Patrice Bertail
email: patrice.bertail@u-paris10.fr

Regenerative Block-Bootstrap Condence Intervals for Tail and Extremal Indexes

Keywords: AMS classication: 60G70, 60J10, 60K20 Regenerative Markov chain, Nummelin splitting technique, Extreme value statistics, Cycle submaximum, Hill estimator, Extremal index, Regenerative-block bootstrap I

A theoretically sound bootstrap procedure is proposed for building accurate condence intervals of parameters describing the extremal behavior of instantaneous functionals {f(X n )} n∈N of a Harris Markov chain X, namely the extremal and tail indexes. Regenerative properties of the chain X (or of a Nummelin extension of the latter) are here exploited in order to construct consistent estimators of these parameters, following the approach developed in [10]. Their asymptotic normality is rst established and the standardization problem is also tackled. It is then proved that, based on these estimators, the (approximate) regenerative block-bootstrap introduced in [7] yields asymptotically valid condence intervals. In order to illustrate the performance of the methodology studied in this paper, simulation results are additionally displayed.

1 Introduction es originlly pointed out in QPD the extreml ehvior of instntneous funtionls f(X) = {f(X n )} n∈N of rrris reurrent wrkov hin X myD just like the symptoti men ehviorD e desried through the regenerE tive properties of the underlying hinF pollowing in the footsteps of this seminl ontriution @see lso PAD the uthors hve reently investigted the performne of regenertionEsed sttistil proedures for estimting key prmeters relted to the extreml ehvior nlysis in the wrkovin setupY see IHF sn prtiulrD speil ttention hs een pid to the prolem of estimting the extremal index of the wekly dependent sequene f(X)D whih mesures to wht extent extreme vlues tend to ome in 4smll lusters4Y refer to ISD IID IV for n ount of this notionF rious extreml index estimtors hve een reently proposed in the sttistil litertureY see ID PQD PID PWD QH for instneF hese estimtors generlly rely on blocking techniquesD where dt segments of (xed @deterministiA length re onsidered in order to E ount for the dependene struture within the oservtionsF elterntivelyD n symptotilly vlid methodology spei(lly tilored for @pseudoEA reE genertive sequenes hs een proposedD sed on dt loks of rndom lengthD orresponding to cycles in etween suessive regenertion timesF roeeding in the sme veinD it hs een estlished in IH tht reE genertive version of the rill estimtorD omputed from the set of cycle sub-maximaD nmely mximum vlues oserved in etween onseutive renewl timesD yields onsistent estimtion of the til index of f(X)9s 1Ed mrginl distriution in the @supposedly existingA sttionry regimeD in the se where the ltter elongs to the préhet mximum domin of ttrtionF st is the purpose of this pper to ontinue this pproh y investigting the prolem of onstruting on(dene intervls for the extreml nd til indexesF e (rst prove the symptoti normlity of the regenertionEsed estimtors onsidered nd then show how to studentize the ltter in order to uild symptoti qussin on(dene intervlsF xextD we propose to extend the rnge of pplition of the (approximate) regenerative block-bootstrap @eEff in revited formAD originlly introdued in V for ootstrpping wrkovin smple mensD to the present settingF esymptoti vlidity of the eff proedureD when pplied to the regenertionEsed index estimtesD is estlished nd empiril simultions hve een rried outD in order to evlute empirilly its performne when ompred to qussin symptoti intervlsF he rtile is strutured s followsF xottions re (rst set out in eE P tion P nd ruil notions relted to the renewl properties of rrris wrkov hinsD tht will e needed throughout the pperD re lso rie)y relledF sn etion QD entrl limit theorems re stted for the regenertive versions of the 4runs4 nd 4loks4 estimtors of the extreml indexF esymptoti normlity of the regenertive rill estimtor is estlished nd the studenE tiztion of these estimtors is lso investigtedF etion R is devoted to the study of the @eAff methodologyD when pplied to the onstrution of on(dene intervls sed on the spei( regenertionEsed estimtors onsideredF pinllyD etion S displys preliminry simultion resultsD omE pring the performne of ootstrp nd qussin intervlsF ehnilities re treted in the eppendixF 2 Preliminaries hroughout the rtileD we will denote y X = {X n } n∈N timeEhomogeneous rrris reurrent wrkov hinD vlued in mesurle spe (E, E) with trnsition proility Π(x, dy) nd initil distriution νY see PV for n E ount of the wrkov hin theoryF e lso denote y P ν @respetivelyD y P x with x ∈ EA the proility mesure on the underlying spe suh tht X 0 ∼ ν @respFD X 0 = xA nd y E ν [.] @respFD E x [.]A the orresponding expetE tionF e strt o' with relling si renewl properties of rrris wrkov hinsD while enhning their onnetion with extreml ehvior nlysisF 2.1 Regenerative chains ell (rst tht the hin X is sid to e regenerative when it possesses rrris reurrent tomD i.e.D rrris set A suh thtX ∀(x, y) ∈ A 2 , Π(x, .) = Π(y, .)F et τ A = τ A (1) = inf {n ≥ 1, X n ∈ A} nd τ A (j) = inf {n > τ A (j -1), X n ∈ A} for j ≥ 2F sn the tomi seD y virtue of the strong wrkov propertyD the sequene {τ A (k)} k≥1 of suessive return times to the tom forms @possiE ly delyedA renewl proess nd more generllyD the dt segmentsD lled regeneration cyclesD determined y the times t whih X forgets its pst re iFiFd rndom vriles vlued in the torus T = ∪ ∞ n=1 E n X B 1 = (X τ A (1)+1 , . . . , X τ A (2) ), . . . , B j = (X τ A (j)+1 , . . . , X τ A (j+1) ), . . . .

Q

e denote y P A the onditionl proility mesure given X 0 ∈ A nd y E A [.] the P A EexpettionF sn the regenertive setupD stohsti stility properties lssilly oil down to heking onditions relted to the speed of return to the regenertive setF st is wellEknown for instne tht X is positive recurrent if nd only if

α = E A [τ A ] < ∞ see heorem IHFPFP in PSD nd its @uniqueA invrint proility distriution µ is then the itmn9s ouption mesure given y µ(B) = α -1 E A [ τ A i=1 I{X i ∈ B}] for ll B ∈ EF he following ssumptions re involved in the susequent nlysisF vet κ ≥ 1 nd ν e ny proility distriution on (E, E)F H(κ) : E A [τ κ A ] < ∞ nd H(ν, κ) : E ν [τ κ A ] < ∞. Cycle submaxima. vet f : (E, E) → R e mesurle funtionF gonsider the sumximum of the instntneous funtionl f(X) = {f(X n )} n∈N over the jEth yleD j ≥ 1X ζ j (f) = mx τ A (j)<k≤τ A (j+1) f(X k ).
st hs een estlished in QPD see heorem QFI thereinD thtD in the posE itive reurrent seD the distriution of the smpling mximum M n (f) = mx 1≤i≤n f(X i ) n e suessfully pproximted y the distriution of the mximum of n/α @roughly the men numer of yles within trjetory of length nA independent reliztions of the yle sumximum s n → ∞D provided tht the (rst @non regenertiveA dt segment plys no role in the extreml ehviorD i.e. P ν (mx 1≤i≤τ A f(X i ) > mx 1≤j≤l ζ j (f)) → 0 s l → ∞F wore preiselyD under these ssumptions we hve

sup x∈R |P ν (M n (f) ≤ x) -G f (x) n/α | → 0 s n → ∞, @IA where G f (x) = P A (mx 1≤i≤τ A f(X i ) ≤ x)
for ll x ∈ RF his shows tht the til ehvior of the yle sumximum9s distriution G f (dx) governs the extreml ehvior of the sequene f(X)F

Regenerative extensions of general Harris chains

elthough the lss of regenertive wrkov hins inludes ll hins with ountle stte spe s well s mny wrkov models used in ypertions eserh for modeling queuingGstorge systemsD the existene of rrris reE genertive set is very restritive ssumption in prtieD tht is not ful(lled y most rrris hinsF rere we rie)y rell theoretil onstrutionD R termed the splitting technique nd originlly introdued in PTD extending in some sense the proilisti struture of generl rrris hinD so s to rti(illy uild regenertion setD together with prtil method for pE proximting the regenertive extensionF st is sed on the notion of rrris small setF ell tht rrris set S ∈ E is small for the hin X if there

exist m ∈ N * D proility mesure Φ supported y SD nd δ > 0 suh tht ∀x ∈ S, ∀A ∈ E, Π m (x, A) ≥ δΦ(A), @PA
where Π m denotes the mEth iterte of ΠF oughly spekingD the smll sets re the ones on whih n iterte of the trnsition proility is uniformly ounded elowF hen @PA holdsD one sys tht X ful(lls the minorization condition M(m, S, δ, Φ)F e point out tht smll sets do exist for rrris hinsD see PPF uppose now tht ondition @PA is stis(edF ther thn repling the originl hin y the hin {(X nm , ..., X n(m+1)-1 )} n∈N D we tke m = 1F he regenertive wrkov hin into whih X is emedded is onE struted y expnding the smple spe in order to de(ne spei( sequene (Y n ) n∈N of independent fernoulli rFvF9s with prmeter δF he joint distriE ution is otined y rndomizing the trnsition Π eh time the hin X hits SD whih ours with proility one @rell tht the hin X is rrrisAF sn order to otin n insight into this onstrutionD oserve (rst thtD when X n ∈ SD the onditionl distriution of X n+1 given X n my e viewed s the following mixture

Π(X n , dy) = (1 -δ) Π(X n , .) -δΦ(dy) 1 -δ + δΦ(dy),
of whih the seond omponent is independent of X n F wore preiselyD the soEtermed split chain {(X n , Y n )} n∈N is uilt the following wyX suppose tht

X n ∈ SD if Y n = 1 @whih ours with proility δ ∈ ]0, 1[AD X n+1 is drwn from ΦD otherwise @i.e. if Y n = 0D whih hppens with proility 1 -δAD X n+1 is drwn from (1 -δ) -1 (Π(X n , .
) -δΦ(.)). glerlyD S × {1} is n tom for the split hinD the ltter inheriting ll the ommunition nd stohsti stility properties from XF sn prtiulr the dt segments in etween onseutive visits to S × {1} re independentF On approximating the regenerative extension. nfortuntelyD the split hin is theoretil onstrution nd the Y n 9s nnot e oserved in prtieF e 4plugEin4 pproh hs een nevertheless proposed in VD in order to generteD onditionlly to X (n+1) = (X 1 , . . . , X n+1 )D rndom vetor ( Y 1 , . . . , Y n ) from @supposedly knownA prmeters (S, δ, Φ) in wy tht its S onditionl distriution pproximtes the distriution of (Y 1 , . . . , Y n ) ondiE tioned upon X (n+1) in ertin sense tht will e spei(ed elowF rere we sE sume tht the onditionl distriutions Π(x, dy) with x ∈ E re dominted y σE(nite mesure λ(dy) of refereneD in wy tht Π(x, dy) = π(x, y)•λ(dy) for ll x ∈ EF his lerly implies tht Φ(dy) is lso solutely ontinuous with respet to λ(dy)D nd tht ∀x ∈ S, π(x, y) ≥ δφ(y), λ(dy) lmost surelyD @QA where Φ(dy) = φ(y)•λ(dy)F qiven the smple pth X (n+1) D the Y i 9s re indeE pendent rndom vrilesF o e more preiseD the onditionl distriution of Y i is the fernoulli distriution with prmeter

δφ(X i+1 ) π(X i , X i+1 ) • I{X i ∈ S} + δ • I{X i / ∈ S}.
@RA e nturl wy of mimiking the xummelin splitting onstrution onE sists of omputing (rst n estimte π n (x, y) of the trnsition density over S 2 sed on the ville smple pth nd suh tht π n (x, y) ≥ δφ(y) FsF for ll (x, y) ∈ S 2 D nd then generting independent fernoulli rndom vriE les Y 1 , . . . , Y n given X (n+1) D the prmeter of Y i eing otined y plugging π n (X i , X i+1 ) into @RA in ple of π(X i , X i+1 )F e point out thtD from prE til viewpointD it tully su0es to drw the Y i 9s only t times i when the hin hits the smll set SD Y i inditing whether the trjetory should e ut t time point i or notF vet l n = 1≤k≤n I{X k ∈ S, Y k = 1}F roeeding this wyD one gets the sequene of approximate regeneration timesD nmely the suessive time points τ S (1), . . . , τ S ( l n ) t whih (X, Y) visits the set S × {1}F yne my then form the approximate regeneration blocks B 1 , ..., B ln-1 D s well s the approximate cycle submaximaX

ζ j (f) = mx 1+ τ S (j)≤i≤ τ S (j+1)
f(X i ) with j = 1, . . . , l n -1. @SA unowledge of the prmeters (S, δ, φ) of ondition @QA is required for implementing this pproximtion methodF e prtil method for seleting those prmeters in fully dtEdriven mnner is desried t length in WF he question of ury of this pproximtion hs een ddressed in VF nder the following ssumptionsD shrp ound for the devition etween the distriution of ((X i , Y i )) 1≤i≤n nd tht of the ((X i , Y i )) 1≤i≤n in the sense of the wllows or sserstein distne hs een estlishedD whih essentilly depends on the rte ρ n of the uniform onvergene of π n (x, y) to π(x, y) over S × SF T A1. he wi of π is of order ρ n when error is mesured y the sup norm

over S 2 X E ν sup (x,y)∈S 2 | π(x, y) -π(x, y)| 2 = O(ρ n ) s n → +∞,
where (ρ n ) denotes sequene of nonnegtive numers deying to zero t in(nityF A2. he prmeters S nd φ re hosen so tht inf x∈S φ(x) > 0F

A3. e hve sup (x,y)∈S 2 π(x, y) < ∞ nd sup n∈N sup (x,y)∈S 2 πn (x, y) < ∞ P ν EFsF F 3 Regeneration-based Extreme Value Statistics sn this setionD we rell how to onstrut estimtors of the extreml nd til indexes sed on the @pproximteA yle sumxim following in the footsteps of IHF por eh estimtor onsideredD symptoti normlity is estlished nd the stndrdiztion prolem is tkledF 3.1 Asymptotically normal estimators of the extremal index e key prmeter in the extreml ehvior nlysis of n instntneous funE tion {f(X n )} n∈N of the hin X is the extremal index θ ∈ (0, 1)D mesuring to wht extent extreme vlues tend to ome in 4smll lusters4Y refer to ISD II nd IV for n ount of this notionF por positive reurrent wrkov hin X with limiting proility distriution µ nd ny mesurle funtion f : (E, E) → RD there lwys exists θ = θ(f) ∈ [0, 1] suh tht

P µ ( mx 1≤i≤n f(X i )) ≤ u n ) ∼ F(u n ) nθ s n → ∞, @TA for ny sequene of rel numers {u n } suh tht n(1 -F(u n )) → η for some η < ∞D denoting y F(x) = (E A [τ A ]) -1 E A [ τ A i=1 I{f(X i ) ≤ x}] the df of f(X 1 )
in stedyEstteD i.e. under P µ F es lredy oserved in IHD positive reurE rent hin is a fortiori strong mixing @cf heorem e in RA nd onsequently stis(es vedetter9s mixing ondition D(u n )Y see PRF sn the reminder of this susetionD the funtion f(x) is (xed nd the index θ is ssumed to e stritly positiveF e point out tht QI hve provedD under n extr tehnil ssumptionD tht the extreml index of ny geometrilly ergodi wrkov hin is stritly positiveY refer to heorem RFI thereinF U 3.1.1 The regenerative "blocks" estimator es originlly shown in QPD it follows from @IA nd @TA thtD for ny sequene

{u n } suh tht n(1 -F(u n )) → η for some η < ∞D θ = lim n→∞ θ(u n )D where θ(u) = © G f (u) Σ f (u) , @UA with Σ f (u) = α © F(u) = E A [ τ A i=1 I{f(X i ) > u}]D denoting y © G(x) = 1 -G(x) the survivor funtion of ny df G(x)D nd the onvention tht 0/0 = 0F
sn the regenertive seD from expression @UAD whih my e viewed s regenertive version of the populr 4loks4 estimtor @see VFIFP in ISAD it hs een proposed in IH thtX

θ n (u) = © G f,n (u) Σ f,n (u) , @VA whereD for ll u ∈ RD G f,n (u) = 1 l n -1 ln-1 j=1 I{ζ j (f) ≤ u} nd Σ f,n (u) = 1 l n -1 ln-1 j=1 S j (u), with S j (u) = τ A (j+1) i=τ A (j)+1 I{f(X i ) > u}D l n = n
i=1 I{X i ∈ A}D nd the usul onvention regrding empty summtion nd 0 0 = 0F ixpetedlyD ounterprt of this quntity in the generl rrris se is oE tined y repling the regenertion yle sumxim y their pproximte versions in @VAX

θ n (u) = 1 -G f,n (u) Σ f,n (u) , @WA whereD for ll u ∈ RD G f,n (u) = 1 ln-1 ln-1 j=1 I{ ζ j (f) ≤ u} nd Σ f,n (u) = 1 ln-1 ln-1 j=1 S j (u)D with S j (u) = τ S (j+1) i= τ S (j)+1 I{f(X i ) > u} for 1 ≤ j ≤ l n -1F
hese estimtors hve een proved onsistent in IH under mild moment ssumptionsD see roposition R thereinF por lrity9s skeD we rell the relted resultF Proposition 1. ([10]) Suppose that θ > 0. Let (r n ) n∈N increase to innity in a way that

r n = o( n/ log log n) as n → ∞. Consider (v n ) n∈N such that r n (1 -G f (v n ))/α → η < ∞ as n → ∞.
V (i) In the regenerative case, suppose that H(ν, 1) and H(2) are fullled.

Then,

θ n (v n ) → θ P ν -almost surely, as n → ∞.
@IHA (ii) In the general case, assume that moment assumptions H(ν, 1) and H(4) are fullled by the split chain and in addition that conditions A 1 -A 3 are satised. Then,

θn (v n ) → θ in P ν -probability, as n → ∞. @IIA
Remark 1. ( On moment assumptions for the split chain) We point out that, in the pseudo-regenerative setup described in 2.2, a sucient condition for condition H(κ) (respectively, for condition H(ν, κ)) to hold is

H(κ) : sup x∈S E x [τ κ S ] < ∞ (resp., H(κ, ν) : E ν [τ κ S ] < ∞).
Practically, drift conditions of the Foster-Lyapounov type are used for checking such moment conditions; refer to Chapter 11 in [25] for further details.

Remark 2. ( On the empirical choice of the threshold sequence)

In practice, the threshold sequence {v n } must be picked by the statistician. A natural choice, based on the available sample, consists of taking

v n = G -1 f,n (1 -η/r n ) in the regenerative case (respectively, v n = G -1 f,n (1 -η/r n )
in the pseudo-regenerative case) and one may easily show that assertion (i) (resp., assertion (ii)) of Proposition 1 remains valid.

he next result revels thtD for (xed threshold u ∈ RD the symptoti distriution of the quntity @VAD respetively @WAD is qussinF he tehnil proof is given in the eppendix setionF Theorem 2. Let u > 0 be xed.

(i) In the regenerative case, under assumptions H(2) and H(ν, 1), there exists a constant

σ 2 f (u) < ∞ such that √ n (θ n (u) -θ(u)) ⇒ N (0, α • σ 2 f (u)) as n → ∞, @IPA
where ⇒ denotes the convergence in distribution.

(ii) In the pseudo-regenerative case, if the moment assumptions H(ν, 1) and H(4) are fullled by the split chain and if conditions A 1 -A 3 are in addition satised, then

√ n θ n (u) -θ(u) ⇒ N (0, α • σ 2 f (u)) as n → ∞.
@IQA W es shown in heorem P9s proofD the symptoti vrine is given y

σ 2 f (u) = σ 2 1 (u) Σ f (u) 2 -2 σ 12 (u) © G f (u) Σ f (u) 3 + © G f (u) 2 σ 2 2 (u) Σ f (u) 4 , @IRA
where

σ 2 1 (u) = © G f (u)(1 -© G f (u)), σ 2 2 (u) = E A   τ A i=1 I{f(X i ) > u} -Σ f (u) 2   , σ 12 (u) = E A I{ mx 1≤i≤τ A f(X i ) > u} -© G f (u) τ A i=1 I{f(X i ) > u} -Σ f (u) .
hese quntities my e strightforwrdly estimted y omputing their empiril ounterprts sed on the @pproximteA regenertion ylesF rowE everD the following result shows thtD for properly hosen threshold sequene {v n }D inresing to in(nity t suitle rteD the seond nd third terms on the right hnd side of @IRA vnishD while the (rst one onverges to (αη

) -1 θ 2 s n → ∞F Proposition 3. Let (r n ) n∈N increase to innity in a way that r n = o( n/ log log n) as n → ∞. Consider (v n ) n∈N such that r n (1 -G f (v n ))/α → η < ∞ as n → ∞.
(i) In the regenerative case, provided that assumptions H(2) and H(ν, 1) are fullled, the following convergence in distribution holds:

n/r n (θ n (v n ) -θ(v n )) ⇒ N (0, θ 2 /η), as n → ∞.
@ISA (ii) In the pseudo-regenerative case, if the split chain satises H(ν, 1) and H(4) and conditions A 1 -A 3 hold, we have the following convergence:

n/r n θ n (v n ) -θ(v n ) ⇒ N (0, θ 2 /η) as n → ∞.
@ITA e point out thtD under the mximum domin of ttrtion @wheA ssumption omined with dditionl tehnil onditionsD the symptoti is my e proved to vnishF sndeedD rell thtD under the ssumption tht θ > 0D the proility distriutions G f (dx) nd F(dx) neessrily elong to the sme wheF uppose for instne tht they elong to the préhet wheF here exists then a > 0 suh tht one my write ©

G f (x) = L 1 (x) • x -a nd © F(x) = L 2 (x) • x -a D
where L 1 (x) nd L 2 (x) re slowly vrying funtionsF sn IH this setupD the extreml index is thus proportionl to the limiting rtio of these two funtionsX

θ(u) = L 1 (u) αL 2 (u) .
essume in ddition tht some seondEorder rllEtype onditions re ful(lledD

L i (x) = lim y→∞ L i (y) + C i • x -β i + o(x -β i ), s x → ∞ where C i < ∞ nd β i > 0D i = 1, 2F henD θ(v n ) onverges to θ t the rte v -β n with β = β 1 ∧β 2 nd v n ∼ r 1/β 1 n F reneD s soon s (r n ) is piked suh tht n/r 1+2β/β 1 n → 0D we hve tht n/r n (θ n (v n ) -θ) ⇒ N (0, θ 2 /η) s n → ∞ in
(1-G f (u n ))/α → η < ∞D we hve θ = lim n→∞ θ (u n ) whereX ∀u ∈ RD θ (u) = P A ( mx 2≤i≤τ A f(X i ) ≤ u | X 1 > u).
@IUA fsed on pth X 1 , . . . , X n D the nturl empiril ounterprt of @IUA in the regenertive setting is

θ n (u) = ln-1 j=1 I{mx 2+τ A (j)≤i≤τ A (j+1) f(X i ) ≤ u < f(X 1+τ A (j) )} ln-1 j=1 I{f(X 1+τ A (j) ) > u} . @IVA
snsofr s @IUA mesures the lustering tendeny of high threshold exeednes within regenertion yles onlyD it should e seen s 4regenertive version4

of the runs estimator

θ(r) n (u) = n-r j=1 I{mx j+1≤i≤j+r f(X i ) ≤ u < f(X j )} n-r j=1 I{f(X j ) > u} , @IWA
otined y verging over overlpping dt segments of (xed length rF sn the pseudoEregenertive seD prtil estimte is uilt y mens of the pproximte regenertion timesX

θ n (u) = ln-1 j=1 I{mx 2+ τ S (j)≤i≤ τ S (j+1) f(X i ) ≤ u < f(X 1+ τ S (j) )} ln-1 j=1 I{f(X 1+ τ S (j) ) > u} @PHA

II

feyond its prtil dvntge @loks re here entirely determined y the dtAD the estimtor @18A my e proved strongly consistent s stted in the (rst prt of the next theoremD while only wek onsisteny hs een estlished for @IWA ut for wider lss of wekly dependent sequenesY see PIF Theorem 4. Let r n increase to innity in a way that

r n = o( n/ log log n) as n → ∞. (i) Assume that H(ν, 1) is fullled. Considering (v n ) n∈N such that r n (1 - F(v n )) → η < ∞ as n → ∞, we then have θ n (v n ) → θ, P ν -almost surely, as n → ∞.
(i ) Similarly, if the split chain fullls moment conditions H(ν, 1) and H(4) and conditions A 1 -A 3 hold, then weak consistency holds in the pseudo regenerative case:

θ n (v n ) → θ, in P ν -probability, as n → ∞.
(ii) In the regenerative case, provided that assumption H(ν, 1) is fullled, the following convergence in distribution also holds:

n/r n θ n (v n ) -θ(v n ) ⇒ N (0, θ 2 (1 -θ)/η), as n → ∞. @PIA (ii )
In the pseudo-regenerative case, if the split chain satises H(4) and H(ν, 1) and conditions A 1 -A 3 hold, we have the following convergence:

n/r n θ n (v n ) -θ(v n ) ⇒ N (0, θ 2 (1 -θ)/η) as n → ∞. @PPA
he lst sttement of the preeding theorem nd roposition Q (i) onE stitute the regenertive versions of heorems Q nd R in QQD who (rst proved the gv for the lssil runs estimtor @sed on loks of (xed lengthD fF @IWAAF he proof of the preeding theorem follows the lines of those of roposition ID heorem P nd roposition QD s skethed in the ppendix setionF IP 3.2 Asymptotic normality of the regeneration-based Hill estimator sn this setionD we ssume tht θ > 0 nd heneD s relled in the previE ous setionD the distriutions G f (dx) nd F(dx) elong to the sme wheF e ssume here tht they elong to the préhet wheF sn the regenertive settingD nturl wy of estimting F9s til indexD proposed in IHD thus onsists in omputing rill estimte of G f 9s til index from the oserved yle sumximX

ξ n,k = k -1 k i=1 log ζ (i) (f) ζ (k+1) (f) -1 , @PQA with 1 ≤ k ≤ l n -1 when l n > 1D denoting y ζ (j) (f) the jEth lrgest sumximumF es l n → ∞, P ν E lmost surely s n → ∞D symptoti results
estlished in the se of iFiFdF oservtions extend strightforwrdly to our settingD see prt @iA of heorem S elowF e point out tht in the iFiFdF setup one my tke the whole stte spe s n tomD i.e. A = ED eh yle omprises then single oservtion nd (23) redues to the stndrd rill estimtorF sn the generl rrris seD one my nturlly uild n estimte y reE pling the yle sumxim y their pproximte versionsX

ξ n, k = k -1 k i=1 log ζ(i) (f) ζ(k+1) (f) -1 , @PRA
with 1 ≤ k ≤ ln -1 when ln > 1 nd denoting y ζ (j) (f) the jEth lrgest pproximte sumximumF st is shown in roposition S of IH tht the pproximtion step does not ompromise the onsisteny of the estimtorD provided tht the estimtor of π(x, y) over S 2 is urte enoughF sn order to estlish rte of onvergeneD we will lso onsider the se where the trnsition estimte used in the pproximtion stge is omputed from trjetory of length N >> n nd will denote y H

(N)
k,n, the orresponding estimtorF he onsisteny nd the symptoti normlity of these estimtors hve een shown in IH under the on wises ondition relled elowY see ropoE sition S thereinF VM assumption. @Von Mises conditionD IWA vet ρ ≤ 0F uppose

© G f (x) = L(x)x -a D lim x→∞ © G f (tx)/ © G f (x) -t -a b(x) = t -a t ρ -1 ρ , t > 0 IQ where b(x) is mesurle funtion of onstnt signD nd withD y onvenE tionD (t -ρ -1)/ρ = log t when ρ = 0F iquivlentlyD if U f (t) = G -1 f (1 -t -1 )D lim x→∞ U f (tx)/U f (x) -t -1/a B(x) = t 1/ξ t ρ/a -1 ρ/a , where B(x) = a -2 b(U f (x))F
rereD we formulte entrl limit theorem in more generl fshionD reveling isEvrine trdeEo' similrly to IQ in the iFiFdF setupF he proof is omitted s it follows y strightforwrd modi(tion of the proof of proposition S in IH nd the referenes thereinF Theorem 5. Assume that F belongs to the Fréchet MDA and the VM assumption holds and consider an increasing sequence of integers {k(n)} such that:

k(n) < n, k(n) = o(n) and log log n = o(k(n)) as n → ∞. Assume further that lim √ kB(n/k) = λ ∈ R, @PSA
(i) then, in the regenerative case, the following convergence in distribution holds

k(l n ) ξ n, k(ln) -ξ ⇒ N ξ 3 λ ρ -ξ
, ξ 2 under P ν , as n → ∞.

@PTA (ii) in the pseudo-regenerative case, if conditions A 1 -A 3 are in addition fullled, let (m n ) n∈N be a sequence of integers increasing to innity such that

m n ρ n /k(m n ) → 0 as n → ∞, then k( l mn ) ξ mn, k( lm n ) -ξ ⇒ N ξ 3 λ ρ -ξ , ξ 2 under P ν , as n → ∞.
@PUA 4 Regenerative block-bootstrap condence intervals sn this setionD we rell the priniple underlying the @pproximteA regenerE tive lokEootstrpD originlly introdued in V for ootstrpping wrkoE vin smple mensD nd estlish its symptoti vlidity when pplied to the estimtors desried in the preeding setionF IR 4.1 The (A)RBB principle rtillyD the @eAff lgorithm pplies to ny sttisti T n = T (B 1 , . . . , B ln-1 )D sed on the @pproximteA yles with stndrdiztion σ n = σ [T (B 1 , . . . , B ln-1 )]F por nottionl simpliityD regenertion yles nd their pproximte versions re here denoted in the sme mnnerF he resmpling sheme onsists of mimiking the underlying renewl struture y drwing dt loks with reE plement until trjetory of roughly length n is uiltF sn this wyD the rndomness in the numer of renewls is reprodued during the proedure ndD onditionlly to the originl dtD the ootstrp series thus generted is regenertiveF Algorithm 1. (A)RBB algorithm 1. (Blocks.) Identify the (pseudo-) blocks B 1 , . . . , B ln-1 from the observed trajectory X 0 , . . . , X n as explained in Section 2.1 (resp. in Section 2.2 in the pseudo-regenerative case) and compute the statistic T n = T (B 1 , . . . , B ln-1 ), and its standard deviation σ n = σ(B 1 , . . . , B ln-1 ).

(Sequential

(Bootstrap CIs.) Bootstrap condence intervals (CI) at level 1-α ∈

(1/2, 1) for the parameter of interest are obtained by computing the bootstrap root's quantiles q * α/2 and q * 1-α/2 , of orders α/2 and 1 -α/2 respectively (in practice, the latter are approximated in a Monte-Carlo fashion by iterating steps 2-3): the basic Percentile bootstrap CI is simply

[q * α/2 , q * 1-α/2 ],
IS the Percentile bootstrap CI is dened as

2 T n -q * 1-α/2 , 2 T n -q * α/2
and the t-Percentile bootstrap CI is given by

T n -t * 1-α/2 σ n √ n , T n -t * α/2 σ n √ n ,
where t * p is the p th quantile of the studentized bootstrap root

T * n -Tn σ * n / √ n .
Remark 3. (Gaussian confidence intervals) These bootstrap CI's can be compared to asymptotic CI's classically built from the statistic and its standardization

T n -Φ -1 1-α/2 . σ n / √ n, T n -Φ -1 α/2 σ n / √ n ,
where Φ -1 p is the p th quantile of the standard normal distribution, or replacing σ n / √ n with a new standardization estimator dened as the empirical standard deviation of T * n given by σ

* 2 = b (T * n -T * n ) 2 /n.

Asymptotic validity of (A)RBB distribution estimates

he results stted elow show tht the ootstrp proedure desried in the previous susetion is symptotilly vlidF vet P * (.) e the onditionl proility given the oserved trjetoryF he following ssertions hold trueF Theorem 6. 1. ("Blocks" estimator) Suppose that the assumptions of Theorem 2 are fullled. Let θ n (u) denote the estimator θ n (u) in the regenerative case, θ n (u) in the pseudo-regenerative case, and let θ * n (u) be its (A)RBB version. Then, we have, as n → ∞:

sup x∈R P * √ n θ * n (u) -θ n (u) ≤ x -P ν √ n θ n (u) -θ(u) ≤ x → 0.
2. ("Runs" estimator) Suppose that the hypotheses of Theorem 4 are satised. Denote by θ n (u) the estimator θ n (u) in the regenerative case, θ n (u) in the pseudo-regenerative case, and let θ * n (u) be its (A)RBB version. Then, we have, as n → ∞:

sup x∈R P * √ n θ * n (u) -θ n (u) ≤ x -P ν √ n θ n (u) -θ(u) ≤ x → 0.
uh results my lso e used to estimte the menEsqure error of √ n(θ n (u) -θ(u)) nd to lirte the level u y minimizing the wiD in the sme spirit s PH or IPD nd s illustrted in the simultion setionF IT 4.3 Markov subsampling and the Hill estimator es limed y the following propositionD the @eAff lgorithm n lso e suessfully pplied to til index estimtion provided tht the sequentil drwing @step P in the previous lgorithmA is repled with susmpling drwing without replementY see IRF roving tht the proedure is still vlid in the sene of susmpling deserves muh more thorough nlE ysisD fr eyond the sope of this pperF e thus introdue the following susmpling vrint of elgorithm IF Algorithm 2. RBB subsampling 1. (Blocks.) As described in step 1 of Algorithm 1.

2. (Subsampling drawing.) Choose a subsampling size m n large enough but small compared to n and compute l mn as the observed number of blocks in a stretch of length m n : typically, l mn is of order [ mn E A τ A ] and is thus asymptotically equivalent to l mn = l n mn n , where ).

Theorem 7. Suppose that the assumptions of Theorem 5 are fullled. Denote by ξ n,k the estimator ξ n,k in the regenerative case, ξ n,k in the pseudoregenerative case, and let ξ * n,k be its subsampling counterpart. Let m n > 1 such that m n → +∞ and m n /n → 0 as n → +∞. If we assume in addition that k( l mn )/k(l n ) → 0, we then have, as n → +∞,

sup x∈R H * n (x) -H n (x)-→ 0,
where

H * n (x) = P * k( l mn ) ξ * mn, k(lm n ) -ξ n, k(ln) ≤ x and H n (x) = P ν k(l n ) ξ n, k(ln) -ξ ≤ x .
sn the susmpling ontextD higher order ury nnot e estlishedF st is thus su0ient to onsider simple form of the stndrdiztion in order to prove the symptoti vlidityF he issue of hoosing the susmpling size m n nd the tuning prmeter k is disussed in the next setionF IU sn this setionD we present illustrtive simultion results to provide empiril evidene of the nie ehvior of the estimtors nd on(dene intervls proE posed in this pperF henever possileD omprison with other estimtors nd on(dene intervls is ondutedF

Regenerative examples

gonsidering witing times of ertin queuing proessesD we ompute nd disE uss the regenertionEsed 4loks4 nd 4runs4 estimtors of the extreml index nd the regenertionEsed rill estimtor of the til prmeterF Regeneration-based extremal index estimators. e (rst onsider the witing times of n wGwGI proess @cf QA with prmeters λ = 0.2, µ = 0.8 nd smple pth length nF es underlined in IHD there exists losed nE lytil form for the extreml index in this seY it is equl to θ = (1-λ/µ) 2 = 0.5625 nd ll the required ssumptions re stis(edF he estimtors θ n (u) nd θ n (u) of the extreml index proposed in this pper re oth de(ned sed on threshold uD supposed to e lrgeF RBB condence intervals. pigures I@A nd I@A show the symptoti nd ootstrp on(dene intervls of the regenertive 4loks4 estimtor nd the regenertive 4runs4 estimtorD respetivelyF hese gs9s re quite similr exept for the lrgest vlues of uF sn the sequelD when ootstrp gs is omputedD it will e the si perentile ootstrp on(dene intervlF he overge proilities of the si ootstrp perentile gs for the wGwGI witing proess is estimted over M = 300 trjetoriesD s shown in pigure PF Choosing the threshold. es mentioned fter heorem TD the threshold u n e hosen y minimizing n estimtion of the menEsqure error of √ n(θ n (u) -θ(u))F he optiml threshold vlue u * n therefore e deterE mined s

u * = rg min u>0 MSE(u), with MSE(u) = σ 2 f (u n + (θ n (u) -© θ * n (u)) 2 D where © θ * n (u)
is the men of the ootstrp sttistisF he sme proess n e pplied to the regenertive 4runs4 estimtorF epplying this to the wGwGI queue yields θ * θ n (u * ) = 0.5263 with gs (0.4431 .6610) @whih inludes the trgeted extreml index 0.5625AF enother possiility @whih does not require ny ootstrpA rises from the ft tht the rtio of the symptoti vrines of our P regenertive IV 0.90 0.92 0.94 0.96 0.98 1.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (a) Regenerative "Blocks" estimator 0.90 0.92 0.94 0.96 0.98 1.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (b) Regenerative "Runs" estimator pigure IX ixtreml index estimtion for witing times of the wGwGI queue with λ = 0.2, µ = 0.8, θ = 0.56 @the xExis gives the perentiles of the simulted (W n )D n = 1000D B = 199 ootstrp smplesD solid red for the regenertive estimtorD solid lk for the men ootstrp estimtorD dshed red for the si perentile ootstrp gsD dotted red for the perentile ootE strp gsD dshed green for the tEperentile ootstrp gsD dshed lue for the symptoti gs sed on the regenertive stndrdiztionD dshed light lue for the symptoti gs sed on the ootstrp stndrdiztionD horizontl lk line is θD vertil dshed red line is the optiml u vlue s determined y minimizing @PVAF IW pigure PX goverge proilities of the si perentile ootstrp gs for the regenertive 4loks4 estimtor nd the regenertive 4runs4 estimtorsF wGwGI queue with λ = 0.2, µ = 0.8, θ = 0.56 @the xExis gives the perentiles of the simulted (X n )D n = 1000D 1-α = 957EgsD B = 199 ootstrp smplesD M = 300D the solid lue urve is tht of the 4loks4 estimtorD the dshed red urve is tht of the 4runs4 estimtorAF estimtors is symptotilly onstnt @σ f (u n ) 2 /σ 2 f (u n ) → 1-θ for properly hosen sequene of thresholds u n D see heorem QD ssertion @iA nd theorem RD ssertion @iiA I FA reneD one my de(ne n optiml threshold vlue u * D nd hene unique estimtor of the extreml indexD y minimizing in u the funtion

σ 2 f (u)/σ 2 f (u) -(1 -θ n (u))
2 , @PVA nd de(ning θ * = θ n (u * )F epplying this proess to the wwI queue yields θ * = 0.5179 with gs (0.4947 .6370) @whih overs the trgeted extreml index 0.5625AF

Alternative estimators. sn IHD the regenertive loks estimtor ws ompred to the intervls estimtor proposed y IT nd to vrious (xed lengths lok estimtors nd runs estimtors @see pigF P thereinAF sts men squred error ws generlly lower thn those of the lterntive estimtorsF es fr s gs9s re onernedD the uthors of IT lso proposed ootstrp proedure sed on n utomti delustering of the proess relying on the estimtion of the extreml index @see setion R thereinAF pigure Q illustrtes 1 The asymptotic variance of the regenerative "runs" estimator for a xed threshold u is σ f (u) 2 = s 2 f (u)/α, with s 2 f (u) given in eq. ( 30) in the appendix section. pigure QX gomprison of our ootstrp si erentile gs to tht proposed in ITF @B = 199D dshed lue for ours nd dshed green for theirsD solid lk is the true θAF tht our ootstrp gs is muh shrper thn theirs on this exmpleF Regeneration-based Hill Estimator. e now onsider the witing times of n wGqGI proess with reto servie timesD with prmeters λ = 0.2 nd a = 3F he susmpling size ws (xed t m n = n/ log(n) F por eh of the M trjetoriesD for eh of the B ootstrp smplesD the regenertive rill estimtor is (rst omputed for vrious vlues of kD from k = 10 to the numer of loks k = l mn F he optiml k is then determined y omputing is orreted rill estimtor @s in TD IUA nd hoosing the vlue k * tht minimizes the estimted wi

MSE(k) = H 2 k, n /k + (H k, n -H k, n ) 2 ,
where H k, n is is orreted version of the rill estimtorF he regenertive stndrdiztion is then omputed s H k * ,n / √ k * . esults of this simultion re presented in le IF xote tht the si perentile gs nd symptoti gs with ootstrp vrine hve the est overge proilities nd re lso very esy to ompute @it does etter thn the symptoti gs whih hs however the dvntge of not requiring the ootstrp resmplingAF egrdE ing the hoie of the susmpling size m n D vrious vlues were tested nd lrger vlues do keep nie overge proility with redued men lengthF he pplition of elgorithm I yields prtiulrly nie results questioning the vlidity of suh proedure for the regenrtive rill estimtor nd hene

PI

= 0.5D 1/a = 1/3D smple pth of length n = 10, 000D m n = [n/ log(n)] = 1, 085D B = 199 ootstrp smplesD M = 300
wonteEgrlo replitions to ompute the overge proilitiesD men lengths of the gs9s nd men squred error of the estimtor @wiA E fsed on the ootstrp vrine E the lst line refers to the tndrd rill estimtor while the rest of the tle refers to the egenertive rill estimtorF the vlidity of the ootstrp of the rill estimtor in the iFiFdF se s well @theoretil work in progressAF Alternative estimator. sn IHD the regenertive rill estimtor is ompred to the stndrd rill estimtor omputed diretly from the longest witing timesD s proposed y PUF he sme is orretion method ws pplied to the stndrd rill estimtor in order to determine the optiml k vlueF sn their pper PUD the uthors do not propose ny on(dene intervl for their estimtor ut one ould ompute ootstrp gs s proposed in S in the iid se @the priniple is to resmple diretly the log di'erenes tht re iid exponentil rther thn the upper sttistisAF his pproh results in very smll gs9s tht fil to ompenste for the ft tht the tndrd rill estimtor is quite d on this exmple nd hene hve null overge proilityY see the lst line of le IF (d) Coverage "runs" estimator pigure RX ixtreml index estimtion for witing times of the eI guhy proess with ρ = 0.8 nd σ = 1, θ = 0.2 @the xExis gives the perentiles of the simulted (X t )D n = 10000D B = 199 ootstrp smplesD M = 100 wonte grlo replitions PR epplition of the helt method (nlly yields @IPAD with

√ n © G f,n (u) Σ f,n (u) - © G f (u) Σ f (u) ⇒ N 0, α • σ 2 1 (u) σ 12 (u) σ 12 (u) σ 2 2 (u) , sine n/(l n -1) → α = E A [τ A ], P ν EFsF s n → ∞D nd with σ 2 1 (u) = © G f (u)(1 -© G f (u)), σ 2 2 (u) = E A   τ A i=1 I{f(X i ) > u} -Σ f (u) 2   , σ 12 (u) = E A I{ mx 1≤i≤τ A f(X i ) > u} -© G f (u) τ A i=1 I{f(X i ) > u} -Σ f (u
σ 2 f (u) = σ 2 1 (u) Σ f (u) 2 -2 σ 12 (u) © G f (u) Σ f (u) 3 + © G f (u) 2 σ 2 2 (u) Σ f (u) 4 .
he demonstrtion of ssertion (ii) relies on similr rguments regrding the symptoti norml ehvior of the ivrite vetor (1-G f,n (u), Σ f,n (u)) otined from the gv stted in heorem IUFQFT of PSF A.2 Proof of Proposition 3 Lemma 8. We have

gonsider (v n ) n∈N suh tht r n © G f (v n )/α → η < ∞ s n → ∞D with
r n Σ f,n (v n )/α → η/θ, r n G f,n (v n )/α → η r n σ 2 1 (v n )/α → η, σ 2 2 (v n ) = O(r -2 n ), σ 12 (v n ) = O(r -3/2 n ).
Proof. gonsider the solution of the oisson equtionX

g(x, u) = E x τ A i=1 I{f(X i ) > u} -Σ f (u) .
yserve tht

σ 2 2 (v n ) ≤ E µ [ g(X 1 , v n ) 2 ] ≤ E A   τ A i=1 I{f(X i ) > v n } 2   + Σ f (v n ) 2 ≤ E A [τ 2 A ] + Σ f (v n ) 2 .
fy guhy hwrzD we yield the lst order of mgnitudeX

σ 12 (v n ) ≤ σ 1 (v n )σ 2 (v n ). xowD de(ne X n = ln-1 i=1 X j,n = ln-1 i=1 I{ζ j (f)>vn}-© G f (vn) √ ln-1σ 1 (vn) D nd oserve tht n/r n (θ n (v n ) -θ(v n )) = √ n √ l n -1 √ r n σ 1 (v n ) r n Σ f,n (v n ) X n - r n σ 2 (v n ) r n Σ f,n (v n ) θ(v n ) √ r n √ n Σ f,n (v n ) -Σ f (v n ) σ 2 (v n ) .
PS qiven the rtes in the preliminry lemmD the seond term on the r is of order O P (r -1/2 n )F gonsequentlyD the symptoti ehvior of n/r n (θ n (v n )θ(v n )) is determined y tht of the (rst termF e diret pplition of the vindeerg peller theorem is not possile here sine l n -1 is not stopping time for (X n )F rowever it n e shown through the pplition of slight modi(tion of the rguments given pge RPS of PS tht if we denote y X n the vetor X n in whih l n is repled with n/α D then n -1/2 |X n -X n | onverges to zero in proilityF hen the symptoti stndrd normlity of X n D nd hene tht of X n D results from the vindeerg peller theorem under the ssumed moment onditionsF e then onlude this proof from the ove lemm tht gives the limit s → ∞ of the term in front of X n D nmely θ/ √ ηF vet us now hek the onditions of the vindeerg peller theorem on X n F H1 he sequene X j,n is symptotilly negligile sine

n/α -1 j=1 P A (|X j,n | ≥ ε) ≤ ( n/α -1)E A [X 4 j,n ] ε 4 ≤ ( n/α -1) -1 ε 4 σ 4 1 (v n ) P →0 sine σ 4 1 (v n ) = O(r -2 n ) = O(log log n/n) nd E A [(I{ζ j (f) > v n } - G f (v n )) 4 ] ≤ 1F H2 he seond ondition n/α -1 j=1 E A [X j,n ] = 0 lso holds sine E A [I{ζ j (f) > v n }] = © G f (v n )F
H3 pinllyD the ondition on the vrine is lso stis(ed sineX

Γ n = n/α -1 k=1 V A (X j,n ) = ( n/α -1)V A (X j,n ) = ( n/α -1)σ 2 1 (v n ) ( n/α -1)σ 2 1 (v n ) P →1,
denoting y V A (.) the vrine under P A (.)F xote tht the ssumptions H(2) nd H(ν, 1) re only needed to ensure tht σ 2 (u) is de(ned for ll uD nd the length of the (rst non regenertive lok hs (nite (rst order momentF gonsequentlyD our regenertionEsed extreml index estimtor does not di'er muh from tht with denomintor

(l n -1)/n n i=1 I{f(X i ) > u}F A.3 Proof of Theorem 4 riting @IUA s the rtio of G 1 f (u) = P A ({mx 2≤i≤τ A f(X i ) ≤ u}∩{f(X 1 ) > u}) nd ©
F 1 (u) = P A (f(X 1 ) > u)D then the regenertive 4runs4 estimtor given PT in @IVA is simply the rtio of the empiril ounterprts of the proilitiesD whih re denoted G 1 f,n (u) nd © F 1 n (u) in the sequelF reneD the proof of @iA nd @iiA extly follows tht of the strong onsisteny of the regenerative blocks estimtorD (8)D provided in IHD nd tht of its symptoti normlE ity given oveD provided tht the next lemm holds true under the stted ssumptionsF Lemma 9. Since ©

G f (u) ∼ G 1 f (u), © F(u) ∼ © F 1 (u), we can state a LIL for G 1 f (u),
a LIL for © F 1 (u) and get asymptotic equivalences similar to those stated in the previous lemma. Let r n ↑ ∞ in a way that

r n = o( n/ log log n) as n → ∞, considering (v n ) n∈N such that r n (1 -F(v n )) → η < ∞ as n → ∞, we have r n © F 1 (v n ) → η/θ.
wore preiselyD we n stte the symptoti normlity of the ivrite vetor

(G 1 f,n (u), © F 1 n (u)) D for ll (xed uD √ n G 1 f,n (u) © F 1 n (u) - G 1 f (u) © F 1 (u) ⇒ N 0, α • V (G 1 f (u), © F 1 (u)) , with V (G 1 f (u), © F 1 (u)) = G 1 f (u)(1 -G 1 f (u)) G 1 f (u)(1 -© F 1 (u)) G 1 f (u)(1 -© F 1 (u) © F 1 (u)(1 -© F 1 (u)) F
xote tht euse of the spei( 4proility4 form of the ovrine terms here @they re ll lower or equl to oneAD we do not need extr moment ssumptions s we did in the se of the regenertive loks estimtorF en pplition of the helt method (nlly yields the following symptoti vrine for ll (xed u

s 2 f (u) = α ×   G 1 f (u)(1-G 1 f (u)) © F 1 (u) 2 -2 G 1 f (u) 2 (1-© F 1 (u)) © F 1 (u) 3 + G 1 f (u) 2© F 1 (u)(1-© F 1 (u)) © F 1 (u) 4   @PWA = α × θ (u) (1 -G 1 f (u)) © F 1 (u) -θ (u) 2 1 -© F 1 (u) © F 1 (u) . @QHA ine r n © F 1 (v n ) → η/θD
we n identify s 2 in @PIA nd @PPA s αθ 2 (1 -θ)F e forml pplition of the vindeergEpeller theorem similrly to the regenerE tive loks estimtor proof @the X n 9s re ivriteA leds to the sme result nd it is esily seen thtD euse only inditor funtions re involvedD no moment ssumption on τ A is neededF por the pseudoEregenertive version @i9A nd @ii9AD it is su0ient to oserve tht under the stted ssumptionsD we n prove similrly to theorem P in IHD tht sup

x∈R | Ĝ1 f,n (x) -G 1 f,n (x)| = O Pν (R n ( πn , π) 1/2 ), s n → ∞, PU nd similrly to vemm TFP in VD tht we hve N(n) × sup u∈R | F1 N(n) (u) - F 1 N(n) (u)| = O Pν (R N(n) ( πN(n) , π) 1/2 ) s N → ∞.
A.4 Proof of Proposition 4.2 he ootstrp version of the 4floks4 estimtor of the extreml index is given y the rtio of the ootstrp version ivrite vetor ( © G f,n (u), Σ f,n (u)) F ine y VD the @eAff @oth in its regenertive nd pseudoEregenertive versionsA is symptotilly vlidD it follows immeditely thtD for (xed uD √ n(θ * n (u) -θ n (u)) hs the sme limiting distriution s √ n(θ n (u) -θ(u)). he sme result remins vlid even if l n is (xed in the ootstrp proedureF he sme rguments my e used for the 4runs4 estimtorF A.5 Proof of Proposition 7 hen using l mn D the proposed proedure oils down to susmpling proE edure in n iFiFd frmeworkF sing ontinuity nd stndrd Esttistis rguments @see IRAD y mimiking the proof of IR @see pF RR thereinAD we otin tht

P * k( l mn ) ξ * mn, k(lm n ) -ξ n, k(ln) ≤ x = P * k( l mn ) ξ * mn, k( lm n ) -a ≤ x + o(1) = P k( l mn ) ξ mn, k( lm n ) -a ≤ x + o(1).
he (rst equlity is strightforwrd onsequene of the ontinuity of the limiting distriution of the rill estimtor nd of the ssumption stting tht k(l n mn n )/k(l n ) → 0. xow using the ft tht m n → ∞ nd the ft tht the rill estimtor so normlized hs nondegenerte distriutionD we get the result of roposition UF

  the regenertive seD nd similr result holds true in the pseudoEregenertive seF 3.1.2 The regenerative "runs estimator" sing the regenertive methodD it hs een proved in QP tht θ my e expressed s limiting onditionl proilityX if n

  drawing.) Draw sequentially and independently bootstrap data blocks B * 1 , . . . , B * k from the empirical distribution of the blocks dened at step 1. until the length of the bootstrap series l *(k) = k j=1 l(B * j ) is larger than n. Let l * n = inf{k ≥ 1, l * (k) > n}.If one is just interested in asymptotic results, one may just draw l n -1 i.i.d blocks (conditionally to the trajectory so that l n is xed in the bootstrap procedure).3. (Bootstrap statistics.) From the bootstrap data blocks generated at step 2, reconstruct a pseudo-trajectory by binding the blocks together, getting the reconstructed RBB sample pathX * (n) = (B * 1 , . . . , B * l * n -1 ) of length n * = l * (l * n -1).Then compute the bootstrap version of the regenerative blocks estimator:

  [x] is the integer part of x. Draw l mn bootstrap data blocks B * 1 , . . . , B * k by sampling without replacement in the blocks B 1 , . . . , B ln-1 . 3. (Subsampling statistics.) Apply steps 3 and 4 of Algorithm 1 to the reconstructed RBB sample path X * (n) = (B * 1 , . . . , B * lm n -1

PH

5. 2

 2 Pseudo-regenerative examplese now turn to exmples for whih regenertive extension must e pproxE imted nd show tht this dditionl step does not dmge the ury of the methodF Approximate regeneration-based extremal index estimator. por the pseudo regenertive seD we onsider (rst order utoregressive model with guhy noiseD with prmeters ρ = 0.8 nd σ = 1D yielding n extreml PP index θ equl to 1 -ρD see IH for detilsD nmely setion SFP therein for preise desription of the onstrution of the pseudoEloksF he ootstrp gs9s nd their overge proilities re shown in pigure RF xote tht the perentiles of X used for the 4runs4 estimtor re lot lower thn those used for the 4loks4 estimtorF he gs9s for the 4loks4 estimtor re etter thn those of the 4runs4 estimtor in terms of overge proilityF Approximate regeneration-based Hill estimator. ith the e@IAE guhy exmple ginD we investigted the estimtion of the til index equl to 1 hereD see IH for detilsF he regenertive rill estimtor ws omputed for M = 100 trjetories of length n = 10, 000D using susmpling size m n = n/ log(n) = 1, 085 nd B = 199 ootstrp replitions in eh seX we otined H n, k * = 1.14 for k * = 104 @sd = 0.111A with si perentile ootstrp gs of (0.592 -1.945)Y overge proility of 947 nd men length of 1.457F eginD when the susmpling size is inresedD the overge proility remins roung the desired 957 while the men length of the gs is drstilly reduedD whih puts questions to the vlidity of the full regenertive ootstrp for the regenertive rill estimtor s proposed in elgorithm I nd used for the regenertive extreml index estimtorsF A Technical Proofs A.1 Proof of Theorem 2 por ssertion (i)D oserve tht θ n (u) is simply the rtio of the omponents of the ivrite vetor ( © G f,n (u), Σ f,n (u)) D whih is symptotilly norml under the spei(ed moment onditions @see the proof of the gv stted in heorem IUFPFP of PSA for the tomi seX

  r n = o( (n/ log log n)F e preliminry step onsists of studying the ehvior of the vrious omponents of σ 2 f (v n ) s n → ∞D s stted in the next lemmF

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  ) .

	1.0					1.0
	0.8					0.8
	0.6					0.6
	0.4					0.4
	0.2					0.2
	0.0					0.0
	0.95	0.96	0.97	0.98	0.99	1.00	0.50	0.52	0.54	0.56	0.58	0.60
		(a) "Blocks" estimator			(b) "Runs" estimator
	1.0					1.0
	0.8					0.8
	0.6					0.6
	0.4					0.4
	0.2					0.2
	0.0					0.0
	0.95	0.96	0.97	0.98	0.99	1.00	0.50	0.52	0.54	0.56	0.58	0.60
	(c) Coverage "blocks" estimator
						PQ