Extremal index, hitting time statistics andperiodicity

Ana Cristina Moreira FreitasCMUP & FEP, Universidade do Porto

joint work with Jorge Freitas and Mike Todd

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Extreme Value Theory

Consider a stationary stochastic process X0,X1,X2, . . . with marginald.f. F .

Let F̄ = 1− F and uF = sup{x : F (x) < 1}.

The main goal of the Extreme Value Theory (EVT) is the study of thedistributional properties of the maximum

Mn = max{X0, . . . ,Xn−1} (1)

as n→∞.

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Extreme Value Laws

DefinitionWe say that we have an Extreme value law (EVL) for Mn if there is anon-degenerate d.f. H : R→ [0,1] (with H(0) = 0) and for all τ > 0,there exists a sequence of levels un = un(τ) such that

nP(X0 > un)→ τ as n→∞, (2)

and for which the following holds:

P(Mn ≤ un)→ H̄(τ) as n→∞. (3)

.

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The independent case

In the case X0,X1,X2, . . . are i.i.d. r.v. then since

P(Mn ≤ un) = (F (un))n

we have that if (2) holds, then (3) holds with H̄(τ) = e−τ :

P(Mn ≤ un) = (1− P(X0 > un))n ∼(

1− τn

)n→ e−τ as n→∞,

and vice-versa.

When X0,X1,X2, . . . are not i.i.d. but satisfy some mixing conditionD(un) introduced by Leadbetter then something can still be said aboutH.

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Condition D(un) from Leadbetter

Let Fi1,...,indenote the joint d.f. of Xi1 , . . . ,Xin , and setFi1,...,in (u) = Fi1,...,in (u, . . . ,u).

Condition (D(un))

We say that D(un) holds for the sequence X0,X1, . . . if for any integersi1 < . . . < ip and j1 < . . . < jk for which j1 − ip > t , and any large n ∈ N,∣∣Fi1,...,ip,j1,...,jk (un)− Fi1,...,ip (un)Fj1,...,jk (un)∣∣ ≤ γ(n, t),where γ(n, tn) −−−→n→∞ 0, for some sequence tn = o(n).

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Extremal Index

Theorem (Leadbetter)If D(un) holds for X0,X1, . . . and the limit (3) exists for some τ > 0 thenthere exists 0 ≤ θ ≤ 1 such that H̄(τ) = e−θτ for all τ > 0.

DefinitionWe say that X0,X1, . . . has an Extremal Index (EI) 0 ≤ θ ≤ 1 if we havean EVL for Mn with H̄(τ) = e−θτ for all τ > 0.

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Linear normalising sequences

The sequences of real numbers un = un(τ), n = 1,2, . . ., are usuallytaken to be one parameter linear families such as un = any + bn,where y ∈ R and an > 0, for all n ∈ N.

Observe that τ depends on y through un and, in fact, depending on thetail of the marginal d.f. F , we have that τ = τ(y) is of one of thefollowing 3 types (for some α > 0):

Type 1: τ1(y) = e−y for y ∈ R,Type 2: τ2(y) = y−α for y > 0,Type 3: τ3(y) = (−y)α for y ≤ 0.

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Characterization of the three types

Theorem (Gnedenko)Necessary and sufficient conditions for τ to be of one of the threetypes are:

Type 1: There exists some strictly positive function g such that, for allreal y,

limt↑uF

1− F (t + yg(t))1− F (t)

= e−y .

Type 2: uF =∞ and limt→∞(1− F (ty))/(1− F (t)) = y−α, α > 0, foreach y > 0.

Type 3: uF 0, for each y > 0.

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CorollaryThe constants an and bn may be taken as follows:

Type 1: an = g(γn), bn = γn;

Type 2: an = γn, bn = 0;

Type 3: an = uF − γn, bn = uF ,

where γn = F−1(1− 1/n) = inf{x : F (x) ≥ 1− 1/n}.

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Examples

1. If F (x) = 1− e−x then τ is of type 1.

2. If F (x) = 1− kx−α, α > 0, K > 0, x ≥ K 1/α, then τ is of type 2.

3. If F (x) = x , 0 ≤ x ≤ 1, then τ is of type 3.

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Hitting Times and Kac’s Lemma

Consider the system (X ,B, µ, f ), where X is a topological space, B isthe Borel σ-algebra, f : X → X is a measurable map and µ is anf -invariant probability measure, i.e., µ(f−1(B)) = µ(B), for all B ∈ B.

For a set A ⊂ X let rA(x) the first hitting time to A of the point x , i.e.rA(x) = min{j ∈ N : f j(x) ∈ A}.

Let µA denote the conditional measure on A, i.e. µA :=µ|Aµ(A) .

By Kac’s Lemma, the expected value of rA with respect to µA is∫A

rA dµA = 1/µ(A).

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Hitting Time Statistics and Return Time Statistics

Definition

Given a sequence of sets (Un)n∈N so that µ(Un)→ 0, the system hasRTS G̃ for (Un)n∈N if for all t ≥ 0

µUn

(rUn ≤

tµ(Un)

)→ G̃(t) as n→∞. (4)

and the system has HTS G for (Un)n∈N if for all t ≥ 0

µ

(rUn ≤

tµ(Un)

)→ G(t) as n→∞, (5)

For systems with ‘good mixing properties’, G(t) = G̃(t) = 1− e−t , inwhich case we say that we have exponential HTS/RTS.

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Stationary stochastic processes arising from chaotic dynamics

Consider a discrete dynamical system

(X ,B, µ, f ),

whereX is a d-dimensional Riemannian manifold,B is the Borel σ-algebra,f : X → X is a map,µ is an f -invariant probability measure, absolutely continuous withrespect to Lebesgue (acip).

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In this context, we consider the stochastic process X0,X1, . . . given by

Xn = ϕ ◦ f n, for each n ∈ N, (6)

where ϕ : X → R ∪ {±∞} is an observable (achieving a globalmaximum at ξ ∈ X ) of the form

ϕ(x) = g(dist(x , ξ)),

where ξ ∈ X , “dist” denotes a Riemannian metric in X and the functiong : [0,+∞)→ R ∪ {+∞} has a global maximum at 0 and is a strictlydecreasing bijection for a neighborhood V of 0..

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Observe that if at time j ∈ N we have an exceedance of the level usufficiently large, i.e. Xj(x) > u, then we have an entrance of the orbitof x in the ball of radius g−1(u) around ξ, at time j .

{X0 > u} = {g(dist(x , ξ)) > u} = {dist(x , ξ) < g−1(u)} = Bg−1(u)(ξ).

and

1− F (u) = µ(

Bg−1(u)(ξ)).

Based on the characterisation of the 3 types in terms of the tail of F ,we may assume that g is of one of the following 3 types:

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Type 1: there exists some strictly positive function p : W → R suchthat for all y ∈ R

lims→g1(0)

g−11 (s + yp(s))

g−11 (s)= e−y ; Example: g1(y) = − log y

Type 2: g2(0) = +∞ and there exists β > 0 such that for all y > 0

lims→+∞

g−12 (sy)

g−12 (s)= y−β; Example: g2(y) = y−1/α

Type 3: g3(0) = D < +∞ and there exists γ > 0 such that for all y > 0

lims→0

g−13 (D − sy)g−13 (D − s)

= yγ ; Example: g3(y) = D − y1/α

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Connection between EVL and HTS

(X ,B, µ, f ) is a dynamical system where µ is an acip.ξ ∈ X are points for which the Lebesgue’s Differentiation Theoremholds.Motivated by Collet’s work, [C01], we obtained:

Theorem (F,F,Todd (2010))

If we have HTS G for balls centred on ξ ∈ X , then we have anEVL for Mn with H = G.

Theorem (F,F,Todd (2010))If we have an EVL H for Mn, then we have HTS G = H for ballscentred on ξ.

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Idea of the proof:

{x : Mn(x) ≤ un} =n−1⋂j=0

{x : Xj(x) ≤ un}

=n−1⋂j=0

{x : g(dist(f j(x), ξ)) ≤ un}

=n−1⋂j=0

{x : dist(f j(x), ξ) ≥ g−1(un)} = {x : rBg−1(un)(ξ)(x) ≥ n}

Thus,

µ{x : Mn(x) ≤ un} = µ{x : rBg−1(un)(ξ)(x) ≥ n}

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Note that

τ

n∼ 1− F (un) = µ

(Bg−1(un)(ξ)

)⇔ n ∼ τ

µ(

Bg−1(un)(ξ))

and so

µ {x : Mn(x) ≤ un} ∼ µ

x : rBg−1(un)(ξ)(x) ≥ τµ(Bg−1(un)(ξ))→ 1−G(τ)

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Consider now a sequence δn → 0. We want to study

µ

({x : rBδn (ξ)(x) <

tµ(Bδn (ξ))

})Choose `n such that g−1(u`n ) ∼ δn. We have that

{x : M`n (x) ≤ u`n} =`n−1⋂j=0

{x : Xj(x) ≤ u`n}

=`n−1⋂j=0

{x : g(dist(f j(x), ξ)) ≤ u`n}

=`n−1⋂j=0

{x : dist(f j(x), ξ) ≥ g−1(u`n )} = {x : rBg−1(u`n )(ξ)(x) ≥ `n}

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As before,

τ

`n∼ 1− F (u`n ) = µ (Bδn (ξ)) ∼ µ

(Bg−1(u`n )(ξ)

)⇔ `n ∼

τ

µ (Bδn (ξ)).

In this way,

µ

{x : rBδn (ξ)(x) <

τ

µ(Bδn (ξ))

}∼ 1− µ{x : M`n (x) ≤ u`n} → H(τ)

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Some history

EVLs for the partial maximum have been studied in[C01, FF08, FF08a, VHF09, FFT10, FFT11, GHN11, HNT12, FFT12,FFT12a, HVRSB12, FLTV11, FLTV11a, FLTV11b, LFW12, K12,AFV12, FFLTV12].

The dynamical systems covered in these papers include:

non-uniformly hyperbolic 1-dimensional maps (in all of them),higher dimensional non-uniformly expanding maps in [FFT10],suspension flows in [HNT12],billiards and Lozi maps in [GHN11]Hénon maps in [FLTV11b, HVRSB12]

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Incomplete list of papers where a standard exponential HTS/RTS lawhas been proved around almost every point:

Markov chains in [P91],Axiom A diffeomorphisms in [H93],uniformly expanding maps of the interval in [C96] ,1-dimensional non-uniformly expanding maps in[HSV99, BSTV03, BV03, BT09]...,partially hyperbolic dynamical systems in [D04],toral automorphisms in [DGS04] ,higher dimensional non-uniformly hyperbolic systems (includingHénon maps) in [CC10] .

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Assuming D(un) holds, let (kn)n∈N be a sequence of integers such that

kn →∞ and kntn = o(n). (7)

Condition (D′(un))

We say that D′(un) holds for the sequence X0,X1, . . . if

lim supn→∞

n[n/k ]∑j=1

P{X0 > un and Xj > un} = 0. (8)

Theorem (Leadbetter)

Let {un} be such that n(1− F (un))→ τ , as n→∞, for some τ ≥ 0.Assume that conditions D(un) and D′(un) hold. Then

P(Mn ≤ un)→ e−τ as n→∞.

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Assuming D(un) holds, let (kn)n∈N be a sequence of integers such that

kn →∞ and kntn = o(n). (7)

Condition (D′(un))

We say that D′(un) holds for the sequence X0,X1, . . . if

lim supn→∞

n[n/k ]∑j=1

P{X0 > un and Xj > un} = 0. (8)

Theorem (Leadbetter)

Let {un} be such that n(1− F (un))→ τ , as n→∞, for some τ ≥ 0.Assume that conditions D(un) and D′(un) hold. Then

P(Mn ≤ un)→ e−τ as n→∞.

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Motivated by the work of Collet (2001) we introduced:

Condition (D2(un))

We say that D2(un) holds for the sequence X0,X1, . . . if for any integers`, t and n

|P {X0 > un ∩max{Xt , . . . ,Xt+`−1 ≤ un}}−P{X0 > un}P{M` ≤ un}| ≤ γ(n, t),

where γ(n, t) is nonincreasing in t for each n and nγ(n, tn)→ 0 asn→∞ for some sequence tn = o(n).

Theorem (F,F (2008))

Let {un} be such that n(1− F (un))→ τ , as n→∞, for some τ ≥ 0.Assume that conditions D2(un) and D′(un) hold. Then

P(Mn ≤ un)→ e−τ as n→∞.

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Motivated by the work of Collet (2001) we introduced:

Condition (D2(un))

We say that D2(un) holds for the sequence X0,X1, . . . if for any integers`, t and n

|P {X0 > un ∩max{Xt , . . . ,Xt+`−1 ≤ un}}−P{X0 > un}P{M` ≤ un}| ≤ γ(n, t),

where γ(n, t) is nonincreasing in t for each n and nγ(n, tn)→ 0 asn→∞ for some sequence tn = o(n).

Theorem (F,F (2008))

Let {un} be such that n(1− F (un))→ τ , as n→∞, for some τ ≥ 0.Assume that conditions D2(un) and D′(un) hold. Then

P(Mn ≤ un)→ e−τ as n→∞.

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Decay of correlations implies D2(un)

Suppose that there exists a nonincreasing function γ : N→ R suchthat for all φ : X → R with bounded variation and ψ : X → R ∈ L∞,there is C > 0 independent of φ, ψ and n such that∣∣∣∣∫ φ · (ψ ◦ f t )dµ− ∫ φdµ∫ ψdµ∣∣∣∣ ≤ CVar(φ)‖ψ‖∞γ(t), ∀n ≥ 0,

(9)where Var(φ) denotes the total variation of φ and nγ(tn)→ 0, asn→∞ for some sequence tn = o(n).

Taking φ = 1{X>un} and ψ = 1{M`≤un}, then

(9)⇒ D2(un),

(with γ(n, t) = CVar(1{X>un})‖1{M`≤un}‖∞γ(t) ≤ C′γ(t) and for the

sequence {tn} such that tn/n→ 0 and nγ(tn)→ 0 as n→∞).(CMUP & FEP) 26 / 38

Periodic points

From here on ξ is a repelling p-periodic point, which implies thatf p(ξ) = ξ, f p is differentiable at ξ and 0 < |det D(f−p)(ξ)| < 1.

Moreover, we assume ξ ∈ X is a point for which the Lebesgue’sDifferentiation Theorem holds.

Note that {X0 > u} ∩ {Xp > u} 6= ∅ and for all u sufficiently large

P({X0 > u} ∩ {Xp > u}) ∼∣∣det D(f−p)(ξ)∣∣P(X0 > u).

Consequently, D′(un) does not hold since

n[n/kn]∑j=1

P(X0 > un,Xj > un) ≥ nP(X0 > un,Xp > un)→∣∣det D(f−p)(ξ)∣∣ τ

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For repelling periodic points as above, let θ = 1− |det D(f−p)(ξ)|.

Then the following condition holds:

Condition (MPp,θ(un))

We say that X0,X1,X2, . . . satisfies the condition MPp,θ(un) for p ∈ Nand θ ∈ [0,1] if

limn→∞

sup1≤j

un|X0 > un) = 0, limn→∞P(Xp > un|X0 > un) = (1− θ)

and limn→∞

supi

P(Xp > un,X2p > un, . . . ,Xip > un|X0 > un)(1− θ)i

= 1.

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Define the event Qp,0(u) := {X0 > u,Xp ≤ u}.

Observe that for u sufficiently large, Qp,0(u) corresponds to an annuluscentred at ξ.

Define the events: Qp,i(u) := {Xi > u,Xi+p ≤ u},

Q∗p,i(u) := {Xi > u} \Qp,i(u) and Qp,s,`(u) =⋂s+`−1

i=s Qcp,i(u).

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Theorem (F, F, Todd (2012))

Let (un)n∈N be such that nP(X0 > un)→ τ , for some τ ≥ 0. SupposeX0,X1, . . . is as in (6) and satisfies MPp,θ(un) for p ∈ N, and θ ∈ (0,1).Then

limn→∞

P(Mn ≤ un) = limn→∞P(Qp,0,n(un)) (10)

First observe that {Mn ≤ un} ⊂ Qp,0,n(un).Moreover, Qp,0,n(un) \ {Mn ≤ un} ⊂

⋃n−1i=0 {Xi > un,Xi+p >

un, . . . ,Xi+si p > un}, where si = [n−1−i

p ].It follows by MPp,θ(un) and stationarity that

P(Qp,0,n(un)\{Mn ≤ un}) ≤n−1∑i=0

P(Xi > un,Xi+p > un, . . . ,Xi+si p > un

)≤ p

[n/p]∑i=0

P(X0 > un,Xp > un,X2p > un, . . . ,Xip > un

)−−−→n→∞

0.

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Condition (Dp(un))

We say that Dp(un) holds forX0,X1, . . . if for any `, t and n∣∣P (Qp,0(un) ∩Qp,t ,`(un))− P(Qp,0(un))P(Qp,0,`(un))∣∣ ≤ γ(n, t),where γ(n, t) is nonincreasing in t for each n and nγ(n, tn)→ 0 asn→∞ for some sequence tn = o(n).

Let (kn)n∈N be a sequence of integers such that kn →∞ andkntn = o(n).

Condition (D′p(un))

We say that D′p(un) holds for the sequence X0,X1,X2, . . . if there existsa sequence {kn}n∈N satisfying (7) and such that

limn→∞

n[n/kn]∑j=1

P(Qp,0(un) ∩Qp,j(un)) = 0. (11)

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Theorem (F, F, Todd (2012))

Let (un)n∈N be such that nP(X > un)→ τ , as n→∞ for some τ ≥ 0.Consider a stationary stochastic process X0,X1,X2, . . . satisfyingMPp,θ(un) for some p ∈ N, and θ ∈ (0,1). Assume further thatconditions Dp(un) and D′p(un) hold. Then

limn→∞

P(Mn ≤ un) = limn→∞P(Qp,0,n(un)) = e−θτ . (12)

Note that

P(Qp,0(u)) = P(X0 > u,Xp ≤ u) == P(X0 > u)− P(X0 > u,Xp > u) =∼ P(X0 > u)− (1− θ)P(X0 > u) = θP(X0 > u),

and so

θ ∼P(Qp,0(u))P(X0 > u)

.

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Applications to specific systems

Examples of systems for which we can prove the existence of an EI0 < θ < 1 at repelling periodic points

Systems with decay of correlations against L1 observables:Uniformly expanding maps on the circle ([H93])Conformal repellers and Markov maps ([FP12])Piecewise expanding maps of the interval like Rychlik maps([FFT12, K12])Higher dimensional piecewise expanding maps like in [S00]([FFT12, K12])

Non-uniformly hyperbolic systems admitting “nice” induced firstreturn time maps:

Maps with indifferent fixed points like Manneville-Pomeau orLiverani-Saussol-Vaienti maps ([FFT12a])

Non-uniformly hyperbolic systems with critical points:Benedicks-Carleson quadratic maps ([FFT12a])

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Decay of correlations against L1 implies D′p(un)

Suppose that there exists a nonincreasing function γ : N→ R suchthat for all φ : X → R with bounded variation and ψ : X → R ∈ L1,there is C > 0 independent of φ, ψ and n such that∣∣∣∣∫ φ · (ψ ◦ f t )dµ− ∫ φdµ ∫ ψdµ∣∣∣∣ ≤ CVar(φ)‖ψ‖1γ(t), ∀n ≥ 0,

(13)where Var(φ) denotes the total variation of φ and nγ(tn)→ 0, asn→∞ for some sequence tn = o(n).

Taking φ = 1Qp(un) and ψ = 1Qp(un), then

(13)⇒ D′p(un),

P(Qp,0(un) ∩Qp,j(un)) ≤ P(Qp,0(un))2 + C′P(Qp,0(un))γ(j) .(τ/n)2 + C′(τ/n)γ(j).

(CMUP & FEP) 34 / 38

Doubling map

(CMUP & FEP) 35 / 38

Rychlik map

(CMUP & FEP) 36 / 38

Intermittent map

(CMUP & FEP) 37 / 38

Benedicks-Carleson maps

(CMUP & FEP) 38 / 38