processes with $\alpha$-Fr\'echet marginal distributions. As these processes are heavy-tailed, we rely on the notion of long range dependence suggested by Kulik and Spodarev (2019) based on the covariance of excursions. Sufficient conditions for the long and short range dependence of $\alpha$-stable moving averages are proven in terms of integrability of the corresponding kernel functions. For max-stable processes, the extremal coefficient function is used to state a necessary and sufficient condition for long range dependence.

Many of these approaches fail for heavy-tailed stochastic processes whose variance does not exist. To circumvent this issue, Kulik and Spodarev (2019) propose to consider the covariance of indicator functions of excursions and introduce Definition 1.2 A real-valued stationary stochastic process X = {X(t), t ∈ T } where T is a subset of R is short range dependent if for any finite measure µ on R, and long range dependent, otherwise. For stochastic processes in discrete time, the integral T dt should be replaced by the summation t∈T : t =0 .
One major advantage of this definition is that the above covariance exists in any case due to the boundedness of the indicators. Nevertheless, its computation might prove to be difficult. We restrict ourselves here to the class of positively associated stochastic processes to at least eliminate the absolute value in (1). The notion of association is given e.g. in Bulinski and Shashkin (2007). Let M(n) denote the class of real-valued bounded coordinate-wise nondecreasing Borel functions on R n , n ∈ N. For a real-valued stochastic process X = {X(t), t ∈ T } and a set I ⊂ T we denote X I = {X(t), t ∈ I}.
Definition 1.3 A real-valued stochastic process X = {X(t), t ∈ T } is positively associated if Cov(f (X I ), g(X J )) ≥ 0 for any disjoint finite subsets I, J ⊂ T and all functions f ∈ M(|I|) and g ∈ M(|J|).
In this paper, we consider two important subclasses of positively associated stationary processes that satisfy certain stability properties. More precisely, we study α-stable moving averages and max-stable processes with α-Fréchet marginals. As these processes are heavy-tailed, the classical definition of long range dependence (Definition 1.1) does not apply. Instead, we check Definition 1.2.
Integrating this relation with respect to t will establish short or long range dependence according to Definition 1.2. Subsequently, we will apply this result to get the long range dependence of symmetric α-stable (SαS) moving averages which are defined as follows.
Here and throughout the paper, we use the notation m ∈ L p (A), p > 0, to imply that A |m(x)| p dx < ∞. Also, we will restrict ourselves to the case α ∈ (1, 2) and leave the case α ∈ (0,1) for future work.
While investigating the long range dependence of the process X given in (2), our main result is In terms of the kernel function m, Theorem 3.5 translates the condition on ρ t to X being short range dependent if m ∈ L α/2 (R). Also, Theorem 3.6 establishes long range dependence if R R (m α (x)∧m α (t)) dx dt = ∞ where a ∧ b is the minimum of a and b.
These results hold also for α-stable linear time series if integrals are replaced by sums.
Besides the case of α-stable processes, we also consider long range dependence of max-stable processes. These are defined as follows.
Definition 1.5 A real-valued stochastic process X = {X(t), t ∈ T } is called a max-stable process if, for all n ∈ N, there exist functions a n : T → (0, ∞) and b n : where the processes X i , i ∈ N, are independent copies of X, and d = means equality in distribution.
If the index set T is finite, X is also called a max-stable vector.
It follows from the univariate extreme value theory that the marginal distributions of a maxstable process are either degenerate or follow a Fréchet, Gumbel or Weibull law. While covariances always exist in the Gumbel and Weibull case and, thus, the classical notion of long-range dependence applies, we will consider the case when X is a stationary max-stable process with α-Fréchet marginal distributions, i.e. P(X(t) ≤ x) = exp(−x −α ) for all x > 0 and some α > 0 and all t ∈ T . Here, covariances do not exist if α ≤ 2.
In combination with Definition 1.2, a well-established dependence measure for max-stable stochastic processes allows for an easily tractable condition for short and long memory, respectively.
More specifically, we use the pairwise extremal coefficient {θ t , t ∈ R} defined via the relation P(X(0) ≤ x, X(t) ≤ x) = P(X(0) ≤ x) θt , which holds for all x > 0, to show that a stationary max-stable process with α-Fréchet marginal distributions is long range dependent if and only if To summarize, our paper is structured as follows: Section 2 establishes the framework to invert the bivariate characteristic functions. In Section 3, we make use of this framework to find conditions for long range dependence of symmetric α-stable moving averages and linear time series, while, in Section 4, we investigate long range dependence of a stationary max-stable process with α-Fréchet marginals. For the sake of legibility, some of the proofs have been left out of the main part of this paper. They can be found in the Appendix.

| From Characteristic Function to Covariance of Indicators
In this section, we express the covariance of indicators of excursions of random variables above some levels u, v through their uni-and bivariate characteristic functions. Notice that for U and V absolutely continuous and identically distributed random variables it holds The proof of the following result can be found in the Appendix.
Theorem 2.1 Suppose U and V are absolutely continuous and identically distributed random variables with marginal and joint characteristic functions of U and V are denoted by ϕ U and ϕ, respectively. Then, for all u, v ∈ R Cov(1{U > u}, 1{V > v}) Proof: This is a very tedious calculation and has thus been moved to the appendix for legibility. Now, it is possible to integrate both sides of (4) with respect to u and v and any finite measure µ.
Theorem 2.2 Suppose U and V are absolutely continuous and identically distributed random variables with marginal characteristic function ϕ U and joint characteristic function ϕ. Then, for a finite measure µ with its Fourier transform denoted by ψ : Proof: Without loss of generality, assume that µ(R) = 1. There exist independent random variables ξ 1 and ξ 2 on the probability space (Ω, F, P) on which U and V are defined such that P(ξ 1 ∈ A) = P(ξ 2 ∈ A) = µ(A) for any Borel set A ⊂ R. Also, ξ 1 and ξ 2 can be chosen to be independent of U and V .
Note thatŨ andṼ are absolutely continuous random variables because U and V are absolutely continuous.
Proof. Equality (6) follows immediately from ϕ U and ϕ being real-valued as characteristic functions of a symmetric random variable and random vector, respectively. Equality (7) follows from Re{xy} = Re{x}Re{y} − Im{x}Im{y} for any x, y ∈ C.
If the stationary real-valued stochastic process X = {X(t), t ∈ R} is positively associated, we can apply Theorem 2.2 and, in the symmetric case, Corollary 2.3 to X(0) and X(t) to check the long range dependence of X.
To do so, let T = R in integral (1).
Lemma 2.4 Let |·| denote the Lebesgue measure on R. The above process X = {X(t), t ∈ R} is short or long range dependent iff X A = {X(t), t ∈ A} is short or long range dependent, respectively, for a set A ⊂ R with |A c | < ∞.
Proof: Show the short range dependence. We simply split up (1) into A and A c As the second integral is finite in any case, the relation (1) holds if and only if the first integral is finite, i.e. iff X A = {X(t), t ∈ A} is short range dependent. Now we give the main result of this section showing the use of characteristic functions to check the short or long range dependence of X.
Theorem 2.5 Suppose we have a stationary real-valued, positively associated stochastic process X = {X(t), t ∈ R} with absolutely continuous marginal distributions. Denote the univariate characteristic function of X(0) by ϕ and the bivariate characteristic function of (X(0), X(t)) by for any finite measure µ with Fourier transform ψ(s) = R exp{isx} µ(dx).
is symmetric for all t ∈ R, then condition (8) rewrites as Otherwise, X is long range dependent. Proof: (a) Take U = X(0) and V = X(t), where t ∈ R in Theorem 2.2. Then, U and V are absolutely continuous and identically distributed random variables. Therefore, the equality in (8)

| Long Range Dependence of α-stable Moving Averages
In this section, we investigate the long range dependence of SαS moving averages in continuous and discrete time.
By Definition 1.4, a symmetric α-stable moving average is defined by t ∈ R where Λ is a symmetric α-stable random measure and kernel function m ∈ L α (R), α ∈ (1,2).
Remark 3.1 (a) Note that the SαS moving average process X = {X(t), t ∈ R} is stationary, X(0) is absolutely continuous and, by Property 3.2.1 from Samorodnitsky and Taqqu (1994), the random vector (X(0), X(t)) is symmetric for every t ∈ R. (c) To exclude the trivial case X(t) = 0 for all t ∈ R we always assume that the Lebesgue measure of the set {x ∈ R|m(x) > 0} is positive.
By Samorodnitsky and Taqqu (1994), Proposition 3.4.2., the characteristic function ϕ : R → C of X(t), t ∈ R, is given by Moreover, the bivariate characteristic function ϕ t : Before we get to our main result, we need to introduce the α-spectral covariance of a stable vector as defined by Damarackas and Paulauskas (2016). Let S 1 = {x ∈ R 2 : x = 1} be the unit circle. Recall that a random vector Z = (X 1 , X 2 ) is symmetric α-stable with parameter α if there exists a finite measure Γ on S 1 , the so-called spectral measure, such that the characteristic function of Z is given by Definition 3.2 Suppose (X 1 , X 2 ) is an α-stable random vector with spectral measure Γ, then the α-spectral covariance of X 1 and X 2 is given by Let us calculate the α-spectral covariance of (X(0), X(t)), t ∈ R, where X is a SαS moving average.
is a SαS moving average process. Then, the α-spectral covariance of (X(0), X(t)), t ∈ R, is given by Proof: Denote m 1 (x) = m(−x) and m 2 (x) = m(t−x), Proposition 3.4.3 in Samorodnitsky and Taqqu (1994) and the the symmetry of Λ yields that (X(0), X(t)) is SαS with spectral measure Γ defined for all Borel sets A ⊂ S 1 by Hereby γ is an absolutely continuous measure w.r.t. the Lebesgue measure with density 1 2 (m 2 1 (x)+ m 2 2 (x)) α/2 . With f (s 1 , s 2 ) = |s 1 s 2 | α/2 sgn(s 1 s 2 ) we get Thus, Now, a sufficient condition for the short range dependence of X can be formulated in terms of Theorem 3.4 Let X = {X(t), t ∈ R} be a SαS moving average process with parameter α ∈ (1,2), nonnegative kernel function m and α-spectral covariance ρ t given in (13). If then X is short range dependent.
Proof: Apply Theorem 2.5 to X for some ε ∈ (0, m α α ) and choose Obviously, the right-hand side of the equality in (9) is bounded by Then, we show in Lemma A.3 that where the finiteness of I 2 clearly follows from condition (14).
By inequality (34) in Lemma A.3 we get Naturally, we may ask what condition (14) means in terms of the kernel function m. This is given by , t ∈ R} be a SαS moving average process with parameter α ∈ (1,2) and nonnegative kernel function m. If m ∈ L α/2 (R), then X is short range dependent.
Proof. By Theorem 3.4 we only need to show the integrability of ρ t . By Fubini's theorem and m ∈ L α/2 (R) we get Now it is natural to ask for sufficient conditions for the long range dependence of X.
Proof: Given in the appendix.
Additionally, if the kernel function m is symmetric and monotonically decreasing on [0, ∞), then we can simplify condition (16) as follows.
Corollary 3.7 Let X = {X(t), t ∈ R} be a SαS moving average process with parameter α ∈ (1,2) and nonnegative kernel function m which is symmetric and monotonically decreasing Proof. We use Theorem 3.6 and compute Now we give an example of a kernel function m ∈ L α (R) and show that the corresponding SαS moving average is long range dependent if m / ∈ L α/2 (R).
Thus, X is long range dependent if δ ≤ 2 α by Corollary 3.7 and is short range dependent if δ > 2 α by Theorem 3.5.
Remark 3.9 This example was given to motivate our conjecture that a SαS moving average is long range dependent iff m / ∈ L α/2 (R). However, its proof is still to be found.
Similar results as above can be obtained for symmetric α-stable linear time series Y .
Notice that Y can be written as a continuous time SαS moving average with parameter α ∈ (1,2) and kernel function m( Thus, Y is positively associated if the coefficients a j are nonnegative for all j ∈ Z. Moreover, the function Remark 3.11 Theorems 3.4, 3.5 and 3.6 as well as Corollary 3.7 apply to linear processes with the obvious substitute of ∞ t=−∞ for R dt. Indeed, let Y = {Y (t), t ∈ Z} be a stationary SαS time series with parameter α ∈ (1,2) and nonnegative coefficients then Y is long range dependent. If coefficients a j are additionally symmetric, i.e. a j = a −j for all j ∈ Z, and monotonically decreasing on Z + then Y is long range dependent if ∞ t=0 t a α t = ∞.

| Long Range Dependence of Max-stable Processes
In this section, we demonstrate that it is possible to use already existing dependence properties to check Definition 1.2 instead of inverting characteristic functions as in the previous sections.
Any max-stable process is positively associated, see for instance Proposition 5.29 in Resnick (2013). Its dependence properties are typically summarized by its pairwise extremal coefficients cf. Schlather and Tawn (2003). By a series expansion of the logarithm, it can be seen that , where θ t = 2 corresponds to the case of (asymptotic) independence between X(0) and X(t) while θ t = 1 means full dependence. Even though the joint distribution of (X(0), X(t)) is not uniquely determined by θ t , this characteristic turns out to be a useful tool for the identification of dependence properties. For instance, Stoev (2008), Kabluchko and Schlather (2010) and Dombry and Kabluchko (2017) provide necessary and sufficient conditions for ergodicity and mixing of a max-stable process in terms of its pairwise extremal coefficients.
Here, we focus on the property of long-range dependence given by Definition 1.2. We obtain bounds for Cov 1{X(0) > u}, 1{X(t) > v} , t ∈ T , u, v > 0, by making use of the following lemma.
Lemma 4.1 Let (X, Y ) be a bivariate max-stable random vector with α-Fréchet margins and extremal coefficient θ ∈ [1,2]. Then, we have Proof. It is well-known that the cumulative distribution function of a bivariate max-stable random vector (X, Y ) with α-Fréchet margins is of the form where (W X , W Y ) is a bivariate random vector with E W α X = E W α Y = 1 cf. Chapter 5 in Resnick (2013), for instance. This so-called spectral vector (W X , W Y ) is closely connected to the extremal Distinguishing between the two cases u ≤ v and v < u, it can be seen that the right-hand side of equation (20) can be bounded from above by where we used the fact that E W α X = E W α Y = 1. This gives the lower bound stated in the lemma. Analogously, we obtain which gives the corresponding upper bound.
Remark 4.2 Note that the lower bound in Lemma 4.1 corresponds to the bound given in Strokorb and Schlather (2015). For the so-called Molchanov-Tawn model, we even have i.e. the bound is sharp in this case.

Using Equation (3), Lemma 4.1 yields the following bounds for Cov
Consequently, one obtains for the integral in (1) that From the lower and the upper bound in (21), we directly obtain a necessary and sufficient condition for long range dependence. .
Proof. First, assume that (22) holds. Choosing the finite measure µ = δ {1} as the Dirac measure, we obtain from the lower bound in (21) and the inequality exp( Conversely, assume that (22) does not hold, i.e.
for all u ≥ 0. Combining this inequality with the upper bound in (21), we have for any finite measure µ on R. Thus, X is short range dependent.
Example 4.4 Here, we consider two popular examples of max-stable processes, namely the extremal Gaussian process and the Brown-Resnick process.
1. The extremal Gaussian process (Schlather, 2002) is a max-stable process based on maxima over an infinite number of copies of a centered stationary Gaussian process {W (t), t ∈ T } with T = R. Its extremal coefficients are given by where ρ t = Corr(W (t), W (0)) denotes the correlation function of the underlying Gaussian process W . Provided that ρ t ≥ 0 for all t ∈ T , we have that θ t ≤ 1 + 1/2, and, consequently, that is, the process is long range dependent by Theorem 4.3.
2. The Brown-Resnick process (Kabluchko et al., 2009) is a max-stable process constructed via maxima over an infinite numbers of copies of {exp(W (t)), t ∈ T }, where W is a Gaussian process with stationary increments on T = R. The extremal coefficients can be expressed in terms of the variogram γ(t) = E[(W (t) − W (0)) 2 ], t ∈ R, of the underlying Gaussian process W via the relation where Φ denotes the standard normal distribution function. Now assume that there exists some constant C > 8 such that γ(t) ≥ C log |t| for |t| being sufficiently large. From Mill's ratio 1 − Φ(x) ∼ x −1 ϕ(x) as x → ∞ with ϕ being the standard normal density function, it follows that is integrable. Thus, by Theorem 4.3, the Brown-Resnick process is short range dependent if lim inf |t|→∞ γ(t)/ log |t| > 8, which is true, for instance, for any fractional Brownian motion W . Note that, in this case, the process also possesses a mixed moving maxima representation, cf. Kabluchko et al. (2009).
If, in contrast, the variogram γ of the underlying Gaussian process W is bounded as in the case of a stationary process, we obtain that sup t∈T θ t < 2. Thus, analogously to the case of the extremal Gaussian process, the Brown-Resnick process can be shown to be long range dependent.
Note that these conditions also appear in the literature when analyzing the existence of a mixed moving maxima (M3) representation of a Brown-Resnick process: In Kabluchko et al. (2009), it is shown that a M3 representation exists if lim inf |t|→∞ γ(t)/ log |t| > 8. In case of a bounded variogram, however, the resulting Brown-Resnick is not even mixing. As sufficient and necessary conditions for the existence of a M3 representation are stated in terms of the asymptotic behavior of the sample paths of the underlying Gaussian process rather than in terms of its variogram (cf. Wang and Stoev, 2010;Dombry and Kabluchko, 2017, for instance), to the best of our knowledge, there is no general treatment of the gap between these two cases. Similarly, for short range and long range dependence, respectively, a detailed analysis of further cases is beyond the scope of this paper.
Remark 4.5 In this section, using known dependency properties allows to avoid complex calculation such that no restrictions on the index set T are required. In particular, all the results are also valid for max-stable random fields, i.e. the case that T ⊂ R d .

A | Appendix
Theorem A.1 Let U and V be two absolutely continuous random variables with marginal and joint characteristic function ϕ U and ϕ, respectively. Then, for all u, v ∈ R (a) the marginal distribution function of U is given by (b) the joint distribution function of U and V is given by Proof: (a) Given in Gil-Pelaez (1951).
(b) Although this seems to be a classical result, we were not able to find its proof in an accessible form. So we have adjusted the proof of the univariate case in Gil-Pelaez (1951) to cover the bivariate case. For u, v, y 1 , y 2 ∈ R denote g(y 1 , y 2 ) = sgn(y 1 − u) sgn(y 2 − v) Further, denote the marginal distribution functions of U and V by F U and F V , respectively.
Proof of Theorem 2.1. We use relation (3) and rewrite the joint and marginal distribution functions using Theorem A.1 which gives us the distribution function of U , V , respectively: Now, we denote the double integral ∞ 0 ∞ 0 . . . ds 1 ds 2 in (23) by C. Then, from (23), (29) and (30) we have Consequently, we can compute the RHS of (3) as First, we compute the product of A and B as By rearranging the terms and factorizing we find that Finally, the claim follows from ϕ U (−s 1 ) = ϕ U (s 1 ), ϕ(−s 1 , −s 2 ) = ϕ(s 1 , s 2 ), e ix = e −ix for all s 1 , s 2 , x ∈ R and Re(z) = z+z 2 for all z ∈ C.
(b) Suppose a < b, then The claim follows analogously if a ≥ b.
is monotonically increasing.
Thus, we get that Lemma A.3 Let X = {X(t), t ∈ R} be a SαS moving average process with parameter α ∈ (0,2], non-negative kernel function m ∈ L α (R), m(x) > 0 on a set of positive Lebesgue measure and α−spectral covariance ρ t . Let ϕ and ϕ t , t ∈ R, be the characteristic functions given in equations (10) and (11). Then, Proof. First, we compute the absolute value of the difference of characteristic functions in I 1 for any s 1 , s 2 > 0: where we have used Lemma A.2(a) in the last inequality.
Consequently, by Definition 1.2 and Fubini's theorem, X is long range dependent if