Tail process and its role in limit theorems Bojan Basrak, University of Zagreb

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1 Tail process and its role in limit theorems Bojan Basrak, University of Zagreb The Fields Institute Toronto, May 2016 based on the joint work (in progress) with Philippe Soulier, Azra Tafro, Hrvoje Planinić 1

2 Stationary regularly varying sequences 2

3 Regular variation regularly varying random variable q x α L(x) p x α L(x) x 3 x

4 Multivariate regular variation σ(s) x α L(x) 4

5 Regularly varying process 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. 5

6 For a stationary regularly varying sequence there exists a tail process (Y t ) t Z such that ( Xt ) x and a spectral tail process t Z X 0 > x d (Y t ) t Z (θ t ) t Z independent of Y 0 such that (Y t ) d t = Y 0 (θ t ) t. Moreover θ 0 = 1 and Y 0 Pareto(α). 6

7 There exists a sequence (a n ) such that ( ) Xt X 0 > a n a n t Z d (Y t ) t Z. 7

8 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 8

9 c) X t = Z t Z t 1, Z t iid RV(α). 9

10 Independent observations Regular variation assumption determines limiting behavior of point processes sums and random walks S n = X X n maxima and other extremes M n = max{x 1,..., X n } records and record times 10

11 Complete convergence theorem 1 simple but powerful cf. Leadbetter-Rootzén, Resnick Theorem For iid X t 0, X 0 is reg. varying is if and only if N n = n 1 δ i n, X i an where N is PRM(Leb d( x α )). d N = i δ Ti,P i, So P (M n /a n u) = P (N n ([0, 1] (u, )) = 0) P (N([0, 1] (u, )) = 0) = e u α 11

12 P ois((b a)u α ) u N a b 12

13 Extremes of dependent sequences cluster 13

14 Strongly mixing observations Mori (1977) showed if then N n d N N = i j δ Ti,P i Q i,j where i δ Ti,P i is a PRM(ϑ Leb d( x α )) with ϑ (0, 1] j δ Qij is an iid sequence of point processes in [ 1, 1], independent of PRM above. 14

15 It was not really clear what is ϑ what is the distribution of j δ Qij what would be a sufficient condition for such a convergence 15

16 16 N

17 Anti-clustering condition or finite mean cluster size condition High level exceedances are not clustering for too long, i.e for some r n and r n /n 0: lim lim sup P m n It implies X i > a n u X 0 > a n u = 0, u > 0. (1) m i r n Y m P 0, as m. 17

18 Main technical lemma Denote L Y = sup i Z Y i and M n = max X 1,..., X n, then under the assumptions above ( rn i=1 δ Xi /M rn, M r n a n Mrn > a n ) ( i δ Yi /L Y, L Y sup j<0 Y j 1 ) with two components on the right hand side being independent B. & Tafro (2016). 18

19 Complete convergence theorem 2 building on Davis & Resnick, Davis & Hsing, Davis & Mikosch tafro,b.(2016) /krizmanić, segers, b. (2012) Theorem Under strong mixing and a.c., as n, N n d N = i,j δ (Ti,P i Q ij ), where i δ Ti,P i is a Poisson process on [0, 1] (0, ] with intensity ϑleb d( x α ) ( j δ Qij ) i is an iid sequence of point processes independent of the process above 19

20 Here ϑ is the extremal index of the sequence X t with representation ϑ = P ( Y i 1) = P ( Y i 1) > 0. i 1 i 1 While cluster shapes satisfy j δ Qj d = i δ Yi /L Y sup j<0 Y j 1 20

21 size Complete convergence theorem time illustrated X t = Z t + 0.9Z t 1.

22 As in the iid case one can prove (functional) limit theorems for partial maxima M nt partial sums S nt under additional conditions and unusual topologies (Avram & Taqqu, B. Krizmanić, Segers, or Jakubowski...) 22

23 However... in the limit there is a loss of information about the order one cannot find the limit of S nt even for some very simple models it is difficult to say much about records or record times 23

24 We introduce a new space Space for ordered clusters l 0 = l 0 / where l 0 = {x = (x i ) i Z : lim i x i = 0} and x y if d(x, y) = 0 with d(x, y) = inf k sup j x j y j+k. Adapting d to l 0 produces a separable and complete metric space. 24

25 Technical lemma in l 0 large deviations result Under a.c. assumption as n ( X1,..., X ) rn Mrn > a n a n ( ) Y i, i Z sup Y j 1 j<0 in l 0 \ {0}. The space is not locally compact, so we need to use w # topology (cf. Daley Vere Jones). Related to Hult & Samorodnitsky (2010) and Mikosch & Wintenberger (2016) large deviations results. As before, conditionally on sup j<0 Y j 1, random variable L Y = sup i Z Y i and random cluster (Y i /L Y ) i are independent. 25

26 Complete convergence theorem 3 very technical but order is preserved

27 Partial sums converge again Under assumptions above (for α (0, 1) plus an additional one for α [1, 2)) we prove functional limit theorem for S nt a n, t 0, but the limit is a process in a new space E[0, 1], M 2. Assume for convenience that X i s are symmetric if α 1. Similarly one can prove the limiting theorem in a more standard space (D[0, 1], M 1 ) for S ns sup, t 0. s t a n 27

28 Space E[0, 1] Whitt 2002 Elements are triples (x, T, {I(t) : t T }), where x D[0, 1], T is a countable subset of [0, 1] with Disc(x) T, and, for each t T, I(t) is a closed bounded interval such that x(t), x(t ) I(t). Moreover, for each ε > 0, there are at most finitely many times t with diameter of I(t) greater than ε. 28

29 Functional limit theorem in the space E planinić, soulllier, b. (2016) Theorem Under assumptions above for α (0, 2) as n, S nt a n d S E, for some S E = (S, T S, {I S (t) : t T S }) with S being an α stable Lévy process. 29

30 Remark In the limit S E = (S, T S, {I S (t) : t T S }), countable set T S includes all the discontinuities of the stable process S, and S can be trivial. All three components of the process S E can be expressed in the terms of the tail process. One can prove that in D[0, 1] with M 1 topology sup s t S ns a n d sup S E (s). s t 30

31 Partial sums for X t = Z t + 0.9Z t 1. sum process

32 Partial sums for X t = Z t 0.7Z t 1. sum process time

33 Running max of sums for X t = Z t 0.7Z t 1. sum process time

34 Sums of X t = Z t 2.5Z t 1 + Z t 2. sum process time

35 Sums of X t = Z t Z t 1. sum process time

36 Record times It is well known (Resnick, 1987) that for any iid sequence from a continuous distribution, point process of record times converges, ie R n = δ i/n I {Xi is a record} i=1 satisfies R n d R = i Z δ Ri where R is a so called scale invariant Poisson process on (0, ) with intensity dx/x. 36

37 Record times are much more difficult to handle when dependence is present, therefore we assume sequence (X n ), maybe after monotone transformation, is regularly varying, strong mixing and a.c. holds. with probability one all nonzero values of the tail process Y i are mutually different. 37

38 The limit of record times planinić, soulllier, b. (2016) Theorem Under assumptions above as n, R n d R = i Z δ Ri κ i, where i Z δ Ri is Poisson process on (0, ) with intensity dx/x and (κ i ) i Z is an iid sequence independent of it. 38

39 Records are broken in clusters. 39

40 Remark The limit R does not depend on ϑ directly. Moreover, κ i have the same distribution as I {Qj >sup i<j Q i e W/α }, j= where W is standard exponential and (( ) (Q j ) d Yi j = L Y i ) sup Y j 1 j<0 40

41 Summary 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 ) with order preserved in the space [0, 1] l 0. random walks with steps (X t ) have an α stable limit for α (0, 2) but in M 2 on E[0, 1] (càdlàg functions are ok only for some special cases). record times have a surprisingly simple compound Poisson structure in the limit. 41

42 Thanks 42