Heavy-tailed time series

Transcription

1 Heavy-tailed time series Spatio-temporal extremal dependence 1 Thomas Mikosch University of Copenhagen mikosch/ 1 Rouen, June

2 2 Contents 1. Heavy tails and extremal dependence - some empirical facts Heavy tails in real-life data Finance Insurance Telecommunications Extremal dependence/independence in real-life data Independence in insurance data Extremal independence in telecommunication data Extremal dependence in financial data Extreme value theory for iid sequences Leadbetter et al. (1983), Resnick (1987, 2007), Embrechts et al. (1997), de Haan and Ferreira (2006) Max-stable distributions (extreme value distributions) Maximum domains of attraction (MDA) The extremal index a measure of the extremal cluster size Definition Examples Modeling heavy tails and serial dependence via regularly varying time series Univariate regularly varying distributions Multivariate regular variation Resnick (1987,2007) Operations on regularly varying vectors Regularly varying stationary sequences Examples 55 Linear process 55 Solution to stochastic recurrence equation 57 Other examples 62

3 Limit representation of a regularly varying stationary sequence Basrak and Segers (2009) The extremal index of a regularly varying sequence revisited The extremogram - an autocorrelation function for serial extremal dependence A motivating example: the tail dependence coefficient Definition Davis, M. (2009) Examples The sample extremogram Bootstrapping the sample extremogram Variations on the theme Davis, M., Cribben (2012ab), Davis, M., Zhao (2013) Cross-extremogram The extremogram of return times between rare events Frequency domain analysis M. and Zhao (2014) Problems and possible extensions Max-stable processes - a flexible model for serial extremal dependence Definition de Haan (1984) Characterization of α-fréchet max-stable processes Stationary max-stable processes The Brown-Resnick process Brown, Resnick (1977) Concluding remarks 132 References 135 3

4 4 1. Heavy tails and extremal dependence - some empirical facts 1.1. Heavy tails in real-life data Finance. log-returns time Figure 1. Plot of 9558 S&P500 daily log-returns from January 2, 1953, to December 31, The year marks indicate the beginning of the calendar year.

5 5 density S&P quantiles normal x empirical quantiles Figure 2. Left: Density plot of the S&P500 data. The limits on the x-axis indicate the range of the data. QQ-plot of the S&P500 data against the normal distribution.

6 6 Hill m Figure 3. Hill plot (dotted line) for the S&P500 data with 95% asymptotic confidence bounds. The Hill estimator approximates the tail index α in the model P(X t > x) cx α as a function of the m upper order statistics in the return sample.

7 lower tail index Upper tail index Hill estimates of the lower and upper tail indices of 442 time series of the S&P 500, , assuming that P(±X t > x) cx α ±.

8 Insurance Figure 4. Danish fire insurance data.

9 Figure 5. Histogram of the logarithmic Danish fire insurance data.

10 Figure 6. Empirical mean excess function of the Danish fire insurance data. u

11 Telecommunications. 11 Teletraffic Data Teletraffic Data n n Figure 7. Time series of transmission durations (BU data).

12 Random Height Random Height time (sec) time (sec) 0.8 Random Height time (sec) Figure 8. Mice and elephants plots (S. Marron).

13 1.2. Extremal dependence/independence in real-life data Independence in insurance data. x(t) 0.0e e e e e e e e e e e e e e e e+07 x(t 1) Figure 9. Scatterplot of US fire insurance losses - independence.

14 Extremal independence in telecommunication data. Teletraffic file sizes X_t 0e+00 1e+06 2e+06 3e+06 4e+06 5e+06 0e+00 1e+06 2e+06 3e+06 4e+06 5e+06 X_(t 1) Figure 10. Scatterplot of file sizes of teletraffic data - extremal independence

15 Extremal dependence in financial data. 15 USD DEM USD FFR Figure 11. Scatterplot of 5 minute foreign exchange rate log-returns, USD-DEM against USD-FRF.

16 16 2. Extreme value theory for iid sequences Leadbetter et al. (1983), Resnick (1987, 2007), Embrechts et al. (1997), de Haan and Ferreira (2006) 2.1. Max-stable distributions (extreme value distributions). A random variable X and its distribution F are max-stable if for every n 2 there exist c n > 0, d n R, such that for iid copies (X i ) of X, c 1 n (M n d n ) = c 1 n ( max i=1,...,n X i d n ) d= X.

17 Any max-stable distribution belongs to the location/scale 17 family of one of the three standard max-stable distributions (also called extreme value distributions): Φ α (x) = e x α, x > 0, α > 0 Fréchet Ψ α (x) = e x α, x < 0, α > 0 Weibull Λ(x) = e e x, x R, Gumbel. The max-stable distributions are the only possible non-degenerate weak limits for standardized maxima of an iid sequence (Fisher-Tippett Theorem 1928, Gnedenko (1943)). The 3 max-stable types can be written as one parametric family (generalized extreme value distribution (GEV)).

18 18 Transformation of max-stable random variables. If X > 0 has a Φ α distribution, logx α has distribution Λ X 1 has distribution Ψ α.

19 2.2. Maximum domains of attraction (MDA). 19 The distribution F of X is in the maximum domain of attraction of the max-stable distribution G {Φ α,ψ α,λ} (F MDA(G)) if there exist constants a n > 0, b n R such that lim n P(a 1 n (M n b n ) x) G(x), x R. F MDA(Φ α ): Regular variation of the right tail F(x) = 1 F(x) = P(X > x) = x α L(x), x > 0, for a slowly varying function L. Then the moments E[X α+δ + ], δ > 0, are infinite. F MDA(Ψ α ): F has finite right endpoint x F. F MDA(Λ): Moderately heavy light tails.

20 20 Examples: MDA(Φ α ): Student with α degrees of freedom, Cauchy (α = 1), infinite variance α-stable distributions, Pareto F(x) = x α, x > 1, log-gamma distribution. MDA(Ψ α ): uniform, β-distribution. MDA(Ψ α ): log-normal distribution, Weibull F(x) = e xτ, x > 0, τ > 0, gamma distribution, normal distribution.

21 3. The extremal index a measure of the extremal cluster size 21 Let (X t ) t Z be a strictly stationary real-valued time series. Its autocovariance and autocorrelation functions do in general not contain information about extremal dependence Definition. The extremal index θ X is a standard measure of extremal dependence in a sequence: 2 for M n = max t=1,...,n X t and a suitable sequence u n x F P(M n u n ) [P(X 1 u n )] nθ X. θ X [0,1] has the interpretation as reciprocal of the expected cluster size above high thresholds. 2See Leadbetter, Lindgren, Rootzén (1983); cf. Embrechts et al. (1997), Section 8.1

22 Figure 12. A sequence of iid random variables Y i (Top) with distribution function F, where F is standard exponential. Bottom: the sequence of pairwise maxima max(y i,y i+1 ) with distribution F. By construction, extremes appear in clusters of size 2. The extremal index is θ = 1/2.

23 3.2. Examples. 23 A Gaussian stationary sequence (X t ) with autocorrelation function ρ X (h) = o(1/logh), h : θ X = 1. No extremal clustering. Leadbetter, Lindgreen, Rootzén (1983) AR(1) model X t = φx t 1 + Z t, φ ( 1,1), (Z t ) iid student with α degrees of freedom: θ X = 1 φ α. Models for log-returns X t = logp t logp t 1 : X t = σ t Z t, (Z t ) iid, σ t > 0 The simple stochastic volatility model: (logσ t ) linear Gaussian, independent of iid student (Z t ): θ X = 1 Davis, Mikosch (2001ab,2009ab) No extremal clustering.

24 24 SV process time GARCH process time Figure 13. Top: Stochastic volatility process X t = σ t Z t for iid student (Z t ) with 4 degrees of freedom, Gaussian ARMA(1,1) process logσ t = 0.5logσ t η t 1 +η t. Bottom: GARCH(1,1) process X t = ( X 2 t σ2 t 1 )0.5 Z t for iid standard normal (Z t ).

25 The GARCH(1,1) model: Bollerslev (1986) X t = σ t Z t, and 25 σ 2 t = α 0 + (α 1 Z 2 t 1 + β 1)σ 2 t 1, (Z t) iid N(0,1). There exists α > 0 such that E [ (α 1 Z β 1) α/2] = 1 and de Haan, Resnick, Rootzén, de Vries (1989) α 2 1 P ( max n 1 n t=1 (α 1Z 2 t + β 1) y 1) y α 2 1 dy =θ σ (0,1).

26 26 The GARCH(1,1) model: X t = σ t Z t, σ 2 t = α 0 + (α 1 Z 2 t 1 + β 1)σ 2 t 1, (Z t) iid N(0,1). There exists α > 0 such that E(α 1 Z β 1) α/2 = 1 and α 2 1 P ( max n 1 n t=1 (α 1Z 2 t + β 1) y 1) y α 2 1 dy =θ σ (0,1). Expressions for the extremal index of a stationary process are often complicated.

27 The GARCH(1,1) model: X t = σ t Z t, 27 σ 2 t = α 0 + (α 1 Z 2 t 1 + β 1)σ 2 t 1, (Z t) iid N(0,1). There exists α > 0 such that E(α 1 Z β 1) α/2 = 1 and α 2 1 P ( max n 1 n t=1 (α 1Z 2 t + β 1) y 1) y α 2 1 dy =θ σ (0,1). Expressions for the extremal index of a stationary process are often complicated. Monte Carlo simulation is not straightforward.

28 28 The GARCH(1,1) model: X t = σ t Z t, σ 2 t = α 0 + (α 1 Z 2 t 1 + β 1)σ 2 t 1, (Z t) iid N(0,1). There exists α > 0 such that E(α 1 Z β 1) α/2 = 1 and α 2 1 P ( max n 1 n t=1 (α 1Z 2 t + β 1) y 1) y α 2 1 dy =θ σ (0,1). Expressions for the extremal index of a stationary process are often complicated. Monte Carlo simulation is not straightforward. Estimation of the extremal index and extremal cluster size distribution is non-trivial; see C. Y. Robert (2009)

29 4. Modeling heavy tails and serial dependence via regularly 29 varying time series 4.1. Univariate regularly varying distributions.. 3 Recall that F MDA(Φ α ) for some α > 0 if and only if F(x) = P(X > x) = x α L(x), x > 0, for some slowly varying function L, (i.e. L(cx)/L(x) 1, x, c > 0) 3See Bingham, Goldie, Teugels (1987) for an encyclopedia on regularly varying functions, Resnick [52, 53] for a modern theory of regular variation for the purposes of applied probability and statistics.

30 30 We call a random variable X R and its distribution F regularly varying with index α > 0 if there exist constants p,q 0 such that p + q = 1 and F( x) qx α L(x) and F(x) px α L(x), x. If e.g. p = 0: F(x) = o(x α L(x)), x. Equivalently, X is regularly varying with index α > 0 and P(X x) P( X > x) q and P(X > x) P( X > x) p, x. Examples. Pareto, student, Cauchy, α-stable, α (0, 2), Burr, log-gamma, Fréchet.

31 Two fundamental operations Convolution. Feller (1971) Let X 1 > 0 be regularly varying with α > 0. Assume X 2 > 0 regularly varying with index α and independent of X 1 OR P(X 2 > x) = o(p(x 1 > x)). Then X 1 + X 2 is regularly varying with index α and P(X 1 + X 2 > x) P(X 1 > x) + P(X 2 > x), x. Multiplication. Breiman (1965) σ > 0, X > 0 independent and Eσ α+δ < for some δ > 0, X regularly varying with index α. Then, as x, P(σX > x) Eσ α P(X > x). 4See Resnick (2007), Jessen and M. (2006)

32 32 Examples. Stochastic volatility model. X t = σ t Z t, t Z, σ t > 0, (logσ t ) Gaussian stationary process, independent of (Z t ) iid regularly varying with index α, Then, by Breiman as x, P(X t > x) E[σ α 0 ]P(Z 0 > x), P(X t x) E[σ α 0 ]P(Z 0 x).

33 Moving average I. 33 X t = θ 0 Z t + θ 1 Z t θ m Z t m, t Z, m 1. Z t > 0 iid regularly varying with index α. Then m P(X t > x) P(θ i Z i > x) i=1 P(Z 0 > x) m θ α i I θi >0, x. i=0

34 34 Moving average II. X t = θ 0 Z t + θ 1 Z t θ m Z t m, t Z, m 1. Z t iid regularly varying with index α and tail balance condition P(Z t > x) p L(x) and P(Z x α t x) q L(x) x α Then P(X t > x) P( Z 0 > x) m θ i α( pi θi >0 + qi θi <0), x. i=0

35 The principle of a single large jump (heavy-tail heuristics): 35 A large value of X t appears in the most natural way, due to a single large summand θ i Z t i. ( m ) P(X t > x) P {θ i Z t i > x} i=1 ( m ) P {θ i Z t i > x,θ i Z t j x,j i} i=1 m P(θ i Z t i > x). i=1 These heuristics remain valid for classes of heavy-tailed distributions other than the regularly varying ones.

36 36 Subexponential distributions: X 1,X 2 > 0 iid P(X 1 + X 2 > x) 2P(X 1 > x). The subexponential distributions are the classical large claim distributions in insurance mathematics: Pareto, Burr, log-gamma, log-normal, Weibull,... see Embrechts et al. (1997) Some subexponential distributions are semi-exponential (e.g. log-normal, Weibull), i.e., they have all (power) moments finite but have infinite moment generating function. Subexponential distributions are long-tailed; see also Foss et al. (2013) for a textbook treatment of long-tailed distributions. P(X 1 > x + y) P(X 1 > x), x, y R.

37 37 How can one model spatio-temporal extremal dependence and heavy tails? One needs to model both the size and the direction of extremes.

38 38 Asymptotic extremal independence Teletraffic file sizes X_t 0e+00 1e+06 2e+06 3e+06 4e+06 5e+06 0e+00 1e+06 2e+06 3e+06 4e+06 5e+06 X_(t 1) Figure 14. Scatterplot of file sizes of teletraffic data.

39 Asymptotic extremal dependence 39 USD DEM USD FFR Figure 15. Scatterplot of 5 minute foreign exchange rate log-returns, USD-DEM against USD-FRF.

40 Multivariate regular variation Resnick (1987,2007). A random vector X R d and its distribution are regularly varying with index α > 0: there exists a random vector Θ S d 1 = {x R d : x = 1} such that for t > 0: P( X > tx,x/ X ) P( X > x) w t α P(Θ ), x. The distribution of Θ: spectral measure of X. Equivalently, P ( x 1 X ) P( X > x) v µ( ), for a non-null Radon measure µ on the Borel σ-field of R d 0 = Rd \{0} with µ(ta) = t α µ(a), t > 0.

41 Equivalently: as x, P( X > tx) P( X > x) t α, t > 0, and ( X ) w P X X > x P(Θ ). 41 A toy example. R,θ independent, θ distributed on [0,2π), P(R > r) = r α, r > 1, Pareto. X = R(cosθ,sinθ) = RΘ, Then X = R and X/ X = Θ = (cosθ,sinθ), P(R > tx) = t α P(R > x), tx > 1, P(Θ R > x) = P(Θ ).

42 42 X_1 X_ X_1 X_ Figure 16. IID vectors X i from the toy model with tail index α = 5. Left: θ is uniform on [0,2π). Right: θ has a discrete uniform distribution on the points 2πi/50.

43 Example 43 Assume X = (X 1,X 2 ) R 2 + iid regularly varying with index α > 0. Choose X = max(x 1,X 2 ). Then, for any ε > 0, P(x 1 X (ε, ) 2 ) P( X > x) = P(X 1 > εx,x 2 > εx) P(max(X 1,X 2 ) > x) = [P(X 1 > εx)] 2 2P(X 1 > x) [P(X 1 > x)] 2 µ((ε, ) 2 ) = 0. µ does not charge sets bounded away from the axes. Hence µ and the spectral measure P(Θ ) are concentrated on the axes.

44 44 In general, if a regularly varying vector X has independent components the measures µ and P(Θ ) are concentrated on the axes. This means: It is very unlikely that two distinct components of X are large (extreme) at the same time. Extremes occur close to the axes. For a vector X with dependent components, this property is sometimes referred to as asymptotic extremal independence.

45 Example. The stochastic volatility model: for s t, iid 45 regularly varying (Z t ), independent of strictly stationary log-normal (σ t ), P( X t > εx, X s > εx) = P(min(σ t Z t,σ s Z s ) > εx) P(max(σ t,σ s )min( Z t, Z s ) > εx) E[max(σ 2α t,σ 2α s )] P(min( Z t, Z s ) > εx) = E[max(σ 2α t,σ 2α s )] [P( Z t > εx)] 2 = o(p( X t > x)), x. Here we used Breiman s result. Hence for any ε > 0, as x, P( X t > εx, X s > εx) P( (X t,x s ) > x) 0 = µ({(x 1,x 2 ) : min( x 1, x 2 ) > ε}).

46 46 This means: the limit measures µ and P(Θ ) of the regularly varying vector (X t,x s ), t s, are concentrated on the axes. or there is asymptotic extremal independence between X t and X s although X t,x s are dependent. More generally: the components of any vector (X 1,...,X n ) are asymptotically independent.

47 Further examples 47 X has iid student(α) distributed components. P(Θ ) is concentrated on the intersection of unit ball and axes. X has a multivariate student(α) distribution. P(Θ ) is supported on the whole unit ball. X is obtained from a Gaussian vector by transforming the marginals to student(α). Then P(Θ ) is concentrated on the intersection of unit ball and axes. X exhibits asymptotic extremal independence.

48 Operations on regularly varying vectors. Lemma. Let X R d be RV(µ X,α) and g : R d R d be continuous, positively homogeneous of order 1. Then P(x 1 g(x) ) v µ X (g 1 ( )). P( X > x) If µ X (g 1 ( )) is non-null g(x) is regularly varying with index α. In particular, if the inverse image of the null set in R d under the mapping g is the null set in R d the limit measure µ X (g 1 ( )) is non-null and g(x) is regularly varying with index α.

49 Examples. Let X be an R d -valued random vector which is 49 RV(µ X,α). If the support of X is the positive quadrant and g(x) = max(x 1,...,X d ), then the inverse image of the null set is the null set and g(x) is regularly varying. The sum g(x) = X X d is regularly varying if µ X (g 1 ( )) o. In this case, for z > 0, µ X (g 1 (z, ]) = µ X ({y R d 0 : y y d > z}) = z α µ X ({y R d 0 : y y d > 1}), µ X (g 1 (, z]) = µ X ({y R d 0 : y y d z}) = z α µ X ({y R d 0 : y y d 1}).

50 50 The right-hand side can be zero, e.g. if X = (Y, Y ) and Y > 0 is regularly varying. Then X 1 + X 2 = 0. The function g(x) = θ X is positively homogeneous for θ S d 1 (relative to the Euclidean norm). If θ has positive components and X is supported on the positive quadrant, θ X = 0 a.s. implies X = 0 a.s. In this case, θ X is regularly varying. Any norm g(x) = X is positively homogeneous and X = 0 implies X = 0 a.s. Hence X is regularly varying.

51 Convolution. Let X i be R d -valued random vectors which are 51 RV(µ Xi,α), i = 1,2. Assume that P( X 2 > x) (4.1) lim x P( X 1 > x) = c 0 for some non-negative constant c 0 and P( X 1 > x, X 2 > x) = o(p( X 1 > x)), x. Then P(x 1 (X 1 + X 2 ) ) v µ X1 + c 0 µ X2. P( X 1 > x) The result remains valid for regularly varying X 1 and any X 2 if (4.1) holds with c 0 = 0.

52 52 Example. Assume (X i ) iid and RV(µ X,α). Then for S n = X X n, P(x 1 S n ) v nµ X ( ). P( X > x) In the univariate case, for iid X i > 0, this property defines the subexponential property: P(S n > x) np(x 1 > x), n, for any (some) n 2. There is no good definition of subexponentiality in higher dimensions.

53 53 Multiplication. (Multivariate Breiman) 5 Let A be a random d d matrix, independent of the R d -valued random vector X which is RV(µ X,α) for some α > 0 and E[ A α+ε ] < for some ε > 0. Here A is any norm of A. Then P(x 1 AX ) P( X > x) v E [ µ X ({y R d : Ay }) ] = E [ µ X (A 1 ) ]. If the limit measure is non-null then AX is regularly varying with index α. The result remains valid if E[ A α ] < and P( X > x) cx α as x. 5See Basrak, Davis, M. (2002).

54 Regularly varying stationary sequences. A real-valued strictly stationary sequence (X t ) is regularly varying with index α > 0 if its finite-dimensional distributions are regularly varying with index α. Equivalently, for every k 1, there exists a non-null Radon measure on R k 0 such that P(x 1 (X 1,...,X k ) ) v µ k ( ). P( X 0 > x) The measures µ k determine the extremal dependence structure of the finite-dimensional distributions. Notice: Normalization P( X 0 > x) does not depend on k.

55 Examples. 55 Linear process. Examples of linear processes are ARMA processes with iid noise (Z t ), e.g. the AR(p) and MA(q) processes X t = Z t + ϕ 1 X t ϕ p X t p, X t = Z t + θ 1 Z t θ q Z t q. A linear process X t = j ψ j Z t j, t Z, is regularly varying with index α > 0 if the iid sequence (Z t ) is regularly varying with index α.

56 56 Under mild conditions on (ψ j ), 6 P(X 0 > x) P( Z 0 > x) j ψ j α (pi ψj >0 + qi ψj <0)= ψ α α, x. Expressions for µ k are complicated. 6See Resnick (1987); cf. Embrechts et al. (1997), Appendix

57 Solution to stochastic recurrence equation. 57 For an iid sequence ((A t,b t )) t Z, A,B > 0, the stochastic recurrence equation X t = A t X t 1 + B t, t Z, has a unique stationary solution X t = B t + t 1 i= A t A i+1 B i, t Z, provided ElogA < 0, Elog + B <. The sequence (X t ) is regularly varying with index α which is the unique solution to E[A κ ] = 1, κ > 0, (given this solution exists) Kesten (1973), Goldie (1991) and P(X 0 > x) c + x α, x.

58 58 Regular variation of the finite-dimensional distributions. Notice that with Π i = A 1 A i, i 1, (X 1,...,X n ) = (Π 1,...,Π n )X 0 + R n We assume E[ B α ] < hence E[ R n α ] < and therefore P( R n > x) = o(p( X 0 > x)), x. Applications of the convolution and multiplication rules for regularly varying vectors (p. 51 and 53) yield the joint regular variation of (X 1,...,X n ): for smooth sets C (0, ) n, P(x 1 (X 1,...,X n ) C) P(X 0 > x) Eµ X0 ({y (0, ) : (Π 1,...,Π n )y C}).

59 But for b > 0, 59 P(x 1 X 0 > b) P(X 0 > x) b α = µ X0 ((b, )), x. Hence Eµ X0 ({y (0, ) : (Π 1,...,Π n )y C}) = 0 αy α 1 P((Π 1,...,Π n )y C)dy Then we also have as x, P(x 1 (X 1,...,X n ) C X 0 > x) = P(x 1 (X 1,...,X n ) C,x 1 X 0 > 1) P(X 0 > x) 1 αy α 1 P((Π 1,...,Π n )y C)dy = P(Y (Π 1,...,Π n ) C),

60 60 where Y has a Pareto distribution with density αy α 1, y > 1, independent of (Π i ).

61 The GARCH(1, 1) process Bollerslev (1986) satisfies a stochastic 61 recurrence equation: for an iid standard normal sequence (Z t ), positive parameters α 0,α 1,β 1, σ 2 t = α 0 + (α 1 Z 2 t 1 + β 1)σ 2 t 1. The process X t = σ t Z t is regularly varying with index α satisfying E [ (α 1 Z 2 + β 1 ) α/2] = 1. Then, by Breiman s result (see p. 31), as x, P(X t > x) = P(σ t Z t > x) E(Z 0 ) α + P(σ 0 > x) E(Z 0 ) α + c +x α, P(X t x) = P(σ t Z t x) E(Z 0 ) α + P(σ 0 > x) E(Z 0 ) α + c +x α. Compare with the tails of a stochastic volatility model on p. 32.

62 Other examples. α-stable stationary processes are regularly varying with index α (0, 2). Samorodnitsky and Taqqu (1994) Max-stable stationary processes with Fréchet (Φ α ) marginals are regularly varying with index α > 0; see Section 6 on p The simple stochastic volatility model (see p. 32) with iid regularly varying noise.

63 Limit representation of a regularly varying stationary sequence. 63 Basrak and Segers (2009) found a useful representation of the limiting measures of the finite-dimensional distributions of a regularly varying sequence.

64 64 Theorem. Assume that (X t ) is an R d -valued strictly stationary sequence. TFAE to regular variation of (X t ) with index α > 0. (1)There exists a sequence (Y t ) of R d -valued random vectors with P( Y 0 > x) = x α, x > 1, such that the following conditional limit relation holds for every h 0: P ( x 1 (X 0,...,X h ) X 0 > x ) w ( P (Y0,...,Y h ) ). (2) X 0 is regularly varying with index α and there exists an R d -valued process (Θ t ) such that for every h 0: P ( X 0 1 (X 0,...,X h ) X 0 > x ) w ( P (Θ0,...,Θ h ) ). Moreover, the process (Y t ) has representation Y (Θ t ), where Y = d Y 0, and Y is independent of (Θ t ).

65 Basrak and Segers (2009) refer to (Y t ) and (Θ t ) as the tail-process and tail spectral process, respectively. As a particular consequence we observe that the distribution of Θ 0 is the spectral measure of X 0. Indeed, we have P ( X 0 1 X 0 X 0 > x ) w P(Θ0 ). Also, the distributions of Θ k, k 0, and Θ 0 are in general not the same. In general, Θ k 1 a.s. for k 0. Also notice that, in contrast to the sequence (X t ), (Y t ) and (Θ t ) are not strictly stationary. 65

66 66 Example. Assume (X t ) iid regularly varying. Then, for k 0, ε > 0, P(x 1 X k > ε X 0 > x) = P(x 1 X k > ε) 0, x. Thus x 1 X k P 0 and (Y 1,...,Y k ) = (Θ 1,...,Θ k ) = 0 a.s. k 1. Example. Assume (X t ) positive strictly stationary regularly varying. Then the following limits exist (extremogram). ρ(h) = lim x P(X h > x X 0 > x) = P(Y h > 1) = P(Y Θ h > 1) = = Emin(1, Θ h α ), h 1. 1 αy α 1 P( Θ h > y 1 )dy

67 The extremal index of a regularly varying sequence revisited. 67 The notion of extremal index originates from Newell (1964), Loynes (1965), O Brien (1974) and was made precise by Leadbetter (1983). The notion of extremal index depends on the definition of a cluster of high level exceedances in the sequence (X t ). Although it is intuitively clear what an extremal cluster means (many unusually large values appear roughly at the same time) an exact definition is not easy. Various probabilistic definitions exist, depending on some asymptotic theory, in particular on point process convergence Leadbetter et al. (1983), Falk et al. (2004), Embrechts et al. (1997), Section 8.1, while for statistical purposes

68 68 (estimation of θ) one has to be pragmatic since one cannot rely on the form of a limiting point process. One way of defining a cluster in a sample X 1,...,X n is to split the sample into blocks of equal size m: X 1,...,X m,x m+1,...,x 2m,...,X (kn 1)m+1,...,X n. k n = [n/m] for the number of (full) blocks. For asymptotic theory, it will be important that m = m n and m = o(n). Now we simply assume that a block constitutes an extremal cluster if there is at least one exceedance of the high threshold u = u n x F in this block, and the expected cluster size is given

69 by ( m E = = t=1 m t=1 m t=1 I {Xt >u n } M m > u n ) P(X t > u n,m m > u n ) P(M m > u n ) P(X t > u n ) P(M m > u n ) = mp(x > u n) P(M m > u n ) = θ 1 n. We define the extremal index θ as the limit of the reciprocal of the expected cluster size of exceedances of u n in a block of size m: θ = lim n θ n = lim n P(M m > u n ) mp(x > u n ), provided this limit exists, and then θ [0,1]. 7 7For positive θ, Leadbetter (1983) showed that this definition is independent of the particular choice of a threshold sequence u n x F. 69

70 70 Our next goal is to reduce the calculation of θ to a problem for finitely many random variables X 1,...,X l. We use an argument from Segers (2005); cf. Lemma 2.8 in Davis, Hsing (1995). Lemma. Assume u n x F = and the anti-clustering condition lim limsup P(M l,m > u n X 0 > u n ) = 0, l n where M s,t = max s i t X i for s t. Then the following relation holds lim limsup θ n P(M l u n X 0 > u n ) = 0, l n and liminf n θ n > 0.

71 The anti-clustering condition prevents the sequence (X t ) from staying too long above the threshold u n. No extremal long-range dependence. We conclude that if θ = lim n θ n exists, lim limsup θ P(M l u n X 0 > u n ) = 0. l n in agreement with O Brien s (1987) characterization of the extremal index as the limit θ = lim n P(M ln u n X 0 > u n ) for a sequence (l n ) with l n = o(n), thresholds u n x F such that nf(u n ) 1 as n, provided an asymptotic independence and other conditions hold. 71

72 72 Example. Basrak and Segers (2009) For a positive strictly stationary regularly varying sequence (X t ), satisfying the anti-clustering condition, we know that the limits exists: lim P(M l u n X 0 > u n ) = P ( max Y t 1 ) n t=1,...,l P ( sup t 1 Hence the extremal index is given by ( ) θ = P supy t 1 t 1 ( = αy α 1 P 1 [ ] = E 1 supθ α t t 1 + [ = E supθ α t sup t 0 t 1 ( = P sup t 1 Θ α t Y t 1 ), l. Y max t 1 Θ t 1 Θ t y 1 )dy ]. )

73 Example. Recall that the solution to the stochastic recurrence 73 equation X t = A t X t 1 + B t, t Z, is regularly varying (see p. 57) and the finite-dimensional distributions satisfy P(x 1 (X 1,...,X n ) C X 0 > x) P(Y (Π 1,...,Π n ) C), where Y has a Pareto distribution with density αy α 1, y > 1, independent of Π i = A 1 A i, i 1. Hence and (Θ 0,Θ 1,...,Θ n ) d = (1,Π 1,...,Π n ) θ = E[sup t 0 Π α t sup t 1 Π α t ].

74 74 5. The extremogram - an autocorrelation function for serial extremal dependence Goals. Find measure of serial extremal dependence in a strictly stationary sequence (X t ). Is there an analog of the sample autocorrelation function for serial extremal dependence? Estimation of this function should be uncomplicated and graphical visualization should be possible.

75 5.1. A motivating example: the tail dependence coefficient. 75 In risk management, 8 the (upper) tail dependence coefficient gained some popularity: λ h = lim x P(X h > x X 0 > x), h 1. If λ h = 0: no upper tail dependence as, for example, in a non-trivial Gaussian stationary sequence. The definition of λ h requires that the limit exists. A sufficient condition is regular variation of (X t ), i.e., the finite-dimensional distributions of (X t ) are regularly varying. 8For example, McNeil et al. [42].

76 76 Recall: Regularly varying strictly stationary time series A strictly stationary R d -valued sequence (X t ) is regularly varying with index α > 0: there exist limit measures µ h o, h 1, such that P(x 1 (X 1,...,X h ) ) P( X 0 > x) v µ h ( ), x. Equivalently, for any sequence (a n ) satisfying P( X 1 > a n ) n 1 there exist limit measures µ h o, h 1, such that np(a 1 n (X 1,...,X h ) ) v µ h ( ).

77 The upper tail dependence coefficient revisited 77 Let (X t ) be a strictly stationary real-valued sequence which is regularly varying with index α > 0. Let A = B = (1, ) and Y h = (X 0,...,X h ). Then λ h = lim x P(X h > x X 0 > x)

78 78 The upper tail dependence coefficient revisited Let (X t ) be a strictly stationary real-valued sequence which is regularly varying with index α > 0. Let A = B = (1, ) and Y h = (X 0,...,X h ). Then λ h = lim x P(X h > x X 0 > x) P(x 1 Y h A R h 1 0 B) = lim x P(x 1 Y h A R h 0)

79 The upper tail dependence coefficient revisited 79 Let (X t ) be a strictly stationary real-valued sequence which is regularly varying with index α > 0. Let A = B = (1, ) and Y h = (X 0,...,X h ). Then λ h = lim x P(X h > x X 0 > x) P(x 1 Y h A R h 1 0 B) = lim x P(x 1 Y h A R h 0) P(x 1 Y h A R h 1 0 B)/P( X 0 > x) = lim x P(x 1 Y h A R h 1 0 )/P( X 0 > x)

80 80 The upper tail dependence coefficient revisited Let (X t ) be a strictly stationary real-valued sequence which is regularly varying with index α > 0. Let A = B = (1, ) and Y h = (X 0,...,X h ). Then λ h = lim x P(X h > x X 0 > x) P(x 1 Y h A R h 1 0 B) = lim x P(x 1 Y h A R h 0) P(x 1 Y h A R h 1 0 B)/P( X 0 > x) = lim x P(x 1 Y h A R h 1 0 )/P( X 0 > x) = µ h+1(a R h 1 0 B) µ h+1 (A R h 0 ).

81 5.2. Definition Davis, M. (2009). 81 For an R d -valued strictly stationary regularly varying sequence (X t ) and Borel sets A, B bounded away from zero the extremogram 9 is the limiting function np(a 1 n X 0 A,a 1 n X h B) µ h+1 (A R d(h 1) 0 B) Extremograms of the type = γ AB (h), h 0. ρ AB (h) = lim n P(a 1 n X h B a 1 n X 0 A) P(a 1 n = lim X 0 A, a 1 n X h B) n P(a 1 n X, h 0, 0 A) for sets A bounded away from zero are of main interest. 9A possible choice of (a n ) is P( X 0 > a n ) n 1.

82 82 Since ncov(i {a 1 n X 0 A},I {a 1 n X h B} ) = n[p(a 1 n X 0 A,a 1 n X h B) P(a 1 n X 0 A)P(a 1 n X 0 B)] = np(a 1 n X 0 A,a 1 n X h B) + o(1) γ AB (h) 0, the matrix function function. ( γaa (h) γ AB (h) γ BA (h) γ BB (h) ) is a covariance matrix One can use the notions of time series analysis to describe the extremal dependence structure in a strictly stationary sequence.

83 One can define long/short range dependence in some 83 meaningful way or the spectral distribution for extremal events in a strictly stationary sequence, or one can use the extremogram to justify the selection of a particular time series model. For example, the autocorrelation functions of a GARCH(1, 1) process and a stochastic volatility model can be very similar, the extremogram of a stochastic volatility model vanishes while it does not for a GARCH(1,1) process.

84 Examples. Take A = B = (1, ). The AR(1) process X t = φx t 1 + Z t with iid symmetric regularly varying noise (Z t ) with index α and φ ( 1,1) has the extremogram γ AA (h) = max(0,(sign(φ)) h φ αh ). Short serial extremal dependence The extremogram of a GARCH(1,1) process is not very explicit, but γ AA (h) decays exponentially fast to zero. This is in agreement with the geometric β-mixing property of GARCH. Short serial extremal dependence

85 The stochastic volatility model with stationary Gaussian 85 (logσ t ) and iid regularly varying (Z t ) with index α > 0 has extremogram γ AA (h) = 0 as in the iid case. No serial extremal dependence A Gaussian stationary sequence is not regularly varying. But its extremogram exists. It vanishes at all lags as in the iid case. No serial extremal dependence

86 The sample extremogram. Let (X t ) be strongly mixing (possibly vector-valued) regularly varying. Assume m = m n and m n /n 0, and a m satisfies P( X 0 > a m ) m 1. Then n P m (C) = m n t=1 I {Xt /a m C} is a consistent estimator of µ 1 (C) = lim m mp(x 0/a m C).

87 In particular, 87 E P m (C) µ 1 (C), var( P m (C)) m n σ2 (C) = m n [ µ 1 (C) + 2 ] τ h (C) for h=1 τ h (C) = µ h+1 (C R d(h 1) 0 C). For µ 1 -continuity sets C bounded away from zero, (n) 1/2 [ Pm (C) mp(a 1 m m X d 0 C)] N(0,σ 2 (C)). (pre-asymptotic central limit theorem). An analogous result holds for finitely many sets C 1,...,C h.

88 88 The ratio sample extremogram ρ AB (h) = = m n n h t=1 I {a 1 m X t+h B,a 1 m X t A} m n n t=1 I {a 1 m X t A} n h t=1 I {a 1 m X t+h B,a 1 m X t A} n t=1 I {a 1 m X t A}, h 0, estimates ρ AB (h) = lim n P(a 1 n X h B a 1 n X 0 A) = µ h+1(a R d(h 1) 0 B) µ h+1 (A R dh 0 ), h 0.

89 Pre-asymptotic limit theory for the ratio estimator follows 89 from the previous central limit theory (n) ( 1/2 m ) ρ AB (i) ρ AB:m (i) i=0,...,h d N(0,Σ), where ρ AB:m (h) = P(a 1 m X h B a 1 m X 0 A).

90 90 extremogram extremogram lag lag Figure 17. The ratiosampleextremogramwithsample size n = 100,000 forthe GARCH(1,1) (left) and stochastic volatility processes (right). A = B = (1, ).

91 91 extremogram lag Figure 18. Ratio sample extremogram with A = B = (1, ) for 5 minute returns of USD-DEM foreign exchange rates. The extremogram alternates between large values at even lags and small ones at odd lags. This is an indication of AR behavior with negative leading coefficient.

92 92 Problems (1) The central limit theorem for the ratio sample estimator is pre-asymptotic. (For applications, the pre-asymptotic centering ρ AB:m (h) = P(a 1 m X h B a 1 m X 0 A) is more relevant than its limit ρ AB (h).) (2) The asymptotic variance-covariance structure of the ratio sample estimator depends on expressions which are unknown. Two methods to overcome (2): random permutations and stationary bootstrap.

93 93 extremogram lag Figure 19. Sample extremogram for the max-moving average MMA(2) process X t = max(z t,z t 1,Z t 2 ) for an iid positive regularly varying sequence (Z t ), A = B = (1, ). The diamonds superimposed on the figure represent the population extremogram values. Confidence bands are based on random permutations of the data.

94 94 extremogram extremogram lag lag Figure 20. The sample extremogram for the lower tail of the FTSE (top left), S&P500 (top right), DAX (bottom left) and Nikkei. The bold lines represent sample extremograms based on random permutations of the data.

95 5.5. Bootstrapping the sample extremogram. 95 The stationary bootstrap Politis and Romano (1994) is well suited for the sample extremogram. Stationary bootstrap setup. Have data X 1,...,X n and construct a bootstrap (BS) sample as follows: Generate K 1,K 2,... iid uniform on 1,...,n L 1,L 2,... iid geometric(p n ) The BS sample is given by the first n values (in the circular sense) in the sequence X K1,...,X K1 +L 1 1,X K2,...,X K2 +L 2 1,... Mean block size 1/p n, mean number of blocks np n.

96 96 The stationary bootstrap ratio sample extremogram is consistent: ( ( P n) ( ) ) 1/2 ρ P AB m (i) ρ AB(i) A Φ0,Σ (A), i=0,...,h for continuity sets A of the limit distribution Φ 0,Σ. David, M., Cribben (2012)

97 97 extremogram extremogram lag lag Figure 21. Left: 95% bootstrap confidence bands for pre-asymptotic extremogram of 6440 daily FTSE log-returns. Mean block size 200. Right: For the residuals of a fitted GARCH(1, 1) model.

98 lag lag lag lag Figure 22. Top left: Sample extremogram for 5-minute GE log-returns. Boxplots at lags 1, 79, 158 using permutations (top right) and bootstrap with mean block size 50 and 200 (bottom). extremogram extremogram extremogram extremogram

99 5.6. Variations on the theme Davis, M., Cribben (2012ab), Davis, M., Zhao (2013) Cross-extremogram. Consider a strictly stationary bivariate regularly varying time series ((X t,y t )) t Z. For two sets A and B bounded away from 0, the cross-extremogram γ AB (h) = lim x P(Y h xb X 0 xa), h 0, is an extremogram based on the two-dimensional sets A R and R B. The corresponding sample cross-extremogram for the time series ((X t,y t )) t Z : ρ A,B (h) = n h t=1 I {Y t+h a m,y B,X t a m,x A} n t=1 I {X t a m,x A}.

100 100 Figure 23. The sample cross-extremograms for the filtered FTSE, S&P, DAX and Nikkei series. For the first row, (Xt) is the filtered FTSE and (Yt) are the filtered S&P, DAX and Nikkei (from left to right). For the second, third and fourth rows, the Xt s are the filtered S&P, DAX and Nikkei series, respectively lag lag lag extremogram extremogram extremogram Nikkei lag lag lag extremogram extremogram DAX extremogram lag lag lag extremogram S&P extremogram extremogram lag lag lag FTSE extremogram extremogram extremogram

101 101 Series LUSDKFR[, 1] LUSDDEM ACF time Lag Series LUSDKFR[, 2] LUSDFRF ACF time Lag Figure minute log-return series of FX data and their sample autocorrelation functions: USD-FRF (top), USD-DEM (bottom).

102 ACF Lag extremogram LUSDKFR[, 1] & LUSDKFR[, 2] lag lag extremogram lag lag Figure 25. The sample cross-correlation function of the FX data (left) and their extremograms (right) extremogram extremogram

103 extremogram extremogram Figure 26. Left: Extremograms for residuals after an AR fit. Right: Extremograms of residuals after an AR-GARCH fit. lag lag lag lag extremogram extremogram extremogram lag lag lag lag extremogram extremogram extremogram

104 The extremogram of return times between rare events. We say that X t is extreme if X t xa for a set A bounded away from zero and large x. If the return times were truly iid, the successive waiting times between extremes should be iid geometric. Using the histogram of waiting times, this hypothesis can be verified. The corresponding return times extremogram is given by ρ A (h) = lim x P(X 1 xa,...,x h 1 xa,x h xa X 0 xa) = µ h+1(a (A c ) h 1 A) µ h+1 (A R dh 0 ), h 0.

105 The return times sample extremogram is then defined as 105 ρ A (h) = n h t=1 I {X t+h a m A,X t+h 1 a m A,...,X t+1 a m A,X t a m A} n t=1 I {X t a m A}, h < n. It is consistent, asymptotically normal, and the stationary bootstrap version has the same properties.

106 return_time return_time Figure 27. Left: Return times sample extremogram for extreme events with A = R\[ξ 0.05,ξ 0.95 ] for the daily log-returns of BAC using bootstrapped confidence intervals (dashed lines), geometric probability mass function (light solid). Right: The corresponding extremogram for the residuals of a fitted GARCH(1, 1) model (right).

107 5.7. Frequency domain analysis M. and Zhao (2014). 107 The extremogram for a given set A bounded away from zero ρ A (h) = lim n P(a 1 n X h A a 1 n X 0 A), h 0, is a correlation function. Therefore one can define the spectral density f A (λ) = cos(λh)ρ A (h) = h=1 h= e iλh ρ A (h). We also introduce the periodogram for λ (0,π): f na (λ) = I m na(λ) P m (A) = n n t=1 e itλ I {a 1 m X t A} 2 m n n t=1 I. {a 1 m X t A}

108 108 One has EI na (λ)/µ 1 (A) f A (λ) for λ (0,π). As in classical time series analysis, fna (λ) is not a consistent estimator of f A (λ): for distinct (fixed or Fourier) frequencies λ j, and iid standard exponential E j, d ( f na (λ j )) j=1,...,h (fa (λ j )E j ) j=1,...,h. ARMA Spectral density Periodogram Frequency Figure 28. Sample extremogram and periodogram for ARMA(1,1) process with student(4) noise. A = (1, )

109 Smoothed versions of the periodogram converge to f(λ): 109 If w n (j) 0, j s n, s n /n 0, j s n w n (j) = 1 and j s n w 2 n (j) 0 (e.g. w n(j) = 1/(2s n + 1)) then for any distinct Fourier frequencies λ j such that λ j λ, j s n w n (j) f na (λ j ) P f A (λ), λ (0,π). These results do not follow from classical time series analysis: the sequences (I {a 1 m X t A} ) t n constitute a triangular array of rowwise stationary sequences.

110 110 Periodogram ARMA Spectral density Periodogram ARMA Spectral density Upper Bound Lower Bound Frequency Frequency Figure 29. Raw and smoothed periodogram for ARMA(1,1) process with student(4) noise. A = (1, )

111 111 Extremogram Periodogram Upper Bound Bank of America Lower Bound Lag Frequency Figure 30. Sample extremogram and smoothed periodogram for BAC 5 minute returns. The end-ofthe day effects cannot be seen in the corresponding sample autocorrelation function.

112 Problems and possible extensions. Frequency based tools for distinguishing between time series models based on their extremes. Goodness-of-fit tests. Zhao (2014) in progress Choice of the threshold (a n ) (depending on the data) Holger Drees; Rafal Kulik, Philippe Soulier The extremogram for spatio-temporal data, estimating max-stable processes with the extremogram,... Richard A. Davis and co-workers, Claudia Klüppelberg, Christina Steinkohl,... Extremogram for functional data.

113 6. Max-stable processes - a flexible model for serial extremal dependence Max-stable processes and random fields have recently attracted some attention for modeling spatio-temporal extremal phenomena. Recall that a max-stable random variable X satisfies 113 ( c 1 n Mn b n ) = d X, n 1, for suitable constants c n > 0 and d n R, iid copies (X i ) of X. We will assume that X has a Fréchet distribution function Φ α (x) = e x α, x > 0. Then c n = n 1/α and d n = 0.

114 114 A Fréchet random variable has the following useful representation. Lemma. Let Γ i = E E i, i 1, be an iid standard exponential sequence (E i ), independent of an iid sequence (V i ) of positive random variables with E[V α ] < for some α > 0. Then sup i 1 Γ 1/α i V i has a Fréchet Φ E[V α 1 ] α distribution. Note. The counting process N(t) = #{i 1 : Γ i t}, t 0,

115 is a unit-rate homogeneous Poisson process. Given N(t) = k, 115 (Γ 1,...,Γ k N(t) = k) d = t(u (1),...U (k) ) for the order statistics of an iid uniform sample on (0,1).

116 116 Proof. Let (U t ) be iid uniform on (0,1), independent of N and (V t ). Using the order statistics property of N, for x > 0, ( P sup i 1 ) Γ 1/α i V i x [ ( )] = lim E P sup Γ 1/α t i V i x N(t) i N(t) [ ( )] = lim E P sup (tu (i) ) 1/α V i x N(t) t i N(t) [ ( )] = lim E P sup (tu i ) 1/α V i x N(t) t i N(t) ( )] = lim E [P N(t) (tu 1 ) 1/α V 1 x t = lim e tp(v 1 α>xα tu 1 ) t = lim t e x α tx α 0 P(V α 1 >y)dy = e x α E[V1 α] = Φ E[V 1 α] (x). α

117 6.1. Definition de Haan (1984). 117 A (positive) max-stable process (Y t ) t T, T R with Fréchet marginals: for iid copies (Y (i) t ) t T, i = 1,2,..., of (Y t ) t T, n 1/α ( max Y (i) d t ) t T = (Yt ) t T, n 1. i=1,...,n Then, in particular, all one-dimensional marginals of the process (Y t ) t T are Fréchet distributed, i.e. Y t has distribution Φ c(t) α for some function c(t) 0, t T. Example (de Haan (1984)). 0 < Γ 1 < Γ 2 <, independent of the iid sequence (U i ) uniform on (0,1), (f t ) non-negative functions with E[f α t (U 1)] <.

118 118 Then the process Y t = supγ 1/α i f t (U i ), t T, i 1 is max-stable with Fréchet marginals. We have to show (max-stability): For any t i T, x i > 0, i = 1,...,m, n 1, P ( Y t1 x 1,...,Y tm x m ) = P n ( Y t1 x 1 n 1/α,...,Y tm x m n 1/α). In view of the Lemma: P ( Y t1 x 1,...,Y tm x m ) = P (sup i 1 Γ 1/α i = e Emax 1 t m(f t (U)/x t ) α ) max (f t(u i )/x t ) 1 1 t m = e 1 0 max 1 t m (f t (u)/x t ) α du.

119 Then the process 119 Y t = supγ 1/α i f t (U i ), t T, i 1 is max-stable with Fréchet marginals. We have to show (max-stability): For any t i T, x i > 0, i = 1,...,m, n 1, P ( Y t1 x 1,...,Y tm x m ) = P n ( Y t1 x 1 n 1/α,...,Y tm x m n 1/α). In view of the Lemma: P ( Y t1 x 1,...,Y tm x m ) = P (sup i 1 Γ 1/α i = e Emax 1 t m(f t (U)/x t ) α ) max (f t(u i )/x t ) 1 1 t m = e n 1 0 max 1 t m (f t (u)/(n 1/α x t )) α du. Max-stability is immediate.

120 Characterization of α-fréchet max-stable processes. This Example already yields an almost complete characterization of the finite-dimensional distributions of a max-stable process. De Haan (1984) gave a complete characterization (we consider the case T = Z only). Theorem. The finite-dimensional distributions of a max-stable sequence (Y t ) t Z with Fréchet marginals with index α > 0 satisfy the relation for x i > 0, i = 1,...,m, m 1, P(Y 1 x 1,...,Y m x m ) = e R n + max t n(y t /x t ) α G m (dy),

121 where G m is the m-dimensional restriction to R m + measure on R +. of a finite 121 Moreover, there exists a finite measure ρ on [0,1] such that (Y t ) has representation (6.2) Y t = sup i 1 Γ 1/α i f t (T i ), t Z, where ((Γ i,t i )) i=1,2,... is an enumeration of PRM(Lebesgue ρ) on (0, ) [0,1], (f t ) are suitable non-negative measurable functions on [0,1] such that E[f α t (T 1)] = 1 0 fα t (x)ρ(dx) <. Kabluchko (2009): any max-stable process (Y t ) t T, T R, with α-fréchet marginals has representation (6.2), f t L α + (E,E,ν), t T, ν a σ-finite measure on the Borel σ-field E of the state

122 122 space E, ((Γ i,t i )) i 1 are the points of a PRM(Lebesgue ν) on the state space R + E. Using the same notation, one can introduce de Haan s (1984) extremal integral E fdm α ν = sup i 1 Γ 1/α i f(t i ), where, as above f is a non-negative function in L α (E,E,ν), and M α ν is an α-fréchet random sup-measure with control measure ν.

123 Stoev (2008) proved that E fdmα ν has various properties similar to the α-stable integrals; see Samorodnitsky and Taqqu (1994). A proof 123 similar to the Example above yields P ( } fdm α ν x) = exp { x α f α dν E E = Φ E fα dν α (x). The integral representation of a max-stable process is convenient. For example, for any f t L α (E,E,ν), x t > 0, t = 1,...,m, m 1, P ( f t dm α ν x t,t = 1,...,m ) E = P ( max (f t/x t )dm α E t=1,...,m ν 1) { } = exp max (f t/x t ) α dν. t=1,...,m E

124 124 We also have for x = (x 1,...,x m ) > 0 and y, (6.3) y [ 1 P ( f t dm α ν y1/α x t,t = 1,...,m )] = yp ( = y E E (y 1/α( 1 exp f t dm α ν E { y 1 E )t=1,...,m [0,x] ) } ) max (f t/x t ) α dν t=1,...,m max (f t/x t ) α dν = µ m,α ([0,x] c ). t=1,...,m Thus the finite-dimensional distributions of a max-stable process (Y t ) t T are regularly varying with index α and limiting measure µ m,α given by (6.3).

125 6.3. Stationary max-stable processes. 125 Recently, strictly stationary max-stable processes (Y t ) t T for T = Z or T = R have attracted some attention. Such a process has again integral representation Y t = E f t dm α ν, t T, where the family of functions (f t ) has to satisfy some particular conditions to ensure strict stationarity, ergodicity, mixing, and other desirable properties; see Kabluchko (2009), Stoev (2008).

126 126 Example. Since (Y t ) is regularly varying with index α one can define its extremogram. For example, the extremogram with respect to the set (1, ) is given by ρ(h) = lim x P(x 1 Y h > 1 x 1 Y 0 > 1) = P(x 1 min(y 0,Y h ) > 1) P(Y 0 > x) { 1 exp x α E = lim min(fα 0,fα h )dν} x 1 exp { x α E fα 0 dν } E = min(fα 0,fα h )dν E fα 0 dν.

127 It is also straightforward to calculate the extremal index of (Y t ). Consider (a n ) satisfying P(Y 0 > a n ) = 1 e a α n E fα 0 dν n 1 i.e. a n n 1/α( E fα 0 dν) 1/α. Then, for x > 0, ( ) { } P a 1 n max Y t x = exp a α t=1,...,n n x α max t=1,...,n fα t dν The limit = 1 θ Y = lim n n is the extremal index of (Y t ). E [ Φ α (x)] n 1 E max t=1,...,nf α t dν / E fα 0 dν(1+o(1)). E max t=1,...,nf α t dν E fα 0 dν 127

128 The Brown-Resnick process Brown, Resnick (1977). Has representation (6.4) Y t = sup i 1 Γ 1/α i e W i(t) 0.5σ 2 (t), t R, where 0 < Γ 1 < Γ 2 < are the points of a unit rate homogeneous Poisson process on (0, ), independent of an iid sequence (W i ) of sample continuous zero-mean Gaussian processes on R with stationary increments and variance function σ 2, e.g. Brownian motions or fractional Brownian motions. The max-stable process (6.4) is stationary; see Kabluchko, Schlather, de Haan (2009). In this paper, the authors also consider the case of max-stable random fields, i.e. W is a mean zero Gaussian