Limit theorems for dependent regularly varying functions of...

Markov chainsLimit theorems for functions of Markov chains

Markov chains with extremal linear behavior

Limit theorems for dependent regularly varying functionsof Markov chains

In collaboration with T. Mikosch

Olivier [email protected]

CEREMADE, University Paris Dauphine and LFA, CREST.

Strasbourg, October 23, 2012

Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains



Motivation: extension of the iid case

Theorem (A.V. and S.V. Nagaev, 1969, 1979, Cline and Hsing, 1998)

(Xi ) 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)

xαand

P(X 6 −x)

P(|X | > x)∼ q

L(x)

xα, x →∞.

Then Sn =∑n

i=1 Xi satisfies the precise large deviations relation

limn→∞

supx>bn

∣∣∣∣ P(Sn > x)

n P(|X | > x)− p

∣∣∣∣ = 0 and limn→∞

supx>bn

∣∣∣∣ P(Sn 6 −x)

n P(|X | > x)− q

∣∣∣∣ = 0 .

1 α > 2 =⇒ bn =√

an log n with a > α− 2,

2 α ∈ (0, 2] =⇒ bn = nδ+1/α for any δ > 0.




Motivation: extension of the iid case

(bn) larger than the rate of convergences in law

1 α > 2 =⇒√

n = o(bn) in the TLC,

2 α ∈ (0, 2] =⇒ n1/αL(n) = o(bn) where L is slowly varying.

Heavy tail phenomena

If nP(|X | > x)→ 0 and p 6= 0

P(Sn > x) ∼x→∞ nP(X > x) ∼x→∞ P(max(X1, . . . ,Xn) > x).

Also true for other sub-exponential distributions (EKM, 1997).

What is happening for dependent sequences for whom extremes cluster?




Outline

1 Markov chainsIrreducibility and splitting schemeRegular variation and drift condition

2 Limit theorems for functions of Markov chainsCentral Limit TheoremRegular variation of cyclesLarge deviations

3 Markov chains with extremal linear behavior




Irreducibility and splitting schemeRegular variation and drift condition

Regeneration of Markov chains with an accessible atom(Doeblin, 1939)

Definition

(Φt) is a Markov chain of kernel P on Rd and A ∈ B(Rd).

A is an atom if ∃ a measure ν on B(Rd) st P(x ,B) = ν(B) for all x ∈ A.

A is accessible, i.e.∑

k Pk(x ,A) > 0 for all x ∈ Rd ,

Let (τA(j))j>1 visiting times to the set A, i.e. τA(1) = τA = min{k > 0 : Xk ∈ A}and τA(j + 1) = min{k > τA(j) : Xk ∈ A}.

Regeneration cycles

1 NA(t) = #{j > 1 : τA(j) 6 t}, t > 0, is a renewal process,

2 The cycles (ΦτA(t)+1, . . . ,ΦτA(t+1)) are iid.





Irreducible Markov chain and Nummelin scheme

Definition (Minorization condition, Meyn and Tweedie, 1993)

∃ δ > 0, C ∈ B(Rd) and a distribution ν on C such that

(MCk) Pk(x ,B) > δν(B), x ∈ C , B ∈ B(Rd).

(MC1) is called the strongly aperiodic case.

If P is an irreducible aperiodic Markov chain then it satisfies (MCk) for some k ∈ N.

Nummelin splitting scheme

Under (MC1) an enlargement of (Φt) on Rd × {0, 1} ⊂ Rd+1 possesses anaccessible atom A = C × {1} =⇒ the enlarged Markov chain regenerates.





Regularly varying sequences

Regularly varying condition of order α > 0

A stationary sequence (Xt) is regularly varying if a non-null Radon measure µd issuch that

(RVα) n P(a−1n (X1, . . . ,Xd) ∈ ·) v→ µd(·) ,

where (an) satisfies n P(|X | > an)→ 1 and µd(tA) = t−αµd(A), t > 0.

Definition (Basrak & Segers, 2009)

It is equivalent to the existence of the spectral tail process (Θt) defined for k > 0,

P(|X0|−1(X0, . . . ,Xk) ∈ · | |X0| > x)w→ P((Θ0, . . . ,Θk) ∈ ·) , x →∞ .





Main assumptions

Assume that (Φt) (possibly enlarged) possesses an accessible atom A, the existenceof its invariant measure π and Φ0 ∼ π.

Assume the existence of f such that:

1 There exist constants β ∈ (0, 1), b > 0 such that for any y ,

(DCp) E(|f (Φ1)|p | Φ0 = y) 6 β |f (y)|p + b 1A(y).

2 (Xt = f (Φt)) satisfies (RVα) with index α > 0 and spectral tail process (Θt).

Remarks

1 it is absolutely (β−)mixing with exponential rate,

2 supx∈A Ex(κτA) for some κ > 1.

3 (DCp) =⇒ (DCp′) for 0 < p′ 6 p.





Illustrations

Time

ts(ser^2)

0 1000 2000 3000 4000 5000

020

4060

80100

Time

ts(garch@

h.t)

0 1000 2000 3000 4000 50000

510

1520

25

CAC40 index Volatility estimated by a GARCH(1,1)





The cluster index

Under (RVα) denote bk(±) = limn→∞ n P(±Sk > an) , k > 1.

Theorem

Assume (RVα) for some α > 0 and (DCp) for some positive p ∈ (α− 1, α). Thenthe limits (called cluster indexes)

b± : = limk→∞

(bk+1(±)− bk(±))

= limk→∞

E[( k∑

t=0

Θt

)α±−( k∑

t=1

Θt

)α±

]= E

[( ∞∑t=0

Θt

)α±−( ∞∑

t=1

Θt

)α±

]exist and are finite. Here (Θt) is the spectral tail process of (Xt).

The extremal index 0 < θ = E[(

supt>0 Θt

)α+−(

supt>1 Θt

)α+

]6 E[(Θ0)α+].




Central Limit TheoremRegular variation of cyclesLarge deviations

Gaussian Central Limit Theorem

Theorem (Samur, 2004)

Assume that (DCp) holds for p = 1, E|X |2 <∞ and EX = 0. Then

1 The central limit theorem n−1/2Snd→ N (0, σ2) where

σ2 = E[( ∞∑

t=0

Xt

)2

−( ∞∑

t=1

Xt

)2]<∞.

2 The full cycles SA(t) =∑τA(t+1)

i=1 f (ΦτA(t)+i ) have finite moments of order 2with EA(S(1)) = 0 and EA[S(1)2] = σ2.





Stable Central Limit Theorem

Theorem

Assume (RVα) with 0 < α < 2, α 6= 1 and (DCp) with (α− 1)+ < p < α then the

Central Limit Theorem a−1n Sn

d→ ξα is satisfied for a centered α-stable r.v. ξα withcharacteristic function ψα(x) = exp(−|x |αχα(x , b+, b−)), where

χα(x , b+, b−) =Γ(2− α)

1− α

((b+ + b−) cos(πα/2)− i sign(x)(b+ − b−) sin(π α/2)

).

Proof: apply Bartkiewicz et al., 2011.





Regular variation of cycles

Theorem

Assume (RVα) with α > 0 and (DCp) with (α− 1)+ < p < α and b± 6= 0 then

PA

(±

τA∑i=1

f (Φi ) > x)∼x→∞ b± EA(τA) P(|X | > x).

Remarks

1 The full cycles SA(t) =∑τA(t+1)

i=1 f (ΦτA(t)+i ) are regularly varying with thesame index α > 0 than Xt ,

2 If τA is independent of (Xt) then PA(SA(1) > x) ∼x→∞ EA(τA) P(X > x),

3 The distribution of the cycles depends of the choice of the atom A.





Sketch of the proof

∣∣∣ P(Sn > bn)

n P(|X | > bn)− b+

∣∣∣ 6 ∣∣∣P(Sn > bn)− n (P(Sk+1 > bn)− P(Sk > bn))

n P(|X | > bn)

∣∣∣+∣∣∣P(Sk+1 > bn)− P(Sk > bn)

P(|X | > bn)− b+

∣∣∣ .1 In the second term, write X truncated X at bn/k2 and Sk = Sk + Sk and deal

with Sk using a lemma from Jakubowski, 1997.2 Using Nagaev-Fuk inequality of Bertail and Clemencon (2009), under (DCp)

limk→∞

lim supn→∞

P(∑n

i=1 Xi1{|Xi |6bn/k2} > bn/k)

n P(|X | > bn)= 0.

3 Use the regeneration scheme to prove that∣∣∣ P(Sn > bn)

n PA(SA > bn)− (EA(τA))−1

∣∣∣→ 0 ,





Precise large deviations

Corollary (Under the hypothesis of the Theorem)

If 0 < α < 1 then limn→∞ supx>bn

∣∣∣∣ P(±Sn > x)

n P(|X | > x)− b±

∣∣∣∣ = 0,

else limn→∞ supbn6x6cn

∣∣∣∣ P(±Sn > x)

n P(|X | > x)− b±

∣∣∣∣ = 0 if P(τA > n) = o(nP(|X | > cn)).

Determination of the constant in LD of Davis and Hsing (1995) valid for α < 2.

Sketch of the proof: Use Sn =∑τA

1 Xi +∑NA(n)−1

t=1 SA(t) +∑nτA(NA(n))+1 Xi .

Under P(τA > n) = o(nP(|X | > cn)) we can restrict to NA(n) > 1.The first and last cycles are negligible using Pitman’s identity.Use Nagaev’s inequality on the iid regularly varying cycles SA(t).




Markov chains with extremal linear behavior (Kesten,1974, Goldie, 1991, Segers, 2007, Mirek, 2011)

Assume (A,B) is absolutely continuous on R+ × R with EAα = 1, Xt = Ψt(Xt−1)with iid iterated Lipschitz functions Ψt with negative top Lyapunov exponent and

At Xt−1 − |Bt | 6 Xt 6 At Xt−1 + |Bt |.

Proposition

If E(X ) = 0 when E|X | <∞, the conclusions of the theorems hold with

b+ = E[(

1 +∞∑t=1

t∏i=1

Ai

)α−( ∞∑

t=1

t∏i=1

Ai

)α]and b− = 0.




Examples

1 Random difference equation Xt = AXt−1 + B.The region [bn, cn] seems optimal (Buraczewski et al., 2011):

P(Sn > x)

nP(|X | > x)∼ b+ +

P(Sn > x , τA > n)

nP(|X | > x)= b+ + r(x)

and r(x) seems not negligible for some x >> cn.

2 Xt = max(AtXt−1,Bt),

3 Letac’s model Xt = At max(Ct ,Xt−1) + Dt .




The GARCH(1,1) model (Bollorslev, 1986, Mikosch andStarica, 2000)

We consider a GARCH(1,1) process Xt = σt Zt , where (Zt) is iid with E[Z0] = 0and E[Z 2

0 ] = 1 and

σ2t = α0 + σ2

t−1(α1Z 2t−1 + β1) = α0 + σ2

t−1At .

Considering the Markov chain (Xt , σt) under conditions of irreducibility, aperiodicityand E(α1Z 2

0 + β1)α/2 = 1, α > 0 and E|Z |α+ε <∞ we obtain

Proposition

If Z is symmetric, the conclusions of the theorems hold with

b± =E[∣∣∣Z0 +

∑∞t=1 Zt

∏ti=1

√Ai

∣∣∣α − ∣∣∣∑∞t=1 Zt

∏ti=1

√Ai

∣∣∣α]E|Z0|α

.




Application to the autocovariance of the squared log-ratios

0 5 10 15 20 25

02

46

810

Lag

ACF

(cov

)

Series log_rendement_FTSE^2

15 21 27 33 39 45 51 57 63 69 75 81 87 93 99

1.0

1.5

2.0

2.5

3.0

3.5

9.95 7.78 6.63 5.80 5.01 4.54 3.85 3.50 3.18 2.92 2.72

Order Statisticsalp

ha (C

I, p

=0.9

5)

Threshold

For α/2 ∈ (0, 2), a−1n

∑n−ht=1 X 2

t X 2t+h

d→ ξα/4 with (Θt)t>0 = (cZ 2t Z 2

t+hΠtΠt+h)t>0,=⇒ b− = 0 and ξα/4 is supported by [0,∞).




Conclusion

Cluster indices b± determine the large deviations of the sums of dependentand regularly varying variables,

Explicit expressions of b± are given for some models,

Under the hypothesis of the theorems

P(Sn > x) ∼n→∞b+

θP(max(X1, . . . ,Xn) > x) for bn 6 x 6 cn

with possibly b+/θ > 1 on models...




Conclusion

Thank you for your attention!


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