limit theorems for dependent regularly varying functions of...
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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
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
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.
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
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?
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
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
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
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.
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
Irreducibility and splitting schemeRegular variation and drift condition
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.
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
Irreducibility and splitting schemeRegular variation and drift condition
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 →∞ .
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
Irreducibility and splitting schemeRegular variation and drift condition
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.
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
Irreducibility and splitting schemeRegular variation and drift condition
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)
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
Irreducibility and splitting schemeRegular variation and drift condition
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+].
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
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.
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
Central Limit TheoremRegular variation of cyclesLarge deviations
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.
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
Central Limit TheoremRegular variation of cyclesLarge deviations
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.
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
Central Limit TheoremRegular variation of cyclesLarge deviations
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 ,
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
Central Limit TheoremRegular variation of cyclesLarge deviations
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).
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
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.
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
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 .
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
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
.
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
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,∞).
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
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...
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains
Markov chainsLimit theorems for functions of Markov chains
Markov chains with extremal linear behavior
Conclusion
Thank you for your attention!
Olivier Wintenberger, CEREMADE and CREST-LFA Limit theorems for dependent regularly varying functions of Markov chains