On heavy tailed time series and functional limit theorems Bojan Basrak, University of Zagreb

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1 On heavy tailed time series and functional limit theorems Bojan Basrak, University of Zagreb Recent Advances and Trends in Time Series Analysis Nonlinear Time Series, High Dimensional Inference and Beyond BIRS, April 2014, Banff based on joint work with Azra Tafro, University of Zagreb Danijel Krizmanić, University of Rijeka Johan Segers, Université catholique de Louvain 1

2 2

3 Stationary regularly varying time series appear in many applications (insurance, econometrics, climate science,etc) it is often important to understand clustering of extremes 3

4 Corresponding functional limit theorem (α < 2) partial answers are known, in the iid case or at least when extremes appear in isolation, the FLT still holds (Tyran Kaminska (2010)). For moving average process, e.g. X n = Z n Z n 1 things can go wrong and FLT in J 1 metric doesn t hold, it can be saved using M 1 topology!?! Avram & Taqqu (1992). 4

5 Regularly varying sequences 5

6 regularly varying random variable q x α L(x) p x α L(x) x x 6

7 A random vector X is regularly varying with tail index α if X is regularly varying, ie P ( X > u) = u α L(u), and for x X X X > x d Θ σ In dimension one Θ ( 1 1 q p ) 7

8 8 u α σ(s)

9 A stationary time series (X n ) n is said to be regularly varying if random vectors (X 0,..., X k ) k 0 are regularly varying for each k. 9

10 Theorem (B., Segers) A stationary process (X n ) n is regularly varying if and only if there exists a process (Y n ) n Z such that for every s t as x, ( ) Xt x X 0 > x d (Y t ) t Z. t Z the process {Y t : t Z} in theorem above is called the tail process. 10

11 distribution of Y 0 where Y 0 = R 0 Θ 0 P (R 0 > u) = u α, 1 u <, i.e. R 0 = Y 0 Θ 0 ( 1 1 q p ). R 0 and Θ 0 are independent. 11

12 spectral tail process there exists a process independent of R 0 = Y 0 such that (Θ t ) t Z (Y t ) t d = R 0 (Θ t ) t 12

13 Examples (for simplicity, assume Θ 0 = 1) a) X t iid RV(α), Θ t = 0, for t 0. b) X t = Z t Z t 1, Z t iid RV(α),..., Θ 1, Θ 0, Θ 1,... {..., 0, 0, 1, 1, 0,... w.p.1/2..., 0, 1, 1, 0, 0,... w.p.1/2 13

14 d) X t = Z t Z t 1, Z t iid RV(α). 14

15 e) X t = Z t 1 2 Z t 1 + Z t 2, Z t iid RV(α). 15

16 e) X t = A t X t 1 +B t, with (A t, B t ) iid satisfying Kesten s (1973) conditions, X t RV(α) and for t = 0, 1, 2,... Θ t = A t A t 1 A 1. f) X t = (Z t, Z t 1 ), Z t iid RV(α), 16

17 There is a subtle and somewhat startling connection between the past and the future of the tail process, so for instance P (Θ t 0) = E Θ t α. 17

18 One can show that there exists a sequence (a n ), a n such that np ( X 0 > a n ) 1. In the sequel normalization depends on the sample size X t a n instead of X t x. 18

19 Point processes 19

20 For our stationary and regularly varying sequence (X i ) consider N n = n δ (i/n,xi /a n ), i=1 on the space E = [0, 1] R \ {0}. 20

21 Theorem For iid X t, X 0 regular variation is equivalent to N n d N, where N is Poisson point process with intensity measure Leb µ. µ(x, ) = px α, µ(, x) = qx α, x > 0 We can always write N = 1 δ Ti,P i 21

22 P ((b a)u α ) u a b 22

23 Extremes of dependent sequences cluster 23

24 Fig. 5. Data for November 1991 at the platform Aqua Alta. Top: time series of wind speed. Middle: time series of SSE; only the meteorological component (i.e. the astronomical tide was subtracted from the data) is shown. Bottom: SWH time series. Solid and dashed lines: ERA-15 downscaled and T106 winds, respectively; circles: observations 24 Heavy tailed weather in Adriatic, Nov 1991, from Lionello et al.: Wind waves and storm-surge climate scenarios, 2003 RMS error shown in Table 1, which were computed 4. WAVES AND STORM-SURGE CLIMATE using the events shown in Fig. 6. The results show that SCENARIOS

25 u N n a b 25

26 Weak dependence condition - part 1 eg strong mixing Main idea: it is possible to break the series into nearly independent blocks of size n r n, r n,, r n so that for k n = n/r n for independent N n d N r n N r n k N r n j d = r n δ (jrn /n,x i /a n ) i=1 =: Ñ n 26

27 Weak dependence condition - part 2 or anti-clustering condition The high level exceedances are not clustering for too long : lim lim sup P m n X i > a n u X 0 > a n u = 0, u > 0. (1) m i r n 27

28 observations can be broken in nearly independent clusters clusters cannot be too large 28

29 Theorem (B., Krizmanić, Segers & B., Tafro) under weak dependence conditions above, there is a point process N with compound Poisson structure such that N n d N 29

30 Here where N d = i i j δ (Ti,P i ) δ (Ti,P i η ij ) is a Poisson process with intensity measure θ Leb µ { } θ = P (A) (0, 1] where A = sup Y k < 1 k<0 and j δ ηij are iid with the same distribution as j δ Yj / sup{y k } A 30

31 31 the shape of the limiting process

32 Functional limit theorems 32

33 Theorem (Skorohod) for (X i ) iid regularly varying with index α (0, 2) there are sequences (a n ), (b n ) and an α stable Lévy process V α S nt nt b n a n d V α (t), in D[0, 1] endowed with the J 1 topology. 33

34 Proof by continuous mapping argument Introduce the sum functional m Ψ m mapping point measures to D[0, 1] m = δ ti,x i Ψ m (t) = x i, t i t i=1 Clearly Ψ Nn (t) = 1 a n S nt = nt i=1 X i a n, t 0. 34

35 Apply contin. map. thm. to show that on K = [0, 1] ( ε, ε) c m n v m = Ψ mn K Ψ m K in appropriate metric, whenever m some part of the space of point measures, say M. problem: Ψ is not continuous at m in J 1 metric unless for all t m(t R) 1. 35

36 Examples a) X t iid reg. var. (α) b) X t = Z t Z t 1, Z t iid reg. var. (α) c) X t = Z t Z t 1, Z t iid reg. var. (α) d) X t = Z t 1 2 Z t 1 + Z t 2 iid reg. var. (α) e) X t = A t X t 1 + B t, with (A t, B t ) iid nonnegative satisfying Kesten s conditions. f) X t = (Z t, Z t 1 ), Z t iid reg. var. (α) 36

37 37 X t = Z t Zt 1

38 38 Problem for J 1 metric

39 M 1 metric: distance between functions f and g is measured by comparing completed graphs Γ f and Γ g and Γ f = {(t, x) : x = f(t) or x [f(t ), f(t)]} d M1 (f, g) = inf λ f,λ g max λ f λ g where infimum is taken over continuous and increasing parametrizations λ f, λ g of Γ f, Γ g. 39

40 40 Solution: use M 1 metric

41 Problem: Ψ is not continuous w.r.t. the M 1 metric, i.e. unless for all t m n v m Ψ mn M 1 Ψ m m(t (0, )) = 0 or m(t (, 0)) = 0 41

42 FLT for dependent regularly varying steps Suppose that (X t ) is a stationary regularly varying sequence with α (0, 2) which is weakly dependent tail sequence is one-sided i.e. P (Y i have the same sign for all i) = 1 Theorem ( B., Krizmanić, Segers 2012) there exist sequences (a n ) and (b n ) such that S nt nt b n a n d V α (t), n in D[0, 1] endowed with the M 1 topology. a), b), c), e). 42

43 problem for M 1 metric: example d) X t = Z t 1 Zt 1 + Zt

44 44 Solution: use M 2 metric

45 FLT for more general MA sequences Assume (X n ) is a stationary regularly varying MA sequence with α < 2 satisfying X n = c 0 Z n + c 1 Z n c m Z n m such that for all j 0 j c i i=1 then the functional limit theorem holds in M 2 metric (B., Krizmanić, 2013). m i=1 c i 45

46 FLT for dependent multivariate sequences For simple multivariate regularly varying time series, such as X t = (Z t, Z t 1 ), Z t iid reg. var. with index α none of the above topologies works. But there is an appropriate topology here as well (B., Krizmanić, 2012, to appear in JoTP). 46

47 47 X t = (Z t, Z t 1)

48 48 Problem for the strong M 1 metric

49 General idea of the proof observe that the point processes N n converge in some sense apply continuous mapping theorem very carefully find conditions which make sum functional Ψ continuous in a chosen topology 49

50 Things we didn t discuss choice of (b n ), problem of approximation when ε 0, topology on p.p. s. of course!?!,... 50

51 Things to remember A stationary regularly varying sequence (X t ) has a tail process (Y t ) the clusters of extremes can be described by (Y t ) point processes N n have a limit characterized by (Y t ) random walks with steps (X t ) have an α stable limit for α (0, 2) but in strange topologies on D[0, 1]. 51

52 THE END 52