f

l[

EXTREMES OF MOVING AVERAGES OF STABLE PROCESSES

by

Holger Rootz~n

Department of MathematiaaZ Statistias

University of Lund

and

Department of Statistias

University of North CaroUna at Chape Z Hi zz.

Institute of Statistics I~eo Series No. 1089

September, 1976

* This research was supported in part by the Office of Naval Researchunder Contract No. N00014-75-C-0809.

EXTREr,lES OF NOVHJG AVERf\GES OF STABLE PROCESSES

byHolger Rootzen*

ABSTRi~CT: In this paper we study extremes of non-norTI1a1 stable moving average

prOpJS3',;-;S)1- i. e. of stochastic processes of the fOUl Xet) = La (A,-t) Z(A) or X(t) =

! a(A-t)dZ(A), 'Where Z(A) is stable with index 0.<2. The extremes are

described as a marked point process $ consist:Lig of the point process of

(separated) exceedances of a level together with TI1arks associated with the

points, a j",1';trk being the nOTI1alized sample path of X(t) around an exceedance.

It is proved that this Inarked point process converges in distribution as the

level increases to infinity. The limiting distribution is that of a Poisson

process 't>lith independent marks \vhich have randor~1 heights but othen-lise are

deterministic. As a byproduct of the proof for the continuous-time case, a

result on sarilple path continuity of stable processes is obtained.

PJIS 1970 subject classifiC2,tion: Prll,~_ry 60FOS, Secondary 60G17.

Key 'ltJOrds and phrases: Extreme values, stable processes, moving average,

sa':1ple path continuity.

* '111is research 'ltJaS sponsored in part by the Office of Naval Research,Contract No. N00014-75-C-0809.

2

1. WTROOUCTION

Adistribution is stable (y~a,e) if it has the characteristic function

(1.1)

with Osy, O<as2, 181~1 and with h(u,a) = tan(na/2) for a~l, h(u,l) =

211"-1 log Iu/. Here y is a scale parameter, a is the inde~, and 8 is

the symmetry parameter of the stable distribution. If 8=0, then the

distribution is sYJl1Iletric, while the distribution is said to be aompZeteZy

asymmetria if /81=1 and a<Z. A stochastic process {X{t); te:T} is stable

with index a if for n=1,2,... ~i.d for arbitrary real numbers al"" ,an

and tl''''' tn e: T, the random variable alX(tl )+ ...+anX(tn) is stable with

index a. In particular, as can be seen from (1.1), a collection of indepen­

dent stable random variables with index a is a stable process.

In this paper, further use of the linear structure is made by restricting

attention to the subclass consisting of moving averages, i.e. to stationary

processes of the form X(t) =r a(A-t)Z(A), where . {Z(A)}~=_oo is anA

independent, stationary stable sequence, or of the form X{t) =fa(A-t)dZ(A)'

where {Z(A); -oo<A.<oo} has independent, stationary stable increments. If

a=2 the process is nonnal, and it can be represented as a moving average iff

its spectral distribution is absolutely continuous. Presently no simple

characterization of the class of moving averages of stable processes is

known for a<2. Of course normal processes are extensively analyzed but,

partly because of the common linear structure, also stable processes with

a<Z constitute a class of probability models that is amenable to analysis.

3

The subject of the present study:.is the asymptotic distribution of

extremes of moving averages of stable processes with a<2. As can be seen

from e.g. Leadbetter et al. (1976), a suitable framework for dealing with

extremes of stationary processes is the theory of point processes as given

in e.g. Kallenberg (1975) used to study the process of exceedances of a

high level. Here we will go one step further and adjoin a Tnar'k to each

point in the process of exceedances, the mark being the entire sample path

of the process, normalized and centered at (a point close to) the upcrossing.

The main results are that both when X(t) has discrete parameter (or

IYtimeil) and when X(t) has continuous parameter, the marked point process

converges in distribution. The limiting distribution is that of a Poisson

process (possibly with multiple points) with independent marks that are

distributed as a(-t) multiplied by a certain random variable.

One of the main differences between the nomal distribution (stable

with a=2) and stable distributions with a<2 is that the tails of the

latter decrease much slower. This leads to a radically different behavior

of extremes. For the nomal distribution the tails are of the order2

e-x /2/x, while for stable distributions with a<2 the tails decrease as-ax This affects extremes of moving averages in two different ways. To

fix ideas, consider e.g. maxi~ of a process X(t) = la(A-t)Z(A) with

discrete parameter. First, extremes increase much slower when a=2 than

when a<2, viz. as (log n)~ compared with nl / a . Secondly, when the

independent sequence {Z(A)} is normal, many of the Z(A)'s, OSAQ1, will

be almost as large as the largest one, and X(t) will be large when many

rather large Z(APS are added. This entails that the lirniting distribution

4

2of Hn = max X(t) only depends on La(A) and that it is the same as iflstSn

{X(t) } were an independent sequence with the same marginal distributions.

On the other hand~ when a<2 the maximum of Z(A) will be much larger

than the typical values ~ and X(t) will be large 1t/hen one very large

Z(A) is multiplied by a large a(A). In this case the limiting distribution

of r!~ depends on max a(A) and on min a(A) aTld is in general not-OO<A<OO -OO<A<OO

the same as if X(t) were an independent sequence with the same marginals.

In an earlier paper (Rootzen (1974)L the limiting distribution of

maxima of moving averages of symmetric stable sequences with a<2 was

obtained? but apart from that there do not seom to'be'any results published

on extremes of stationary stable processes with a<2.

The plan of this paper is as follows. Section 2 deals with convergence

in distribution of marked point processes. In Section 3 rather complete

asymptotic results on extremes of moving averages of stable sequences are

obtained. Section 4 contains preliminaries concerning moVing averages of

continuous parameter stable processes. In particular some conditions that

ensure sample path continuity are fotmd~ which may be of independent

interest. Finally, in Section 5 results for continuous parameter processes

corresponding to those of Section 3 are established for a~l~ under some

restrictions on a(A).

2. CONVERGENCE m OISTRIBUTIOf,1 OF i'IARKED POUlT PROCESSES.-._-----_._------_._----~ .._--

We are interested in the times Ostl<t2<•.• of occurrence of extreme

values of a stochastic process {X(t)} and in the behavior of the smnple

5

paths of {X(t)} near the ti's, and will describe them as a marked point

process. In this section we introduce some notation and develop techniques

needed in the remaining sections to prove convergence in distribution of marked

point processes. Unfortunately the notation is somewhat cumbersome, but

nevertheless we think it is well warranted, considering the completeness

of results it makes possible.

Write N for the space of integer-valued and locally finite Borel

measures on R+ and define a metric on N in the following way: Let00 + +,

F = {f.}. 1 be a sequence of functions in Cc = {f: R -+ R ; f is continuous1~ .

with compact support} such that any f E: Cc can be miformly approximat~d

00 -iby functions in F. For 1l,V E: N put P(l-hV) = li=l 2 Pi (ll,vL where

Pi(ll~V) = min(l, /ffidll-ffidvl). 111en P is a metric on N that generates

the topology of vague convergence (llnE:N converges vaguely to llE:N if

ffdlln -+ ffdll for all fE:Cc ); see e.g. Bauer (1972), p. 241. A point+

process in R is defined to be a (Borel measurable) random variable with

values in (N,p). As soon as we have (Borel measurable) random variables

in a metric space we may of course consider convergence in distribution,

using the theory of convergence in distribution in metric spaces as given

in e.g. Billingsley (1968). For further information on convergence in

distribution of point processes see [6]. We regard the tlines O~tI<t2<'"

of occurrence of extremes of {X(t)} as a poL~t process N by putting+N(B) = #{tiE:B} for allY Borel set B cR.

With each of the ti~s we associate a mark Yi ,where Yi

is the

entire sample path of {X(t) }, normalized and centered to show the behavior

close to t i • If {X(t)} is a discrete parameter process, Yi

is a

6

random variable in a space Rm

= {( ..•x_1,xO'x1,···); xifR, i=O,±l, ••• }.-Iii ~

We consider Rm as a metric space \lTith the metric o(x,y) = 4=_m2 0i(x,y)

that generates the product topology~, where <\ (x,y) = min(1/3, IXi-YiD. If

X(t) has continuous parameter we will impose conditions that make Yi a

random variable in D(-m,m), the space of functions on (-m,~) which are

right continuous and have lefthand limits. On D(-m~m) we use (a slight

modification of) the metric given by Lindvall (1973). Since there is no

risk of confusion we will use the same notation as for the metric on Rm

.

Thus for x,y f D(-m,m) we let o(x,y) = r~=_oo 0i(x,y) where 0i(x,y) =

min(1/3, h(c1.x,c.y)) and h(c.x,c.y) is the quantity given on p. 113-115111

of Lindvall's paper, modified to D(-m,m) i..Tlstead of D(O,oo) in the way

proposed on p. 121.

TIle marked point process is the vector n = (N',YpYZ'.'.) with values00 00"

in 5 = NxR xR x.. • (or in S = NxD ( -co:~:oo}xD( -oo:;.oo}x. •• ) which we again

consider as a metric space given a product metric d(x,y) = }:~=O Z-idi(x,y),

where for x = (v,x1,xZi ".) 2nd y = (v'Yl'YZ' ... ) we put dO(x,y) = p(v,~)

and di (x,y) = o(xpYi) 7 i~1. Our aim is to prove convergence in distribu­

tion of marked point processes, and to this end we need the following simple

criterions, which we state as lemmas for easy reference.

LE!'fMA 2.1. 1 Let Tln = (l~n'Ynl'YnZ 9 ••• ) and n = (N,Yl'Y2' ... ) be random

variables in the product space (5,d) . Suppose that Nn ,YrJ.'YnZ ' . • . are

1 The results of this section are fOTInulated in tems of a discrete parameter,n, which tends to infinity. They of course remain valid if the parametertends to Winite in a continuous manner, and they will be used accordinglyin Section 5.

7

independent for each n and that N,YI'Y2' . • . are independent. Then

T1n ~ n if (and only if) Nn ~ N and Yni ~ Yi ~ i~l.

PROOF. Since S is a product of separable spaces this follows as on p. 21

of [3]. o

LFl"I:JiA 2. 2. Let n~ = (N~,Y~l 'Y~2 1•.• ) 1 n~11 k~11 be random variables inkd k kd

(S,d). Suppose that for k~l, 11n ~ n as n+oo and that n ~ n as

k-kXl. Suppose further that

(2.1)

Then

lim lim sup P(di (n~,nn»e:)k-+oo n+oo

dnn ~ n as 1'):+00.

PROOF. For given e:>0 choose i to make Z-i < e:/2 and thus

\'00 -j kL· ·+1 2 d.(n~n) < e:/2. ThenJ=l. J n n

P(dCn;:.n"l>E) ~ j~O phCn;:.n"l > E/(ZCi+ll))

and by (2.1) we thus have

which by Theorem 4.2 of [3] proves the lemma. 0

For i~l, di ({~Tln) = o(Y~i'Yni) and repeating the above argument

once more we see that (2.1) holds for i~l if

(2.2) lim lim sup p(o. (yk . 'Ynl.·»e:) = 0, Ve:>O, j~l.k-+oo n+oo J Ill.

In the discrete-pararr~tercase (2.2) is easy to check, but when the parameter

8

is continuous further simplification is needed.

LElVlv1A. 2.3. Suppose that for each i~l there are random variables

with lim lim sup P( IEkl >x) = O~ "x>O, and such thatk-+oo n~ n

(2.3) lim lim sup p( sup /Yni (t)-Y~. (t+E~) I>x) = O~ Vx>O, .2.>0,k-+oo n-+oo _R,sts.2o 1!

and that furthennore lim lim sup Pl sup IYni (t) I>u) = 0, V.2o>O. Then~ ~ -.2ostsR,

(2.2) holds and thus (2.l)~ for i~l.

PROOF. Using the time transformation A(t) = {l-e-E-e-E/t} it is seen

that 0j(Y~z) s sup {ly(t)-Z(t+E)I+Ely(t)I} + E if OSEsl, and the-j-2stsj+2

lennna follows. 0

It is also possible to give a simpler condition for

lim lira sup P(do(~'Tln»E) = 0.k-+oo n+oo

LErc!NA. 2.4. Let ost~lst~2s... be the atoms (repeated according to their

multiplicities) of N~ and similarly let Ostn1stn2~'" be the atoms of

l~. Suppose that

(2.4) lim lt~ sup P(lt~.-tnil>£) = 0, V£>O, i=I,2, .••k-+oo n-+oo 1

and that in addition Nk~ Nk as n-+oo and that l'f ~ 1'1 as ~. Thenn

(2.1) holds for i=O.

PROOF. As above, it is enough to prove

(2.5) P = lim lim sup p(PJ.(~,Nn»£) = 0, V£>O, j=1,2, ..•k-+oo J'.t+OO

where Pj(N~,Nn) s IJfjdN~-Jfjd.Nnl with f j 6 Cc ' Suppose that the

9

support of f j is contained in [O~T]. For 0>0 there is a K with

P(N(£6~T+l]).>R}> 0 and thus lim sup p(rJe([O,Tj»K} ,( 0 andk-+oo

lim sup lim sup P(Nk([O,'l']»K) >'0. Furthennore take a step ftmctionk-+oo Jl'+<Xl n

get) = IT=i a i I(Ti<t~Ti+l)' with O~Tl<···<TmsT that approximates f j

unifonnly, with sup IfJo(t)-g(t)/ < e:/(2K). LettLllg Ostlst2s ... be theOst

atoms of N, we assurile without loss of generality that

(2.6) P(ti-Tj ) = 0, j=l,. 80m, i~l.

On the set {N~([O,T])~K, Nn([O:;T])sK} .we have

IffodNk_ff.ill~ I < IfgdNk_fgill~ I + 2Ke:/(2K).J n J n n n

Hence

P s limk, sup lim sup {p(N~(rO',T1)1K) +P(NIi([O,TJ»K) + ~p{1 fg~~f·gdNnl >O)}

"+00 11-+00

sLice p(lfg~-fgill~nl>O) + 0 by (2.6) and the hypothesis of the lemma.

However, 0>0 is arbitrary~ so (2.5) follows. 0

Finally, it should perhaps be stressed that the convergence nn ~ ~

that is to be proved in the following sections only says somet.~ing about

dthe sample paths near extremes: Namely, Yni -+ Yi where Yni , Yi are

random variables in (Roo, 0) if and only if (Yni (-k) , ... 'Yni (k)) ~

(Yi (-k), ... 'Yi (k)) as n+oo, for each k2:l. Similarly, since the limits

of Yni € D(-oo,oo) which \'le obtain are continuous, the convergence in

D(-oo,oo) is equivalent to convergence in D(-T,T) for each T>O.

10

3. EXTRENES IN DISCRETE THiJE

Let {Z(A)}~=_oo be a sequence of independent stable (l~a~a) random

variables. It is immediate that t:oo a(A)Z(A) converges in distribution

if and only if

00

(3.1) r la(A)l a < 00 and in addition~ for <l=l, a~o~A=-oo

~breover, if (3.1) is satisfied then, since the Z(A)'S are independent~

t:oo a(A)Z(A) converges with probability one also. The limiting distribution

is stable with index a, scale parameter (~~oo la(A)I<l)l/<l and with

symmetry parameter 8 r~oo {a+<CA)<l-a- (A) <l}ft:oo Ia(A) ,<l s where a+(A) =

max(O,a(A)) and a-(A) = max(O,-a(A)). Further;"if a=l the distribution

is translated by an amount -8 ZIT- l r:oo a(A)logla(A)I.

Given {a(A)}~=_oo satisfying (3.1) a moving average process {X(t) }f=-oo.

is obtained by putting X(t) = r~=_oo a(A-t)Z(A). Let x>O be fixed~ take

a sequence {h(n)}~=l with h(n)too and h(n)/n -+ 0 but otherwise arbitrary,

and define the separated exceedances of xnl/a recursively by putting

t nl = inf{t~h(n); X(t»xnl/a} and t ni = inf{t~tn,i_l+h(n); X(t»xnl/a}~

for h2. The reason for using separated exceedances is that we want to

count several exceedances which are iia fixed distance apart" as one event

only. At the end of the section, also ordinary exceedances will be considered.

The time-nonnalized point process N of separated exceedances is then- n <

defined by Nn(B) = #{tni/n<€ lH for 3QY03?~ sets BcR+. Further, for a

given sequence hni}~=l the mark Yni at exceedance no i is defined as

11

and we then have a marked point process 1'1n = (Nn'Ynl~Yn2"")'

In order to find the limiting distribution of 1'1n it is convenient

to consider completely asymmetric processes first. Let ca = n- l r(a)sin(arr/2)

and put A = ma4 _a+ (:>") . Further let Yi have the distribution of the-00<:>,,<00

vector (•.. ,a(l)Zl'a(O)z,a(-l)Z, ... ) in Roo l' where Z is a random

variable with distribution function F(z) =1 - xaA-az-a l' Z ~ xA-l .

Then the limiting distribution is that of

(3.2) (N,Yl'Y2l>O") "Ihere the components are independent,

N is a Poisson process with intensity ~ = 2c~ax-a

and where the Yi's have the distribution given above.

LE-friA 3.1. Suppose that {Z(:>..) }~oo are independent and stable (l,a,l),

that- {a(:>..)}oo satisfies (3.1) with A = max a(:>..) > 0, and that-co· .~ ~"

-00<:>"<00X(t) = !:oo a(:>"-t)Z(:>..). Then there are time points {Tni; n~l, i~l} with

{tni-Tni}~=l tight for each i~l (Le. lim lim sup P(ltni-Tnil>k)=O, i~l )k-+oo n-+oo

such that if Tln is the marked point process of separated exceedances of

mIla defined abovel' then Tln ~ 1'1 with the distribution of 1'1 given

by (3.2).

REIvJARICo It would seem more natural to center the marks at the tni's

instead of at some Tni'S ~ihich are not explicitly defined in terms of

X(t), but tmforttmately the limiting distribution then becomes much more

complicated. However, using the entire observed structure of the sample

path near extremes it is possible to find the centering. For instance,

if the maximum of {a(:>..)} is l.mique then l'Ve may take Tni as the time

12

point when {X(t); t~[tnPtni+h(n)]} first assumes its maximum. (The

validity of this claim is verified at the end of the proof of the lemma.)

PROOF. The essential facts we will use are that X is a moving average

and the following simple estimates of the tails of Faa ' the stable

(l,a,S) distribution (see Bergstrom (1953));

(3.3) l-Fcd(z)-a- 2c z as x~a

.Fa1 (z) = o(lzl-a) as z+-oo

(where f-g means f =g(l+o(l)) ) and

(3.4) F (z) + (l-F o(z)) S k,..z-a, Z>Oaa af.) .....

for some constant ka •

Define OSTn1<Tn2<... as the times when Z(A) > xA-ln1/ a ~. put

N~(B) = #i'rn/n E: B} for Borel sets BcR+ , and let Y~i(t) = Z(Tni)/nl / a

for t=O and Yni(t) = 0 for t~O. Further, let ~ have the distribution

obtained from (3.2) by putting a(O)=A, a(A)=O, A~O in the definition

of Yi .

(3.5)

The first step is to prove that for ?;;n = (N~,Y;U,Y~2 ' ... ) we have

d~ -+ ~ as ~.

Obviously S1 has independent components, so according to Lemma 2.1

it is sufficient to prove that each of the components converges. From

(3.3) we have

which by Theorem 3.2 of Leadbetter (1976) proves that N~ ~ No Furthennore,

13

P(Y~i (O)sz) = 1 - P(Y~i (O»z)

= 1 - p(Z(O) > nl/azlz(O) > nl/axA-l )

ex -a"" 1 - nx L = 1 _ xCJ.A-cxz-cx

-anz

and it follows that Y~i ~ Yi and thus that (3.5) holds.

TIle next step is to prove that {tni-Tni}:=l is tight for i~l~ Le.

Now~ putting "\i = {jtni-Tnil>kl, we have P('\.i) s P(~,i-l~i) + PC'\.,i-1L

(defining Aiio = n L and by recursion (3.6) follows if we prove

(3.7) lim li.m sup P(~ i-lAni) =0, i=I,2, •••.~ n-+oo ~

Let N be a positive integer, put Bn = {Tni>nN} and put Cn =

{z(t)€nl / CtA-1(x-2e:,x] for some t€{O,nN)}. Taking x/3 > e: > 0 we have,

for n such that hen) > 2k, that

A* . l{t .<T .-k} c {X(t .»nl/ax, Z(t)snl / CtxA-1 for It-tnl.·!sk, t .<.nN-k} uB·11, l.- nl. nl. nl. nl. n

c {X(tni»nl/Ctx, Z(t)snl / cxA-1Cx-2e:) for It-tnilsk,

tni<~l-k} U Bn U Cn •

Let Dn be the event that z(t) < -nl/cxe:(2k+l)-I(InaiiG.-a*O.))--lfor someA

t € (O,nN), and let % be the event that there are time points t', t"

(O<t' ,t"<nN) with It'-tlll s 2k+l and z(t'),z(t") > nl/ae:(2k+i)~IA-l •

t .<nN-k}n1

14

Further introduce Xk(t) = L~::k+t a(A-t)z(A) and write Fn for the event

that sup{IX(t)-Xk(t)l; O<t~nN} exceeds nl / a e. Then

{X(tni»nl/ax, Z(t)snl /aA- l (x-2e) for It-tnil~k,

c {Xk(tni»nl/a(x-e), Z(t)Sn1/ aA-1(x-2e)

where the last inclusion follows from the fact that if D~ occurs, iftk 1/0.. 1/0. -1Xk(tni) = LA=-k a(A)z(A+tni) > n (x-e), and 1f z(A+tni) ~ n ~ (x-2e)

for IAI s k, then for at least two values of A with IAI ~ k the

summands a+(A) Z (A+tniL which are not larger than Az (A+tni), have to

exceed n1/ ae(2k+l)-1. rt follows .that

(3.8)

We proceed to estimate the probabilities of the events in the right-

hand side of (3.8). From (3.5)

i-I jP(B) = P(N'(O,N)Si-l) + r~e-~N

n n j=O J.

as n+oo and, using Boole's inequality and (3.3),

Peen) S nNP (z (0) €nl/aA-1 (x- 2e ,x])

-a ( -Ct -ex)+ No 2o ca A (x-2e:)-x

as n+oo, and similarly

Again by Boole' s inequality and by independence and (303) we have

15

peEn) s; ([nIlJ/ (2k+1) ]+1) P ({jt v ; tiS ~ Os;t V~t"s;,~k+2, i (t V) >.nl / ae:.(2k+1) -1A-1 ~

Z(tll»111/()e:(2k+l) -lA-I})

s; ([nN/ (2k+1) ]+1) (2k+1) (4k+3){P (Z (0) >n1/ ae:{2k+1)-lA-1) }2

+0

as fl-l'CO. Finally, X(t) -Xk(t) is stable with index a and scale parameter

q:!XI>k1a(X,)la}1/a so (3.4) gives

P(Fn) s; ru~(IX(O)-Xk(O)I>nl/ae:)

s; nNk e:-a J. la(A) lan-la Ix'T>k

=k He:-a I la(x')l a •a IAI>k

Hence, by (3.BL

-a+ k Ne:a

so

and since E>O and N are arbitrary (subject to X/3>E>0) we get

Simi1ar1y~we can show

16

the main change of the proof being that in the definition of En we have

to consider tV ~tii satisfying ltV-ell s hen) + 2k (instead of ItV-t11Isk)

and thus have to use h(n)/n+O ·to prove PeEn) -+ O. Nmv (3.7) and thus

(3.6) follows.

To prove the remainder of the theorem, introduce yk. =nl

(•. oO~a(k)z(Tni)~•.•a(-k)z(Tni)'O~••. )~ put N~ = N~ , and let

~ = (N~,y~ ,Y~2' ••• ) • Since the function that maps ~ into n~ is

t · (3 5) . l' h 1< d k 1 k ha h d' 'b .con muous ~ • ~111p les t at nn -.0+ n , W lere n . s t e lstrl utlon

that is obtained from (3.2) by putting a(A)=O for IAI>k. Furthennor~'

it is immediate from Lemma 2.1 that nk~)- n. Thus~ by Lerr.ma 2.2, nn ~ n

follows if we prove that (2.1) holds. The atoms of ~ are Tnl/n,Tn2/n~.••

and the atoms of Nn are tn1/n,tnZ/n, ••• and thus, since it follows

from (3.6) that P(!'t"n/n-tn/nl>e:) -+ 0, as ~ Ve:>O, the hypothesis of

Lemma 2.4 is satisfied so (2.1) holds for i=O. Next, by definition

OJ(Yni'Y~i) s IYni(j)-~i(j)1 = IX(j+'t"ni)-a(-j)Z(Tni)ln-l/

cts

IX(j+'t"ni)-Xk(j+'t"ni)ln-1/ct+!Xk(j+Tni)-a(-j)Z(Tni)ln-1/ct if jsk. Hence

{O.(Y .,yk.»2e:} c {IXk(j+T .)-a(-j)Z('t" .)I>nl/cte:,J nl In nl nl

and as in the proof of (3.7) we obtain

17

i.e. that (2. 2) holds for i~L Since this implies that also (2.1) holds

for i~l~ it completes the proof that T1n ~ n.

Finally, let T~ be the first time when {X(t); t~[tni,tni+h(n)]}

assumes its maximum and suppose that for AD satisfying a(AO) = A we

have min (a(Ao)-a(A)) = 2e: > O. To verify the claim of the remark weA;tAo

show that it is then possible to replace {Tni} by {T~i-AO} in the

statement of the lemma. To do this s it is sufficient to prove

(3.9) P(T' . -Ao;t't .) + 0, as n-+oo.n~ n~

Let T~i be defined from Xk(t) = r~::k+t a(A-t)z(A) in the same way

as 'tnv; is defined from X(t). Now {T'.-AO;t't .} c {'tv.;t'tk., 'tk.-AO='t .} u.... n~ n~ n~ n~ n~ n~

h~i-Ao;t'tni} and for k ~ IAol \ve have h~i;t't~P T~i-AO=Tni} c Dn U E~ U

Fn u {'tni>nN-k-AO-h(n)} where E~ is defined in the same \-lay as ~

except that It'-t"l S 2k+l is replaced by It'-tnl S h(n)+2k. Furthennore

{T~i-Ao;tTni} c E~ u {'tni>nN-k-AO-h(n)} and tllUs (3.9) follows as in the

proof of (3.8). o

The general result follows rau'1er easily from Lemma 3.1. Recall the

+ - a -anotation A = max a ·(A) , put a = max a (A) and set II v = c A (1+S)x ,A A a

lltl = caaa(l-S)X-a . Further let Z' and ZH be independent with distribution

functions F1(z) = l_xaA-a(l+S)-lz-a, Z ~ x-1A(1+S)1/a and

a -a -1 -a -1F2(z) = I-x a (l-p) Z , z ~ x a(l-S) respectively and let

Yi = (...a(l)Z', a(O)Z', a(-l)Z' ,H') with probability ll'/(ll"+lliV) and

Yi = (•.• -a(l)ZII, -a(O)ZI1, -a(-l)Z", ...) otherwise. Then the limiting

distribution is that of the marked point process

(3.10) (N~Yl'Y2~... ) where the components are independent~

N is a Poisson process with intensity lJ = lJv+lJ" and

where the Y.~s have the distribution given above.1.

18

THEOREM 3.2. Let {a(A)}:oo satisfy (3.1) and let {Z(A)}:oo be a."l independent,00

stable (l,ct~S) sequence. Suppose that the moving average sequence {X(t)}_oo

is given by X(t) = roo a(A-t)Z(A). Then there exist h ni} with

{t . -"C .}oo 1 tight for each i2l, such that n ~ n, where n is thenl nl n= 'n 'n

marked point process of separated exceedances of nl/ctx by {X(t); t20}

and where the distribution of n is given by (3.10).

PROOF. It is immediate from (1.1) that if X and Yare independent and

(1+2!3~ l!aX ._ ..(12- 13) l!cty is stable

stable (l,a,l) with a~l, then J

(l,a,B). If a =1 a constant has to be added to

1:hlS representation, but since this introduces only trivial complications

';fe asstmle ct;.el for the remainder of the proof. Let {Zv(A)}:oo and {ZiY(A)}:oo

be independent stable (1,0:,1) sequences and put xv (t) = la(A-t) (li13j l/ctzv (A)

and X"(t) = Ia(A-t) (li 13)l/O:t'(A). Then the stochastic process

{XV (t)-X'i(tJ};=_oo has the same distribution as {X(t)}~=_oo and since we

are interested only in distributional properties, we may thus consider

XV (t) -X"(t) instead of X(t).

Let OS"C~<"C~2<••• be the times when {Zi(t); t20} exceeds

nl / ctA-1(li13'\ -l/o.x a:l(!. let O<"C;; <"Cll < . be t"'e tiry~·'~ 1':' 0.'1' {Z"("'")' t20}') - nl nZ ..., -" .' '"'~ ._l...·_A ... J

ed l/a -l(l-S,-l/o:exce s n a TJ x. Using h~i} and h~}, define marked point

processes of separated exceedances of the level nl/ctx, ~ = (N~~Y~l'Y~2' ... )

from {XV (t)} and ~ = (lf~,Y~l'Y~2'''') from {Xii (t)}. Further let

n' and nll be independent and with the distributions obtained from (3.2)

by replacing a(A) with a(A) (Ii (3) 1/0. and with a(A) ei13) 1/0. respectively.

19

From Lemma 3.1 we have n~ ~ n' and ~ ~ nil ~ and since n~ and r£dare independent, (n~~~) ~ (n' ~nl1) (using the product metric on SXS).

Let N~ = N~+N~ , let O~Lnl~Lni::;... be the atoms of N~, and if Tni=T~k

for some k put Y~i =Y~ , otherwise put Y~i = YVril< for the k that

satisfies Lni = L~(. Then the function that maps (n~,n~) into n~ is

continuous except on the set where N~ and N~ have common atoms. Since

this set has (n~ ~n") probability zero it follows that {~n, where the

distribution of n is given by (3.10). Finally, using the independence of

0<i;(t)} and {X"h(t)} and similar (but easier) arguments as in the proof of

Lemma 3.1, it follows that P(d(T1n,~»E) + 0 as n~, for all E>O, and

the theorem is proven. o

Theorem 3.2 gives a rather complete description of the asymptotic

behaviour of extremes of linear stable processes, but it is somewhat

complicated, and we will spend the rest of this section on some (simpler)

corollaries to it.

The point process I'~ gives the separated exceedances of :>.enl/a , but

+also ordinary exceedances are interesting. For Borel sets BcR put

En(R) = #{t/n€B; X(t»xnl/a}, let v+(z) = #{A; za(A»x} and

v- (z) = {A; -zaO»x}, and let the point process E have the following

distribution: atoms occur according to a Poisson process with intensity

1..I = l..I'+l..I"~ the multiplicities of different atoms are independent and with

the distribution of v, where v = v+(Z') with probability l..I'/(l..I'+l..IH)

and v = v- (ZI1) othenvise. (1..I!, 1..I" and the distributions of Z' and

z" are given on p. 27.)

20

COROLLARY 3.3. Suvpose that {X(t)}~=_oo satisfies the assumptions of

Theorem 3. 2. Then En.SL. E~ where En is the point process of exceedances

of xnl / a and where the distribution of E is given above.

kPROOF. We only sketch the proof. Let En have the same atoms as 1\ ~

but with the multiplicity of the atom at t ni equal to #{t€[tnptni+k];

X(t»xnl / a}. According to the theorem nn ~ n and 1"ith probability one

n is of the form (v?Yl~Y2~ ... ) where v is a locally finite measure00

and where yi € R is of the fom Yi (t) = zia(-tL t=O, ±l ~ ... • However,

for vectors of this fonn~ the function that maps nn into ~ is continuous

except if zia(-t) = x for same i~l. As the set of such vectors has n

probability zero, it follows that Ek converges in distribution to somenk kdpoint process~ say E . It is easily seen that E -+ E as k-l<lO, a.nd the

proof can be finished~ using similar methods as in the proof of Lemma 3.1,Ie

by approxiraating ~ by En. 0

COROLLARY 3.4. Suppose that {X(t)}:oo satisfies the hypothesis of Theorem

3.2 and that the Borel set B c R+ has boundary "nth Lebesgue measure zero

as 11-1'00~ where the vi ~ s are independent and with the same distributions

as v. Similarly, if Bp ... BR, c It are disjoint and have boUJ.idaries with

Lebesgue measure zero, then P(En(B1)=kp ...En(BR)=k~) tends to the product

of the corresponding probabilities.

21

Also the joint limiting distribution of heights and locations of the

exceedances can be obtahled from Theorem 3.2, but since the limiting

distributions are complicated we only give the siinpiest result.

COROLLARY 3.5. Suppose that {XCt)}:oo satisfies the hypothesis of Theorem

3.2 and let I'.'k = max X(t). Thenl::s;t::s;n

p~1n/nl/a::s;x) ~ exp{-Ca(AaCl+s)+aaCl-S))x-Ct}

as ~.

PROOF. Obviously PO''h/nl/a::s;x) = P(En([O~l])=O), and the latter probability

converges to exp{_cCt(ACtCl+S)+aCtCl_S))x-Ct} by Corollary 3.4. 0

In Theorem 3.2 it is assumed that the independent variables have a

stable distribution. tIm'lever , as was noted in the proof of Lennna 3.1, the

essential property of the stable distribution is that the tails decrease

polynomially. Thus results similar to those above hold for moving averages

as S0011 as the tails of the distribution of the independent variables

decrease as negative powers of x, e.g. if the independent variables belong

to t..lJ.e domain of nonaal attraction of a stable law with exponent et<2.

4. P10VL·jG AVERJ\GES OF STP,fJlE PROCESSES HI COHTINUOUS TIHE

Consider a stocl1astic process {Z(A); A€R} that has stationary

independent increments with ZCO)=O and ZCl) stable (l,a,S). In the

sequel 'we will assume that a:tl. 'fL.e usual way of obtaining an integral

!a(A)dZ(A) is to first define it for step functions of the form

aCA) = L~=l aiICb.,c.](A) with -00 < bl < cl ::s; bZ < ••• < ck < 00 by putting1 1

22

la(A)dZ(A) = t~ l' a. (zec.) -Z(b.));", Then, as'ht easily seen. from (1.1) fa(l)aZ(AJ: £1= 1 1 1

is stable with index a~ scale parameter {fla(l)ladl}l/a ~ and symmetry

parameter {f(a+(l)a-a-(A)a)dl}/fla(l)ladl. Next~ if a(l) is (Lebes~~e)

measura:.bleT:tand satisfies

(4.1) (O<a<l or 1<a<2),

we can find a sequence {an(l)}:=l of step functions with fla(l)-an(l)ladl-+O

as n-+oo. Then~ for In = fan(l)dZ(lL the scale parameter of In-1m is

Ulan(l)-am(A)ladl}l/a which finds to zero as min(m,n)-+oo. Hence {In}:=l

is a Cauchy sequence in the sense of convergence in probability and there is

a random variable I with In r.. I. The integral is then defined (uniquely

a.s.) by !a(A)dZ(A) = I.

The object of study is moving averages~ Le. processes of the form

X(t) = fa(A-t)dZ(A) with a(l) satisfying (4.1). We always assume that a

separable version has been chosen. Our approach is to approximate a(l)

by step functions ak(l) = raiI(i2-k<l~(i+l)2-k) arid thus to approximate

X(t) by Xk(t) =Lai {Z((i+l)2-k+t)-Z(i2-k+t)}. The necessary estimates are

given by the following two lennnas.

LB,~~ 4.1. Suppose X(t) = rai{Z((i+l)2-k+t)-Z(i2-k+t)}~ where rlaila < 00

and where {Z(l); AE:R} has stationary independent increments with Z(O)=O

and Z(l) stable (l~a,l) ~ and put X = sup IX((.R-+t)Z-k) -X(.R-2-k) I. IfR. O~t~l

O<a<l then~ for some constant ka.'

23

•(4.2) P( max X~>x) s KtxIA~Nx-a

Os~s2kN-l

where A. = max la k I. If 1<a<2 then1 OsjS2k i2 +j

(4.3) P( max X~>x), s I~Ilail~{x-a+(2-k+allaila) (2-a)/ax-2}

os~s2~-1for x > Ca.(z-kllaila)l/a ,where Ca. is a constant.

PROOF. We have

(4.4) X((~+t)Z-k)-X(~Z-k) =Iai{Z((i+l+~+t)2-k) -Z((i+R,+t)2-k)}

- Iai{Z((i+l+Jl,) 2-k) - Z( (i+R,) 2-k) }

= lCai_l-aiH Z((i+Jl,+t)Z-k) - Z((i+Jl,)2-k)}.

Suppose O<a<l. Then {Z (t.) } has nondecreasing sample paths and hence for

•

It follows that

•

•24

00

The sequence {z(i+1)-z(i)}._ is independent and stable (l~a~l) and1--00

thus, by (3.4),

P( max XR,>x) s Z~a ! (Ai +Ai +1)ax-aOSR,SZk_1

Hence

P( max Xj/,>x)OSj/,sN2k-1

S NP( max Xj/,>x)OSR,SZk_1

S Z 0 Z~ ! Ac:r«-aa 1

•

•

and (4.2) holds with Ka = Z 0 zaka .

Next suppose that 1<a<2. Put Yet) = X(tZ-k)-X(O) and set b. = a.-a. l'1 1 1-

It follows from (4.4) that {yet); OStsl} has stationary independent

increments with Y(O)=O and Vel) stable with index a~ scale parameter

Y = (Z-I<rlbila)l/a and synnnetry parameter ~ = r((b.~)~:(bj)a)/IIbila. In+ -analogy with the proof of Theorem 3.2, \ve represent Yet) as Y (t)-Y (t) ~

+ - +. -where Y (0) = Y (0)=0 and the processes {Y (t)} and {Y (t)} are

independent and have stationary independent increments with y+ (1) stable

(Y((l+a)/z)l/a,a,l) and Y-(l) stable (Y((l-a)/z)l/a,a,l). Further,

y+ (t) = Y~(t)+Y~ (t) where Y~ (t) is the sum of jumps of y+(t) of size

~ 1 in [O,t] and Y-(t) =Yo(t)+Yi(t) where Viet) is the sum of

jumps of Y-(t) of size ~ 1 in [O,t]. Let ~(u~x) = (eiux_1_iux)lxl- l -a •

Using the Levy representation of the characteristic function of Y+(t) as

~t(u) =exp{taCaya(l+a)f~~(u,x)dx} it follows (see Levy (1954)~ Breiman

(1968)) that {Y~(t)} and' {Y~(t)} are independent and that Y~(t) has

the characteristic runction exp{taCaya(l+~)(f~~(u,x)dx-iUf~x-adx)}.

Similarly {YO(t)} and {Viet)} are independent and Yo(t) has the

25

• characteristic ftmction expftaca.Y(l-S)(f~ w(u,x)ebe+iu/i: xClebe)}'

Hence

Xo = sup IY(t)/ s sup /Y~Ct)l+ sup IY~(t)l+ sup IYoCt)1+ sup IYiCt)IOstsl Ostsl Ostsl Ostsl Ostsl

= sup IY~(t)I+Y~(I)+ sup IYo(t)I+Yi(l)Ostsl Ostsl

s 2 sup IY~(t)I+Y~(1)+2 sup lyo(t)I+Y~(l).OstSl Ostsl

•

+The process {YO(t); Ostsl} has moments of all orders (Feller (1971),

p. 570) and, if IDa and va are the mean and variance in the distribution1 00 -a

with characteristic ftmction expfacaUOw(u,x)d.x-iuflx ebe)}, then

E(Y~(t)) = tY(I+/3)l/ama. and V(Yb(t)) = ·ty2(1+~)2/ava.' . For ';/')(1 > 16ylmal

KolmogorovVs inequality gives

p(z sup IY~(t)l>x/4) s p(Z sup IY~(t)-tYCl+S)I/am J >X/4-Y(1+S)I/alma IJOstSl Osts1 a

s p( sup IY~Ct)-tY(l+S)l/~ l>x/16)Ostsl a

s y Z22v 162x- Z •a

Similarly, for x> 16Ylmal,

Thus, for x > 16m y,a.

P(2 sup IYo(t)l>x/4) s yZZZv 162x- 2Ostsl a

P(Y;CI»x/4) s ko,ya CI+/3)4a.x-a. and p(Y~(l»X/4) sBy (3.4) we have

kayo,CI-I3)4ax-a .

•P(XO>x) S p(Z sup IY~(t)l>x/4)+P(Y+(I»x/4)+P(2 sup IYo(t)I>x/4)+P(Y-(1»x/4)

Ostsl Ostsl

s K 2-aya.Cx-a.+y2-ax-2)0,

•26

if ~ = 2a rruax(4ol62Va,ka4a)~ so since ya = 2-k r Ibil aand bi = ai-l-ai

we have

P( max X.Q,>X) s ZkNP(Xo>X)OSJ/,sNZk-l ..

Z-a= Ka2-aLlbil~(x-a+(Z-krlbila)~x-2)

2-aS I~!lail~{x-a+(Z-k+aLlaila)~x-2},

which proves (4.3). 0

° -kaId = a(1.Z ). One possibility is to require

In order to apply Lemma 4.1 to X(t) = fa(A-t)dZ(A) some conditions

are needed. Let l\i = sup a(A) , bId = inf a(A) and putA2k€(i i+l] AZk€(i,i+l]

• (4.5) a (A) is unifonnly continuous,O<a<l.

roo a° Blo < 00 and1=-00 1.

The second part of this condition is of course equivalent to 2~=-00~i < 00,

for all k~l. Another possibility is to require

(4.6) a(A) is unifonnly continuous,,

exist 0>0 and K such that

and l<a<Z.

•

Obviously, this condition implies that Llaki 1a < 00 for all k~l. The

latter part of the condition perhaps needs some motivation. Suppose that

a(A) is continuously differentiable, except possibly at the points

•27

{1°2-k+l}ooo d f I '(')1 Th / /1=-00 ' an put ki = sup a A. en aki-ak-l ,[i/2] s

A2ke: (i-l,i]-k tl la -k(a-l)t~-kf ki2 and hence L aki-ak-l ,[i/2] S 2 LIki2. Thus the latter

part of (4.6) holds with 0 =a-I if e.g. rfki2-k converges as k-+ro, and

to require that this holds is rather close to requiring lla'(A) IadA < 00.

LBiY1A 4.2. Suppose that X(t) = !a(A-t)dZ (A), where a(A) is non-negative

and satisfies (4.1) and {zeAl; Ae:R} is as in Lemma 4.1. Furthermore put

Xk(t) = Iaki{Z((i+l)2-k+t) - Z(i2-k+t)} (the sum converges, by (4.5) or

by (4.6)). If a(A) satisfies (4.5) then, for some constant K'-a

Aki = max IB k -b k I· If a(A) satisfies (4.6) thenosts2k k,i2 +j k,iZ +j

p( sup IX(t)-Xk(t)/>x) s K'2-ko/~~(x-a+x-2),Ost~J a

k -alarge enough to make 2 > Caex ,where Cae is a constant.

00

p( sup IX(t)-Xk(t)l>x) s K~ or AkiNx-a •OstsN 1=-00

(4.7)

where

• (4.8)

for k

•

PROOF. Suppose O<a<l and let Xk(t) = I~i{Z((i+l)2-k+t)-Z(iZ-k+t)},

~(t) = Ibki{Z((i+l)Z-k+t)-Z(iZ-k+t)}. Since ZeAl has nondecreasing

sample paths, I1«t) = Xk(t) -~(t) ~ IX(t) -Xk(t) I. Using the same methods

as in the first part of Lermna 4.1 it is easily seen that p( sup l\:(t»x) sOstsN

roo -aK.. 0 _ Ak10Nx and thus (4.7) follows with K' =K •

u 1=-~ a a

Now consider l<a<Z. In this case let Dk(t) = Xk(t)-Xk_l(t) and put

dki = aki-ak-l,[i/2] ,n~\lilZ Dklt) = Idki{Z((i~1)2-k+t)-Z(i2-k~t)}.

Further let Dt = sup 1I\((Jl,+t)2-~ ~I\(tZ-k) I and use (3.4) and LennnaOstsl

4.1 to obtain

28

p( sup Dk(t»x) :::; p( max /Dk(R,Z-k) I>x/2)+p( max D >x/Z)O:::;t:::;N~. R,

O:::;R,:::;Z N-l 0:::;R,s2nN-l

:::; kaNLldki/ax-a+XaNrl~ila{x-a+(2-k+alldki/a)(2-a)/ax-2}

for x>Ca(Z-k21~ila)1/a. Hence by (4.6)

(4.9) p( sup Dk(t»x)O:::;t:::;N

Kt~{(k +K )Z-kox·a+y K(Z-a)/azZ-az-k{Z+2o-a}/a -2}aa ....a x ~

'(

for x >'C (re z-k(l+o))l/a. Let x.·= z-(i-l)o/(2a) ci-2':<5/ (2a))x~ soa 1

tOO tOO -(k+i)o -a -ko -athat Li=l xi = x, Li=l 2 xi:::; constant x 2 x ~ and

t~ Z-·(k+i) (2+2o-a)/a x:2 :::; co tant x Z-lc(2+20-a)/a.-2 '. s.-L1=1 1 ns X... IDce

we have

00

(4.10) p( sup IX(t)-Xk(t) I>x):::; L p( sup Dk+· (t»xi) ~O:::;t::::J.'J . i=l O:::;t~N 1

and if 2k > ci_2-o/(2a)(lC~Kx-a then xi > Ca(K 2(k+i) (i+o))l/cx , for

i~l, and (4.8) .follows from (4.9) anq (lL 10). 0

The conditions used above imply that X(t) has continuous sample

paths, and. although we do not need this result for the sequel, it is

interesting in its own right.

TIIEOREM 4.3. Let' {Z(A): AE:R} have stationary independent increments,

which are stable with index a. Further suppose that a(A) satisfies

(4.1) and that both a+(A) and a-(A) satisfy either (4.5) or (4.6).

Then the moving average X(t) = fa(A-t)dZ (A) has continuous sample paths.

PROOF. Obviously it is no restriction to assume that Z(O)=O and Z(l)r + _

is stable (l~cx,~), and since fa(A-t)4Z(A) = fa (A-t)dZ(A)+!a (A-t)dZ(A),

29

it is enough to show that each of the tenns is contirruous~ Le. we may

also assume a(A)~O.

Let {Z I (A); A€R} and {Zli (A): A€R} be independent and have stationary

independent increments with Z1(0) = Z11(O) = 0 and Z1 (1), Z"(l) stable

(l~a~l). Then fa(A-t)dZ(A) has the same distribution as

eiS)l/afa (A-t)dZ'(A) - ei(3)l/afa(A-t)dZ~~J))_~ and thus we may further

assume 13=1.------~

The proof proceeds by approximating X(t) by Vk-(tJ-=.-fak(A-t)dZ(~,_

where ak(t) is defined by the requirement that ak(t)=O, Itl~k', for

-k -kk' = kICk) to be specified later, that ak(~2 ) = a(~2 ),

~ = 0,±1, •.• ±k'zk_1, and that ak(t) is linear between these points.

Using the definition of the integral as a limit of sums and Abelian summation,

it is seen that ("partia1 integrationli)

Iak(A-t)dZ(A) =Jak(A)dZ(A+t)

= -Iai(A)Z(A+t)dAk -k

k'2 -1 '{'n(' -k) . -k} I(i+l)2= - I .. !!.l(1+!)2 _1c-a (12 ) Z(A+t)dA, (here

i=-k'Zk 2 iZ-k a(k') = a(-k') = 0)

where the integrals are defined as limits in probability of sums. However,

Z(A) € D(-oo,oo) (see e.g. Breliruan (1968), p. 306), and is thus locally("+l)Z-k

Rieman integrable and hence f 1_k Z(A+t)dA is a.s. a Rieman integrali2

and is thus a.S. continuous in t, and it follows that also

Vk(t) = fak(A-t)dZ(A) is continuous in t a.s.

Hence, if we prove e.g.

30

(4.11)

then the desired result fo11ows~ since there is then a sequence {~} of

integers with p( sup IX(t)-Vk (t)1 ~ 0 as n+oo) = l~ i.e. X(t) isOstsl -n

a.s. a unifonn (in [0,1]) limit of continuous ftmctions and is thus

We have

continuous in [O~l] and hence~ by stationarity, in all of R. Now, let

Xk(t) be as in Lemma 4.2 and put Xl~(t) = L a. .{Z(Ci+l)Z-k+t)-Z(iZ-k+t)}.lizkl<k ' 1(1

C4.1Z) p( sup IXCt)-Vk(t)l>x) s p( sup IX(t)-Xk(t)l>x/3)Ostsl ~ Ostsl

+ p( sup IXk(t)-Xk(t)l>x/3)+P( sup IXk(t)-Vk (t)l>x/3).OstSl OstSl

TI1US~ if O<a<l, Lemmas 4.1 and 4.Z give that

Choosing e.g. k ' (k) ::: k it follows from (4.5) and the dominated convergence

theorem that the righthand side of (4.13) tends to zero as k~~ and thus

(4.11) holds for O<a<l.

It is no loss of generality to assume oSa in (4.6), and then it can

be seen that, regardless of the value of k ' $ ak(A) satisfies (4.6) with

K not depending on k and with the same 0 as a(A), and thus if l<a<Z

o

it follO\vs from Lerrnna 4.Z that the first and the third terms of (4.1Z) are

bounded by K' Z-ko/a3ax·a for large k. Furthermore, by (3.4) the seconda

term is bounded by Ka. L I~i Ia3ax -a ~ and since LIaki Ia < co byIiZk, >k '

(4.6) , k 1 can be chosen large enough to make L 1 I~ila ~ 0 asliZ <I>k '

k~, and it follows that (4.11) holds also for l<a<Z.

31

5. EXTREMES IN CONTINUOUS TIME

Let {X(t); t€R} be a moving aveTa.ge~ and in analogy with Section 3

define recursively tTl = inf{t~h(T); X(t»xTl/ a

}$ t Ti =inf{t~tr$i_l+h(T);

X(t»xTl/ a} for i~Z. For a given sequence {-rTi}~=l put ¥Ti (t) =

X(t+'rTi)/Tl / a ~ let Nr(B) = #{tTik€ B} for Borel sets BcR+ and consider

the marked point process Tlr = (NT'YTI 'YT2 $.•• ) of separated exceedances

of xTl / a • Furthennore$ put A = sup a+(A)~ let II and Z be as definedA€R

on p. 17$ and let Yi have the distribution of the random variable

{Za(-t); t€R} in D(-oo$oo). If ZeAl is completely asymmetric the limiting

distribution will be that of

(5.1) (N$Yl$YZ$.•• ) where the components are independent$

N is a Poisson process with intensity ll, and where

the Yi$s have the distribution given above.

LEM'JIA 5.1. Suppose that {Z(A); A€R} has stationary independent increments $

with Z(O)=O and Z(l) stable (l$a,l). Further suppose that a(A)

satisfies (4.1), that both a+(A) and a-CAl satisfy either (4.5) or

(4.6L and that A>O. Then there exists {-rTi} such that {tTi-'rTi ; T~ll

is tight for each i~O and such that the marked point process nT of

sep~ratcd exceed'mces of xTl/a by X(t) = !a(lt-t)dZ(A) converges in

distribution to n. where the distribution of n is given by (5.1).

PROOF. Without loss of generality we assume that a(O)=A. First suppose

that O<a<l. Recall the definitions of Xk(t) and ~(i from Lemma 4.2

and put Xk(t) = Xk(Z-k[Zkt ]) so that Xk(iZ-k) = ~(iZ-k) and Xk(t)

32

is constant for t € [i2-k,(i+l)Z-k]. We have

by Lenma (4. Z) applied to both a+(A) and a- (A) and by Lenma (4.1).

From (4.5) it follows that ~~ ,e. -+ 0 when k~ and hence[,1=-00 -]a

(5.3) I~~ lim sup p( sup !X(t)-Xk(t)I>T1/ax) = O.k~ T~ O;s;t;s;T

k.· k k k . fNm1, let ~T = (l,!~YTI ,YTZ ' ••• ) be the marked pomt process 0 h(T) -

separ-"ted UPCl"OSSingS of xT1!a by the discrete process . {Xk(iZ-k) }~=_OO ,

k - -k k . k .where YTi(t) = Xk(tZ +TTi)' t=O,±I, .•. ,w1th TTi the t1me of the

. 1/a -k ( -k) 001'th exceedance of xT /A by the sequence {Z(iZ )-2 (i-I)Z }i=O.

According to Lemma 3.1 ~~ converges in distribution, to ~k say, as

T~. Furthermore, if n~ is the marked point process of upcrossings of

xTl / a by the continuous-time process {Xk(t); t€R}, with the marks centered

at the T~. 's, then the function f: NxRooxRoox ••. -+ NxD(-oo,oo)xD(~oo,oo)x~ ••11

that maps z;;i into n~ is continuous and hence n~ ~ f(Z;;k) = nk, say.

It is easily seen that the distribution of nk is obtained from (5.1) byk

replacing a(A) with ak(A) =I~2-Ik akiI(iZ-k<A;S;(i+l)Z-k). Thusk k __k k . 1=-k2 .

n = (N ,ri,Y2' ••• ) has mdependent components, and 1£ n = (N,Y1 'YZ' ••• )

has the distribution given by (5.1) then Nk has the same distribution as

33

kdN and Y. -+ y. ,since su~ la(A)-ak(A)I + 0 by (4.5). By Lemma 2.11 1 A€R

it follows that nk~ n as n~.

The appropriate centering for the Pth mark, l'Ti' of nT is the

time of the i' th jump larger than TI/ o.x/A in the process {z (t); t2:0} 0

To show this we first prove

(5.4) lim P(ll'Ti-1'~il>2-k) = O.T~

To do this, let E € (0, x/(2A)) and write Z(t) = ZI(t)+Z2(t)+Z3(t)+Z4(t),

where Zl(t) is the sum of jumps by Z(t) in [o,t] of size larger

than Tl!o.(x/A+E), where Z2(t) is the sum of jumps of size belonging

to Tl!o.(X/A-E, x/A+E], and where Z3(t) is the sum of jumps of size

belonging to TI/o.(E/n, x/A-E], (n>O). Further, for t=I,2,3, let

Et be the point process which has its atoms at the times of jumps of

zt(t). We recall that, putting Zj = Z(jZ-k)-Z((j-I)Z-k), l'~i is equal

to Z-k times the location of the i'th exceedance of TI/o.x/A by the

{ }oo Q, t. -k t (. -k) t4 tsequence Zj j=l. Put Zj = Z (J2 )-Z CJ-I)Z so that Zj = Lt=l Zj ,

let N be a positive number and denote the event that E1([O,TN]) < i by2Ay , the event that E ([O,TN]) > 0 by Br ' the event that

EZ(z-kCj-l,j]) > 1, for some j € [1, 2kNT], by Cr, and the event that

E3(Z-kCj - l ,j ) > 1, for some j € [1, 2~], by DT . If 11'Ti-1'~il > 2~k

and ~ n B~ occurs, then at least one of the following three events

must happen: either Zj > Tl!o.x/A and EI (2 -k(j -1, j]) = 0, for some

j € [1, Z~], or Zj $. TI/o.x/A and El(Z-kCj-l,j]) > 0, for some

j € [1, zl)IT], or else Cr occurs. Moreover, if B¥ n C¥ n ~ occurs,

then the first two events both imply that Rr = {IZ11>T1/o.E, some

j € [1, 21<mn happens. Thus we have proved

and..

34

Let l-Il = caCl.-1 (x/A+f:.ra , 112 = CCl.CJ.:-~{ (x/A-.e:J -Cl.. (x/A+e:f~! "'011d

. -l{( I ,-CI. (fA )-a} 'Th"'"" +0"" 0=1· 2 3 BQ, ·is B. poisson 'J'113"\' ::; 'C a e: .n) '.- Yo' !'.-: e:' :'. . ,,,.• j> -- ,<' N ,', ',~. .., • , • '. '.' ,

CI. .

pJ:'OC€SS i'lith ii'1t.ensity 11Q,/T (see e.g. Breimcm (1968)) and thus

i-I -N11 (N11)jP(A_) =F(E l ([O,1N])<i) = reI j7

-T j=O'

2 -N112PCBr) =P(E ([O,TN])~l) =l-e •

Furthennore"

-+ 0,

as T~" and similarly P(DT) -l: a.Further, by differentiating the Levy

representation of the characteristic function of zi (c.f. the proof

of Lemma 4.1) it is seen that B((Zi)2) ~ K2-k(e:/n)2-ay2/a-l , for some

constant K, and then Chebychev's inequality gives

?,~) ~ zkNrp(IZil>Tl/Cl.f:.)"

~ zkNrE(IZiIZ)T-Z/Cl.e:- Z

'(/~T -a 2-a~ !'U~e: n ,

as T~. Hence

i-I -NJ,l (NJ,ll)j -1'4llim sup P(ITTi-T~il>2-k) ~ I e 1 " + l-e 2 + KNe:-an2-a ,T~ j=O J.

and inserting the values of 111 and J,lZ and letting first n-1-OO, then

e:-+O, and tJ.i.im.' '}'l'-+co, this proves (5 .4) .

Now we are in a position to show that' {tri-TTi: T~l} is tight, or

equivalently to prove

•

•

•

35

(5.5) lim lim sup P(ltyi-LTil>y) = O•y-)OO T-)OO

Let {~i} be the h(T)-separated upcrossings of Tl/a(x-AE) by Xk(t)

d {~ ~ -an let tTi} be the h(T)-separated upcrossings of T (x+k) by Xk(t).

Further, let !~i be the location of the i'th exceedance of Tl!a(X/A-E)

-k l/aand TTi the location of the i'th exceedance of T (xlA+E) by the

d' . { -Ie ( -k)}ooJ.screte process Z(j2 )-Z (j-l)2 j=l' write FT for the event that

sup IX(t)-Xk(t)I > Tl/aAE and, changing the notation slightly, letO~t~rif_k ~ k NkAr = {tTi<NT-y} • On the event ~ n F~ we have .tTi ~ 'tri ~ tTi and

thus { It Ti-LTi I>y} c' (t~i-LTi>y} u (rTi-~i>Y} uAy uFT· Let Gr be

the event that Zj € Tl/aCx/A-E, x/A+E:] for some j E: [1, 2kNT]. If

I""k ,.., I k k' -k d hft..!;. n (1.cl' n {LT' -Lr,' ~} occu1~s;\ then ;[1;' '= 1'T; = LTi an t us£ ]. IJ. k _ ~._

,..,k k -Ie k k -Ie} Ie-k{ltTi-TTil>y} c {tTi-TTi>y-2 } u {LTi-tri>y-2 u {I LTi-LTi l>2 } u Ar u FT

,..,,..,k -kk k -k k I -kc {tri-LTi>y-2 } u {Iri-tTi>y-2 } u {ILTi-LTi >2 } u Ar u FT

uGr·i-I -N~l ~N~l)j k-k

Here P(/L) -+ L--O· e • I , P(ITr "-LT·I>2 ) -+ 0 as T-l-OO, and""1' J- J. ]. l.

P(Gy) ~ 2~ffi>(zlET1/CI.(x/A-E,x/A+e:]) ,.., 2 NCa{(X/A-Efa - (X/A+Efa

}. Hence

"'k -k -k k k -k~ lim lim suP{P(tTi-TTi>y-2 ) + P(!Ti-.tTi>y-2 )}

"'roo T-+00 "

i-I -N~ (I'Jll)J+ I e 1 .f + lim lim sup P(FT)

j=O J. y-+oo T-+oo

+ 2 Nc {(X/A-E)-a_(x/A+EfCl.},CI.

and the first term is zero by Lemma 401, the third term tends to zero as

k-+oo by (5.3) and the remaining biO tems tend to zero as first E;-+O and

then N-+oo, and thus (5.5) follows .

•

•

36

dTo complete the proof that TlT ~ Tl it is enough to show that the

conditions of Lemmas 2.3 and 2.4 are satisfied. However~ the atoms of NT_Je k k

are tTI/T~ ty2/T~ ••. and the atoms of NT are tTI/T? tT2/T~ •.• ~

and (tri-~i)/TL 0 follows from (5.4) and the facts that {cn-TTi ; T~I}

and {t~i-T~i: T~l} are tight~ so the hypothesis of Lemma 2.4 is satisfied.

Further~ let ~ = TTi-T~i' Then, by (5.4L lim lim sup P(IE~I>x) = 0 fork / k-+oo T-+oo k k

x>O. Since 'l:.T. (t) = Xl,(t+~T· )/rl a we have { sup IYT· (t) -YT· (t+~) I>x} =1 ~ 1 _n<t<n 1 1

H

l1/ N_ -N 1/{ sup IX(t+TT.)-~ (t+TTi)I>T ax} c { sup IX(t)-.~c(t)I>T ax} u Ar and-JI,~t~JI, 1 ( -JI,~t$NT+JI,

from (5.3) it then follows that

lim lim sup p( sup /¥r- (t)-~. (t+Enc) f>x) ~ lim lim sup PCAr)k~ T-+oo -R.;s;t~JI, 1 1 k-+oo T-+oo

i-I -NJ.lI (NJ.lI)j= L e "

j=O J.

and since N is arbitrary it follows that (2.3) holds. Further,

lim lim sup p( sup IYni (t) I>u) = 0 fol10\'ls easily from Lemma 4.2, and thusl}'+<Xl T-+oo -JI,<t<JI,the hypothesis of Lemma 2.3 is satisfied. This concludes the proof of the

lemma for the case O<a<l.

If instead l<a<2 . we have to use the second parts of Lemma 4.1

•

and 4.2 instead of the first ones to prove (5.3), but apart from that, the

lemma follows in precisely the same way as above also in this case. 0

Since the restriction that zeA) is cmapletely asymmetric can be

removed L'I1 precisely the same way as Theorem. 3.2 is obtained from Lemma 3.1,

we omit the details of this proof and only state the result.

Recall the notation A = sup a(A) , put a = sup a-CA) 9 letAeR AeR

ll' = c AaCl+S)x-a , llH = c aa CI-13)x-a9 let Z' and Zii have the distributionsa a

•37

given on p. 27 and let Yi (t) = Z'a(-t) , t€R, lvith probability ll' / (ll' +1lV!)

and Yi (t) = -ZHa(-t), t€R, otheoose. Then the limiting distribution of

ny is that of the marked point process

(5. 6) (N,Y1'Y2' .•• ) where the components are independent,N is a Poisson process with intensity II = 1l1+1l2and the Yi's have the distribution given above.

THEOP~~ 5.2. Let a(A) satisfy (4.1) and let both a+(A) and a-(A)

satisfy either (4.5) or else (4.6). Further let {Z(A); A€R} have

stationary independent increments with 2(0)=0 and 2(1) stable

(l,a,~) and let X(t) = fa(A-t}dZ(A). Then there exist {TTi; i=I,2, ••• ,T~1}

with {tTi-TTi; T~l} tight for each i~l such that TlT ~ n as T-+oo, where

T1r is the marked point process of separated exceedances of TI/ax by

• {X(t); t~O} and linere the distribution of n is given by (5.6).

SiIlrl.1arly as for the discrete time case, various corollaries concerning

the behavior of extremes can be deduced from Theorem 5.2. Here we only give

the very simplest result, concerningr1r = sup X(t).O~t~T

COROLLARY 5.3. Suppose that X(t) satisfies the hypothesis of Theorem

5.2. Then

as T-+oo.

We have not treated the case a=l above. However, using methods rather

similar to those for l<a<2, conditions for the result of Theorem 5.2 to hold

can be obtained also for a=l.

•

38

• Finally we note that it is easy to see that all of the limit theorems

of this paper are mixing in the sense of Renyi (the first result in this

direction is proved in [11]). Hence they can be extended to cases where the

a level is random~ and possibly depending on the process X(t).

\;

I am grateful to Stamatis Cambanis? Ross Leadbetter, and Georg Lindgren

for some stimulating conversations on the subject of this paper.

REFERE~JCES

Bauer, H.: FPobabiUty TheoPy. Berlin, De Gruyter 1972•

Bergstrom, H.: Theorie der stabilen Verteiltmgsftmktionen. AT'chiv deT'Mathematik, 4~ 380-391 (1953).

[4]

[5]

Billingsley, P.: ConveT'gence of FPobabiZity MeasUT'es. New York, Wiley 1968.

Briem.'ln, 1.: ProbabiUty. Reading~ lYBSS. ~ Addison-Wesley 1968.

Feller, W.: An Introduction to FPobabiUty TheoT'y and Its AppUcations,VoZ. II. New York~ Wiley 1971.

[6] Kallenberg, 0.: Random MeasUT'es. Berlin, Akademische Verlag 1975.

[7] Leadbetter, M.R.: Weak convergence of high leYel exceedances by a stationarysequence. z. WahPsaheinZiahkeitsteoT'ie ve~. Gebiete 34, 11-15 (1976).

[1]

• [2]

[3]

[8] Leadbetter, M. R., Lindgren, G., de Mare, J., Rootzen , H.: Lectures onextreme value theory. To appear.

[9] Levy, P.: TheoT'ie de ZVaddition des VaT'iabZes AZeatoiT'es. Paris, Gauthier­Villars, second edition 1954.

[10] Lindvall, T.: Weak convergence of probability measures and random ftmctionsin the function space D[O,oo). J. AppZ. PT'ob. 10, 109-121 (1973).

•[11] Rootzen, H.: Some properties of convergence in distribution of sums and

maxima of dependent random variables. Report 1974:7, Dept. of Math.Statist., Univ. of Lund.