Limit theorems for dependent regularly varying functions of Markov chains

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1 Limit theorems for functions of with extremal linear behavior Limit theorems for dependent regularly varying functions of In collaboration with T. Mikosch Olivier Wintenberger CEREMADE, University Paris Dauphine and LFA, CREST. Strasbourg, October 23, 2012 Limit theorems for dependent regularly varying functions of

2 Limit theorems for functions of with extremal linear behavior Motivation: extension of the iid case Theorem (A.V. and S.V. Nagaev, 1969, 1979, Cline and Hsing, 1998) (X i ) iid random variables with regularly varying (centered if α > 1) distribution: p, q 0 with p + q = 1 and a slowly varying function L such that P(X > x) P( X > x) p L(x) x α and P(X x) P( X > x) q L(x) x α, x. Then S n = n i=1 X i satisfies the precise large deviations relation lim sup P(S n > x) n n P( X > x) p = 0 and lim sup P(S n x) n n P( X > x) q = 0. x b n x b n 1 α > 2 = b n = an log n with a > α 2, 2 α (0, 2] = b n = n δ+1/α for any δ > 0. Limit theorems for dependent regularly varying functions of

3 Limit theorems for functions of with extremal linear behavior Motivation: extension of the iid case (b n ) larger than the rate of convergences in law 1 α > 2 = n = o(b n ) in the TLC, 2 α (0, 2] = n 1/α L(n) = o(b n ) where L is slowly varying. Heavy tail phenomena If np( X > x) 0 and p 0 P(S n > x) x np(x > x) x P(max(X 1,..., X n ) > x). Also true for other sub-exponential distributions (EKM, 1997). What is happening for dependent sequences for whom extremes cluster? Limit theorems for dependent regularly varying functions of

4 Outline Limit theorems for functions of with extremal linear behavior 1 Irreducibility and splitting scheme Regular variation and drift condition 2 Limit theorems for functions of Central Limit Theorem Regular variation of cycles Large deviations 3 with extremal linear behavior Limit theorems for dependent regularly varying functions of

5 Limit theorems for functions of with extremal linear behavior Irreducibility and splitting scheme Regular variation and drift condition Regeneration of with an accessible atom (Doeblin, 1939) Definition (Φ t ) is a Markov chain of kernel P on R d and A B(R d ). A is an atom if a measure ν on B(R d ) st P(x, B) = ν(b) for all x A. A is accessible, i.e. k Pk (x, A) > 0 for all x R d, Let (τ A (j)) j 1 visiting times to the set A, i.e. τ A (1) = τ A = min{k > 0 : X k A} and τ A (j + 1) = min{k > τ A (j) : X k A}. Regeneration cycles 1 N A (t) = #{j 1 : τ A (j) t}, t 0, is a renewal process, 2 The cycles (Φ τa (t)+1,..., Φ τa (t+1)) are iid. Limit theorems for dependent regularly varying functions of

6 Limit theorems for functions of with extremal linear behavior Irreducibility and splitting scheme Regular variation and drift condition Irreducible Markov chain and Nummelin scheme Definition (Minorization condition, Meyn and Tweedie, 1993) δ > 0, C B(R d ) and a distribution ν on C such that (MC k ) P k (x, B) δν(b), x C, B B(R d ). (MC 1 ) is called the strongly aperiodic case. If P is an irreducible aperiodic Markov chain then it satisfies (MC k ) for some k N. Nummelin splitting scheme Under (MC 1 ) an enlargement of (Φ t ) on R d {0, 1} R d+1 possesses an accessible atom A = C {1} = the enlarged Markov chain regenerates. Limit theorems for dependent regularly varying functions of

7 Limit theorems for functions of with extremal linear behavior Regularly varying sequences Irreducibility and splitting scheme Regular variation and drift condition Regularly varying condition of order α > 0 A stationary sequence (X t ) is regularly varying if a non-null Radon measure µ d is such that (RV α ) n P(a 1 n (X 1,..., X d ) ) v µ d ( ), where (a n ) satisfies n P( X > a n ) 1 and µ d (ta) = t α µ d (A), t > 0. Definition (Basrak & Segers, 2009) It is equivalent to the existence of the spectral tail process (Θ t ) defined for k 0, P( X 0 1 (X 0,..., X k ) X 0 > x) w P((Θ 0,..., Θ k ) ), x. Limit theorems for dependent regularly varying functions of

8 Limit theorems for functions of with extremal linear behavior Main assumptions Irreducibility and splitting scheme Regular variation and drift condition Assume that (Φ t ) (possibly enlarged) possesses an accessible atom A, the existence of its invariant measure π and Φ 0 π. Assume the existence of f such that: 1 There exist constants β (0, 1), b > 0 such that for any y, (DC p ) E( f (Φ 1 ) p Φ 0 = y) β f (y) p + b 1 A (y). 2 (X t = f (Φ t )) satisfies (RV α ) with index α > 0 and spectral tail process (Θ t ). Remarks 1 it is absolutely (β )mixing with exponential rate, 2 sup x A E x (κ τ A ) for some κ > 1. 3 (DC p ) = (DC p ) for 0 < p p. Limit theorems for dependent regularly varying functions of

9 Illustrations Limit theorems for functions of with extremal linear behavior Irreducibility and splitting scheme Regular variation and drift condition ts(ser^2) ts(garch@h.t) Time Time CAC40 index Volatility estimated by a GARCH(1,1) Limit theorems for dependent regularly varying functions of

10 The cluster index Limit theorems for functions of with extremal linear behavior Irreducibility and splitting scheme Regular variation and drift condition Under (RV α ) denote b k (±) = lim n n P(±S k > a n ), k 1. Theorem Assume (RV α ) for some α > 0 and (DC p ) for some positive p (α 1, α). Then the limits (called cluster indexes) b ± : = lim k+1(±) b k (±)) k = [( k ) α ( k ) α ] lim Θ t Θ t k ± ± t=0 t=1 [( ) α ( ) α ] = E Θ t Θ t ± ± t=0 exist and are finite. Here (Θ t ) is the spectral tail process of (X t ). [( ) α ( ) α ] The extremal index 0 < θ = E sup t 0 Θ t sup t 1 Θ t E[(Θ 0 ) α +]. + + t=1 Limit theorems for dependent regularly varying functions of

11 Limit theorems for functions of with extremal linear behavior Gaussian Central Limit Theorem Central Limit Theorem Regular variation of cycles Large deviations Theorem (Samur, 2004) Assume that (DC p ) holds for p = 1, E X 2 < and EX = 0. Then 1 The central limit theorem n 1/2 S n d N (0, σ 2 ) where [( ) 2 ( ) 2 ] σ 2 = E X t X t <. t=0 2 The full cycles S A (t) = τ A (t+1) i=1 f (Φ τa (t)+i) have finite moments of order 2 with E A (S(1)) = 0 and E A [S(1) 2 ] = σ 2. t=1 Limit theorems for dependent regularly varying functions of

12 Limit theorems for functions of with extremal linear behavior Stable Central Limit Theorem Central Limit Theorem Regular variation of cycles Large deviations Theorem Assume (RV α ) with 0 < α < 2, α 1 and (DC p ) with (α 1) + < p < α then the Central Limit Theorem a 1 n S n d ξα is satisfied for a centered α-stable r.v. ξ α with characteristic function ψ α (x) = exp( x α χ α (x, b +, b )), where χ α (x, b +, b ) = Γ(2 α) 1 α Proof: apply Bartkiewicz et al., ( ) (b + + b ) cos(πα/2) i sign(x)(b + b ) sin(π α/2). Limit theorems for dependent regularly varying functions of

13 Limit theorems for functions of with extremal linear behavior Regular variation of cycles Central Limit Theorem Regular variation of cycles Large deviations Theorem Assume (RV α ) with α > 0 and (DC p ) with (α 1) + < p < α and b± 0 then P A ( ± τ A i=1 ) f (Φ i ) > x x b ± E A (τ A ) P( X > x). Remarks 1 The full cycles S A (t) = τ A (t+1) i=1 f (Φ τa (t)+i) are regularly varying with the same index α > 0 than X t, 2 If τ A is independent of (X t ) then P A (S A (1) > x) x E A (τ A ) P(X > x), 3 The distribution of the cycles depends of the choice of the atom A. Limit theorems for dependent regularly varying functions of

14 Limit theorems for functions of with extremal linear behavior Sketch of the proof Central Limit Theorem Regular variation of cycles Large deviations P(S n > b n ) n P( X > b n ) b + P(S n > b n ) n (P(S k+1 > b n ) P(S k > b n )) n P( X > b n ) + P(S k+1 > b n ) P(S k > b n ) b +. P( X > b n ) 1 In the second term, write X truncated X at b n /k 2 and S k = S k + S k and deal with S k using a lemma from Jakubowski, Using Nagaev-Fuk inequality of Bertail and Clemençon (2009), under (DC p ) P lim lim sup k n ( n ) i=1 X i1 { Xi b n/k 2 } > b n /k = 0. n P( X > b n ) 3 Use the regeneration scheme to prove that P(S n > b n ) n P A (S A > b n ) (E A(τ A )) 1 0, Limit theorems for dependent regularly varying functions of

15 Limit theorems for functions of with extremal linear behavior Precise large deviations Central Limit Theorem Regular variation of cycles Large deviations Corollary (Under the hypothesis of the Theorem) P(±S n > x) If 0 < α < 1 then lim n sup x bn n P( X > x) b ± = 0, P(±S n > x) else lim n sup bn x c n n P( X > x) b ± = 0 if P(τ A > n) = o(np( X > c n )). Determination of the constant in LD of Davis and Hsing (1995) valid for α < 2. Sketch of the proof: Use S n = τ A 1 X i + N A (n) 1 t=1 S A (t) + n τ A (N A (n))+1 X i. Under P(τ A > n) = o(np( X > c n )) we can restrict to N A (n) 1. The first and last cycles are negligible using Pitman s identity. Use Nagaev s inequality on the iid regularly varying cycles S A (t). Limit theorems for dependent regularly varying functions of

16 Limit theorems for functions of with extremal linear behavior with extremal linear behavior (Kesten, 1974, Goldie, 1991, Segers, 2007, Mirek, 2011) Assume (A, B) is absolutely continuous on R + R with EA α = 1, X t = Ψ t (X t 1 ) with iid iterated Lipschitz functions Ψ t with negative top Lyapunov exponent and A t X t 1 B t X t A t X t 1 + B t. Proposition If E(X ) = 0 when E X <, the conclusions of the theorems hold with [( b + = E 1 + t ) α ( t ) α ] A i A i t=1 i=1 t=1 i=1 and b = 0. Limit theorems for dependent regularly varying functions of

17 Limit theorems for functions of with extremal linear behavior Examples 1 Random difference equation X t = AX t 1 + B. The region [b n, c n ] seems optimal (Buraczewski et al., 2011): P(S n > x) np( X > x) b + + P(S n > x, τ A > n) = b + + r(x) np( X > x) and r(x) seems not negligible for some x >> c n. 2 X t = max(a t X t 1, B t ), 3 Letac s model X t = A t max(c t, X t 1 ) + D t. Limit theorems for dependent regularly varying functions of

18 Limit theorems for functions of with extremal linear behavior The GARCH(1,1) model (Bollorslev, 1986, Mikosch and Starica, 2000) We consider a GARCH(1,1) process X t = σ t Z t, where (Z t ) is iid with E[Z 0 ] = 0 and E[Z 2 0 ] = 1 and σ 2 t = α 0 + σ 2 t 1(α 1 Z 2 t 1 + β 1 ) = α 0 + σ 2 t 1A t. Considering the Markov chain (X t, σ t ) under conditions of irreducibility, aperiodicity and E(α 1 Z β 1) α/2 = 1, α > 0 and E Z α+ɛ < we obtain Proposition If Z is symmetric, the conclusions of the theorems hold with [ Z0 E + t=1 Z t α t i=1 Ai t=1 Z t α] t i=1 Ai b ± = E Z 0 α. Limit theorems for dependent regularly varying functions of

19 Limit theorems for functions of with extremal linear behavior Application to the autocovariance of the squared log-ratios Series log_rendement_ftse^2 Threshold ACF (cov) alpha (CI, p =0.95) Lag Order Statistics For α/2 (0, 2), an 1 n h t=1 X t 2 Xt+h 2 d ξ α/4 with (Θ t ) t 0 = (czt 2 Zt+h 2 Π tπ t+h ) t 0, = b = 0 and ξ α/4 is supported by [0, ). Limit theorems for dependent regularly varying functions of

20 Limit theorems for functions of with extremal linear behavior Conclusion Cluster indices b ± determine the large deviations of the sums of dependent and regularly varying variables, Explicit expressions of b ± are given for some models, Under the hypothesis of the theorems P(S n > x) n b + θ P(max(X 1,..., X n ) > x) for b n x c n with possibly b + /θ 1 on models... Limit theorems for dependent regularly varying functions of

21 Limit theorems for functions of with extremal linear behavior Conclusion Thank you for your attention! Limit theorems for dependent regularly varying functions of