Sample ACF of Multivariate Stochastic Recurrence ... - Semantic Scholar

Sample ACF of Multivariate Stochastic Recurrence Equations With Application to GARCH Richard A. Davis1

Colorado State University Thomas Mikosch

University of Groningen Bojan Basrak2

University of Groningen Draft April 23, 1999

Short Title: Sample autocorrelations

Abstract

We study the weak limit behaviour of the sample autocorrelation function (ACF) of non-linear stationary sequences with regularly varying nite-dimensional distributions. In particular, we consider the sample ACF of solutions to multivariate stochastic recurrence equations (SRE's). The latter includes the important class of the squares of GARCH processes. Point process convergence is exploited to derive weak limits of the sample ACF. The limits are functionals of in nite variance stable random vectors. It turns out that the closer the distribution of the process is to having an in nite second moment the slower the rate of convergence of the sample ACF. In the case of an in nite second moment, the sample ACF converges to a non-degenerate limit law.

This research supported in part by NSF DMS Grant No. DMS-9504596. This research supported by an NWO PhD grant. AMS 1991 Subject Classi cation: Primary: 62M10 Secondary: 62G20 60G55 62P05 60G10 60G70 Key Words and Phrases. Point process, vague convergence, multivariate regular variation, mixing condition, stationary process, heavy tail, sample autocovariance, sample autocorrelation, GARCH, nance, Markov chain.

1

2

1 Introduction The aim of this paper is to study the weak limit behaviour of the sample autocovariances and autocorrelations of non-linear time series models with regularly varying nite-dimensional distributions. This research is primarily motivated by a variety of models which has been used in the econometrics literature for describing highly volatile real-life nancial time series such as the logreturns of foreign exchange rates, share prices and stock indices. The \folklore" in this area says that such nancial time series, over a suitable time horizon, can be considered as a realisation of a strictly stationary process whose marginal distributions have possibly a nite variance but an in nite 4th or 5th moment. Morover, the observed time series exhibit a complicated dependence structure which is seen by clusters of high and low level exceedances. See Embrechts et al. [18] for an introduction to heavy-tailed phenomena in nance. Telecommunications is another area where non-linear time series models with regularly varying nite-dimensional distributions might have some future applications. Empirical evidence shows that le sizes and on/o periods of le transmissions have distributions with extremely heavy tails, implying that the marginal distributions have in nite variance; see for example Willinger et al. [41]. Although classical queuing models with iid input have been applied in the context of telecommunations it is generally conjectured that the dierences between service times and interarrival times of successive customers in such models are dependent and non-linear. Therefore time series models with heavy tails and dependence are called for. In the nancial context, it is believed that models of the form (1.1)

Xt = t Zt ; t 2 Z ;

are adequate to describe much of the observed behaviour of real-life data. Here (Zt ) is an iid noise sequence, often assumed to be symmetric, and (t ) is a sequence of random variables which is adapted to the ltration generated by the past variables Zt?1 ; Zt?2 ; : : :. The latter class of models contains the celebrated GARCH processes; see Section 4 for the de nition and properties of this class of processes. Under mild conditions on the noise sequence (Zt ), the nite-dimensional distributions of ((Xt ; t )) are regularly varying; see the Appendix for various de nitions and properties of multivariate regular variation. It is surprising that (Xt ) can have heavy tails even though the noise sequence (Zt ) is light-tailed, for example Gaussian. This fact follows from a classical result by Kesten [25]; see Theorem 2.7 below. The weak limit behaviour of the sample autocovariance function (ACVF) and sample autocorrelation function (ACF) for linear processes with regularly varying nite-dimensional distributions was studied by Davis and Resnick [14, 15]. They found that the sample ACF is a consistent estimator of the autocorrelations of a Gaussian time series with the same coecients as the underlying linear process. This result is surprising insofar that it does not matter whether the underlying

nite-dimensional distributions of the linear process have nite or in nite variance. Moreover, the smaller the index of regular variation the better the rate of convergence in the distributional limit of the sample ACF. For more than a decade this result has been considered as a basis for the statement that \the sample ACF is in principle ne (robust) even if there are very large values in the time series". However, in contrast to linear processes, the weak limit behaviour of the sample ACF and sample ACVF of non-linear processes with regularly varying nite-dimensional distributions can be quite unusual. This was shown rst by Davis and Resnick [16] for a bilinear process with regularly varying noise sequence. They showed that the sample ACF of such processes with in nite 4th but nite second moments has a rate of convergence to the true ACF that becomes slower the closer the marginal distributions are to an in nite second moment. Moreover, the sample ACF has a non-degenerate limit distribution whenever the time series has an in nite second moment. The same kind of weak limit behaviour was later discovered for a large variety of non-linear processes; see Davis and Mikosch [13] for a general theory of the sample ACF and applications to ARCH(1) processes, Basrak et al. [2] for a simple bilinear process with light-tailed noise, Mikosch and Starica [32] for a GARCH(1,1) process, see also the recent review paper by Resnick [36] which gives some more insight and various examples of non-linear processes with heavy-tailed marginal distributions. In this paper we focus on the limit behaviour of the sample ACF and sample ACVF of non-linear processes with regularly varying nite-dimensional distributions which uni es previous work and also simpli es it. One of the keys tools is the embedding of such processes in stochastic recurrence equations (SRE's) of the form Xt = At Xt?1 + Bt ; t 2 Z ; where ((At ; Bt )) is an iid sequence, the At 's are d  d-matrices and the Bt 's are iid d-dimensional random vectors. In Section 2 we summarize the basic probabilistic properties of the processes (Xt ). In particular, we discuss conditions for the existence of a strictly stationary solution, its mixing properties and where the regular variation of the nite-dimensional distributions comes from. The latter property is a crucial one for the sample ACF behaviour; it follows from the aforementioned Kesten theorem. We show in Section 4 that the conditions of Kesten's theorem are met for a large class of squares of GARCH processes. There we also collect some more properties of this class of processes which is relevant in the context of nancial time series analysis, in particular, for the prediction of such time series. As in Davis and Mikosch [13], the weak limit behaviour of the sample ACF and sample ACVF for processes with regularly varying nite-dimensional distributions is based on point process convergence for weakly dependent processes. The necessary point process theory is provided in Sections 3.1 and 3.2. The corresponding general limit theory for the sample ACF of solutions to SRE's is then given in Section 3.3. The sample ACF of a GARCH is the subject of Section 4. We found it 2

convenient to collect and compare in the Appendix dierent de nitions of multivariate regular variation. There we also consider situations when functions of regularly varying vectors are regularly varying.

2 Basic theory for stochastic recurrence equations Consider a d-dimensional time series (Xt ) satisfying the stochastic recurrence equation (SRE)

Xt = At Xt? + Bt ; t 2 Z ; where the sequence ((At ; Bt )) is iid, At are random d  d-matrices and Bt are random vectors. We assume that (Xt ) is a causal strictly stationary solution of (2.2). By j
j we denote any norm in Rd , and by k
k the corresponding operator norm, i.e., for any d  d-matrix A, kAk = sup jAxj :

(2.2)

1

jxj=1

2.1 Existence of a stationary solution There exist various results about the existence of a strictly stationary solution to (2.2); see for example Kesten [25], Vervaat [40], Bougerol and Picard [8]. Below we give a sucient condition which remains valid for ergodic sequences ((An ; Bn )) (see Brandt [10]) and which is close to necessity (see Babillot et al. [6]). Before we formulate this result we introduce the notion of Lyapunov exponent. For an iid sequence (An ) of iid d  d matrices,  1 (2.3)

= inf E ln kA


A k ; n 2 N

n

n

1

is called the top Lyapunov exponent associated with (An ). If E ln+ kA1 k < 1, an application of the subadditive ergodic theorem (see Kingman [26]) or results in Furstenberg and Kesten [20] yield that 1 ln kA


A k a:s:

= nlim (2.4) 1 n !1 n In most cases of interest, cannot be calculated explicitly when d > 1. Then relation (2.4) oers an alternative for determining the value of , via Monte-Carlo simulations of the random matrices An. Work by Goldsheid [23] even allows one to give asymptotic con dence bands through a central limit theorem. Now we can formulate sucient conditons for the existence of a stationary solution.

Theorem 2.1 Assume E ln kA k < 1, E ln jB j < 1 and < 0. Then the series +

(2.5)

1

Xn = B n +

+

1 X k=1

1

An


An?k Bn?k +1

3

converges a.s., and the so-de ned process (Xn ) is the unique causal strictly stationary solution of (2:2).

Remark 2.2 Notice that < 0 holds if E ln kA k < 0. 1

The condition on in Theorem 2.1 is particularly simple in the case d = 1 since then 1 E ln jA


A j = E ln jA j = :

n

1

n

1

Corollary 2.3 Assume d = 1, ?1  E ln jA j < 0 and E ln jB j < 1. Then the unique stationary solution of (2:2) is given by (2:5).

1

+

1

2.2 The strong mixing condition The Markov chain (Xn ) satis es a mixing condition under quite general conditions as for example provided in Meyn and Tweedie [39]. Recall that a Markov chain (Yn ) with state space E  Rd is said to be -irreducible for some measure  on (E; E ) (E is the Borel - eld on E ), if

X

n>0

pn(y; C ) > 0 for all y 2 E , whenever (C ) > 0.

Here pn (y; C ) denotes the n-step transition probability from y to C . If the function

E (g(Yn ) j Yn?1 = y) ; y 2 E ;

(2.6)

is continuous for every bounded, continuous g on E , then the Markov chain is said to be a Feller chain. The Markov chain (Yn ) is said to be geometrically ergodic if there exists a  2 (0; 1) such that ?n kpn (y;
) ? (
)kTV ! 0 ; where  denotes an invariant measure of the Markov chain and k
kTV is the total variation distance. The following result is Theorem 1 in Feigin and Tweedie [19].

Theorem 2.4 Suppose that (Yn) is a Feller chain, that there exists a measure  and a compact set C with (C ) > 0 such that

 (Yn ) is -irreducible.  There exists a non-negative function g : E ! R satisfying g(y)  1 for y 2 C ; and for some  > 0,

(2.7)

E (g(Yn ) j Yn?1 = y)  (1 ? ) g(y) ; y 2 C c : 4

Then (Yn ) is geometrically ergodic.

A particular consequence of geometric ergodicity is that the Markov chain is strongly mixing with geometric rate, i.e. there exist constants K > 0 and a 2 (0; 1) such that (2.8)

sup

A2(Ys ; s0) ; B2(Ys ; s>k)

jP (A \ B ) ? P (A) P (B )j =: k  K ak :

This follows, for example, from Theorem 16.1.5 in Meyn and Tweedie [39]. The function k is called the mixing rate function of (Yt ).

Proposition 2.5 For the SRE in (2:2), suppose there exists an  2 (0; 1] such that E kAk < 1 and E jBj < 1. Then there exists a unique stationary solution to (2:2), the Markov chain (Xn ) is geometrically ergodic and hence strongly mixing with geometric rate.

Remark 2.6 The condition E kAk < 1 in some neighbourhood of zero is satis ed if E ln kAk < 0 and E kAk < 1 for some  > 0. Indeed, the function h(v) = E kAkv then has derivative h0 (0) = E ln kAk < 0, hence h(v) decreases in a small neighbourhood of zero, and since h(0) = 1 it follows that h() < 1 for small  > 0. On the other hand, E kAk < 1 for some  > 0 implies that E ln kAk < 0 by an application of Jensen's inequality. Proof. Stationarity follows at once from Theorem 2.1. Alternatively, one can establish that the in nite series in (2.5) converges a.s by showing that the sum of the th moments is nite. For geometric ergodicity we apply Theorem 2.4. An application of Lebesgue dominated convergence ensures that (2.6) is continuous in y and hence the Markov chain is Feller. Choosing  to be the stationary measure (i.e.,  is the distribution of the in nite series in (2.5)), it follows that 1 X

n=1

pn(x; C ) > 0 ;

whenever (C ) > 0. Hence (Xn ) is -irreducible. Now choose

g(x) = jxj + 1 ; x 2 Rd ; for the  2 (0; 1] given in the assumptions. We have to check that (2.7) holds:

E (g(Xn ) j Xn?1 = x)  E jAxj + E jBj + 1 ;

 E kAk jxj + E jBj + 1 =: E kAk g(x) + L : Choose C = [?M; M ]d and M > 0 so large that

E (g(Xn ) j Xn?1 = x)  (1 ? )g(x) ; jxj > M ; for some constant 1 ?  > E kAk . This proves (2.7) and concludes the proof. 5

2

2.3 Regular variation of X A d-dimensional random vector X is said to be regularly varying with index  > 0 if there exists a sequence of constants (an ) and a random vector  with values in Sd?1 a.s., where Sd?1 denotes the unit sphere in Rd with respect to the norm j
j, such that for all t > 0,

n P (jXj > t an ; X=jXj 2
) !v t? P ( 2
) ; as n ! 1 :

(2.9)

This is the same as for all t > 0, P (jXj > t x ; X=jXj 2
) !v t? P ( 2
) ; as x ! 1 ; P (jXj > x) cf. de Haan and Resnick [21]. The symbol !v stands for vague convergence on S d?1; vague convergence of measures is treated in detail in Kallenberg [24]. Note that regular variation of X = (X1 ; : : : ; Xd ) in Rd implies regular variation of jXj in R. For further information on multivariate regular variation we refer to Resnick [35]. There exist numerous extensions of regular variation to the multivariate setting. In the Appendix, we describe some of these extensions and show that they are equivalent at least when restricted to non-negative valued random vectors. Under general conditions, the stationary solution to the SRE (2.2) satis es a multivariate regular variation condition. This follows from work by Kesten [25] in the general case d  1; for an alternative proof in the case d = 1 see Goldie [22]. We state a modi cation of Kesten's fundamental result (Theorems 3 and 4 in [25]). In these results, k
k denotes the operator norm de ned in terms of the Euclidean norm j
j.

Theorem 2.7 Let (An) be an iid sequence of d  d matrices with non-negative entries satisfying:  For some  > 0, E kAk < 1.  A has no zero rows a.s.  The event (2.10) fln (An


A ) : is dense in R for some n and An


A > 0g has probability 1, where (C) is the spectral radius of the matrix C and C > 0 means that all 1

1

entries of this matrix are positive.

 There exists a  > 0 such that 0

(2.11) and

(2.12)

0 E@

1 d X Aij A  d = min i ;:::;d 0

0 2

=1

?

j =1




E kAk ln+ kAk < 1 : 0

6

Then there exists a unique solution 1 2 (0; 0 ] to the equation 1 E ln kA


A k : (2.13) 0 = nlim n 1 !1 1

n

If (Xn ) is the stationary solution to the SRE in (2:2) with coecient matrices (An ) satisfying the above conditions and B has non-negative entries with E jBj < 1, then X is regularly varying with index 1 . 1

Remark 2.8 De ne

Qd = i=1min ;:::;d

Condition (2.11) is implied by

d X j =1

Aij d?1=2 :

P (Qd > 1) > 0 :

Remark 2.9 In the case d = 1, all the conditions of the rst part of Theorem 2.7 are automatically satis ed provided A is a non-negative valued random variable with a density with support [0; 1), E ln A < 0, 1  EA and EA ln A < 1 for some  > 0. Then the equation (2.13) can be rewritten in the form E jAj = 1 which has a unique solution  > 0. Moreover, if EB  < 1 then 0

0

+

0

1

1

1

X is regularly varying with index 1 .

Remark 2.10 There are extensions of Theorem 2.7 to general A and B. Without the positivity constraints, the required conditions can be quite cumbersome. See Kesten [25] and Le Page [29].

Remark 2.11 Resnick and Willekens [37] considered SRE's under slightly dierent conditions than those imposed in Theorem 2.7. They assume that Xt = At Xt? + Bt , where ((At ; Bt )) is iid with At and Bt independent. If B is regularly varying with index  and E kAk  < 1 for some  > 0, then X is also regularly varying with index . Notice that the moment condition for B in 1

+

Theorem 2.7 is not satis ed for this model.

Corollary 2.12 Under the conditions of Theorem 2.7 the nite-dimensional distributions of the stationary solution (Xt ) of (2:2) are regularly varying with index  . 1

Proof. First note that we can write (X1 ; : : : ; Xm ) = (A1 ; A2 A1 ; : : : ; Am


A1 )X0 + Rm ; where the components of Rm have lighter tails than the components of X0 . The regular variation of the vector (X1 ; : : : ; Xm ) is assured by Proposition 5.10. 2 7

3 Some point process theory for solutions of stochastic recurrence equations

3.1 Preliminaries on point processes

We consider a strictly stationary sequence (Xt )t2Z of random row vectors with values in Rd . For simplicity, we write X = X0 = (X1 ; : : : ; Xd ). We follow the point process theory in Kallenberg [24]. The state space of the point processes considered is R d nf0g, where R = [?1; 1]. Notice that a Borel set of R d nf0g is bounded if it is bounded away from the origin. Write M for the collection of Radon counting measures on R d nf0g P n " , where with null measure o. This means  2 Mnfog if and only if  is of the form 1 i=1 i xi ni 2 f1; 2; : : :g, the points xi are distinct, and #fi : jxi j > yg < 1 for all y > 0. In what follows we formulate various results on the weak convergence of the point processes

Nn =

n X t=1

"Xt =an ; n = 1; 2; : : : ;

where X is regularly varying in the sense of De nition 5.8 and (an ) is a sequence of positive numbers such that (3.1) n P (jXj > an) ! 1 ; n ! 1 : The proofs follow from the results in Davis and Mikosch [13]; they are variations on results for d = 1 in Davis and Hsing [12]. De ne





f =  2 M : (fx : jxj > 1g) = 0 and (fx : x 2 Sm? g) > 0 ; M

(3.2)

1

f) be the Borel - eld of M f. and let B(M Theorem 3.1 Assume that the following conditions are satis ed:  The stationary sequence (Xt ) is strongly mixing with mixing rate n; cf. (2:8).  All nite-dimensional distributions of (Xt ) are jointly regularly varying with index  > 0. To be speci c, let (?k ; : : : ; k ) be the (2k + 1)d-dimensional random row vector with values in the unit sphere S(2k+1)d?1 that appears in the de nition of joint regular variation of (X?k ; : : : ; Xk ), k  0.

 The following condition holds: there exist two integer sequences rn; mn ! 1 such that nmn =rn ! 0, rnmn =n ! 0 and (3.3)

1 0 _ jX j > a yA = 0 ; y > 0 : @ j X j > a y lim lim sup P t n n k!1 n!1 kjtjr 0

n

8

Then the limit

Wk j k j  . E j k j j j



= klim E j0(k) j ? !1

(3.4)

+

exists and is the extremal index of the sequence (jXt j).

 If = 0 in (3:4), then Nn !d o :  If > 0, then Nn !d N 6= o, where N =d

P1 " i P =1

i

( ) 0

( )

=1

1 1 X X i=1 j =1

"Pi Qij ;

is a Poisson process on R+ with intensity measure

 (dy) = y??1 dy : This process is independent of the sequence of iid point processes f; B(M f)), where Q is the weak limit of distribution Q on (M

E



j k j ? Wkj=1 jj(k)j ( ) 0



+

I


P

jtjk "tk

P1 " , i  1, with joint Q j

 . 

( )

=1

E j(0k) j ?

ij

Wk j k j  j j ( )

=1

+

as k ! 1 which exists.

Remark 3.2 The strong mixing condition can be weakened to the condition A(an) in Davis and

Mikosch [13]. However, since the processes we consider below are strongly mixing, we will assume this well-known condition. If n decreases at a geometric rate, i.e. n  const an for some a 2 (0; 1), then one can choose rn = [n] and mn = [n ] for any positive  and  with  +  < 1; see the discussion of mixing conditions in Leadbetter and Rootzen [31], Lemma 2.4.4.

3.2 Point process convergence In this section we study the weak convergence of point processes generated by the stationary solution (Xt ) to the SRE (2.2). For m  0, let

Xt (m) = (Xt ; : : : ; Xt m ) : +

Theorem 3.3 Assume the conditions of Theorem 2.7 hold and let (an ) be a sequence of constants

satisfying (3.5)

n P (jX1 (m)j > an ) ! 1 :

Then the conditions of Theorem 3.1 are met, and hence

Nn =

n X t=1

"Xt (m)=an

!d

N=

1 1 X X i=1 j =1

"PiQij ;

where (Pi ) and (Qij ) are de ned in the statement of the theorem.

9

Proof. We check the conditions of Theorem 3.1. The regular variation of all nite-dimensional distributions follows from Corollary 2.12. Proposition 2.5 implies that (Xt ), and hence (Xt (m)), are strongly mixing with geometric rate. In view of the de nition of Xt (m), it suces to verify

(3.3) for m = 0 and to show that in (3.4) is positive. Iterating (2.2), we obtain for t > 0,

Xt =

Yt

j =1

Aj X + 0

Xt Yt

j =1 m=j +1

Am Bj =: It; X + It; : 1

0

2

and hence

P (jXt j > an y j jX0 j > an y)

(3.6)

 P (jX jkIt; k > an y=2 j jX j > an y) + P (jIt; j > an y=2) : 0

1

0

2

Choose  > 0. Then, using Markov's inequality and Karamata's theorem, the lim supn!1 of the rst term on the right of (3.6) is bounded above by E [jX jI(an y;1) (jX0 j)] lim sup E kIt;1 k (2=y) a 0P (jX  C (E kAk )t : j > a y ) n!1 0 n n Here C is a constant independent of t. Now choose  2 (0; 1) such that E kAk < 1. This is always possible in view of Remark 2.6. As for the second term in (3.6), we have

t j? t j? ? 1 jY X XY Y X d kA k jB j = Y a:s: kA k jB j " A B  jIt; j = j m m j j m m j j m m j 1

1

1

2

=1

=1

=1

=1

=1

=1

for some random variable Y . Thus we obtain by Markov's inequality and the same   1,

P (jIt;2 j > an y=2)  P (Y > an y=2)  a?n (2=y) E jBj

1 X j =1

(E kAk )j  const a?n  :

According to Remark 3.2 we can take rn  n for any small  > 0. Choosing  so small that rna?n  ! 0 and combining the bounds for the terms in (3.6), we obtain,

0 1 _ lim lim sup P @ jXt j > any jX j > anyA k!1 n!1 kjtjr X 0

n

 klim lim sup !1 n!1  klim (const) !1 = 0:

kjtjrn

1 X t=k

P (jXt j > an yjjX0 j > an y)

(E kAk )t

2

This completes the veri cation of (3.4). 10

3.3 Limit theory for the sample ACF Using the point process theory of the previous section, it is possible to derive the asymptotic behaviour of the sample cross-covariances and cross-correlations of the stationary solution (Xt ) to the SRE (2.2) satisfying the conditions of Theorem 2.7. For ease of exposition we concentrate on the sample autocovariances of the rst component process (Yt ) say of (Xt ). De ne the sample autocovariance function

n;Y (h) = n?1

(3.7)

nX ?h t=1

Yt Yt+h ; h  0 ;

and the corresponding sample autocorrelation function

n;Y (h) = n;Y (h)/ n;Y (0) ; h  1 :

(3.8) We also write

Y (h) = EY0 Yh and (h) = Y (h)= Y (0) ; h  0 ;

for the autocovariances and autocorrelations, respectively, of the sequence (Yt ) if these quantities exist. Mean-corrected versions of both the sample and model ACVF can also be considered|the same arguments as above show that the limit theory does not change.

Theorem 3.4 Assume that (Xt ) is a solution to (2:2) satisfying the conditions of Theorem 2.7. (1) If 1 2 (0; 2), then



n1?2= n;Y (h) 1

 h=0;:::;m

!d (Vh)h

;:::;m ;

=0

d (Vh =V0 )h=1;:::;m ; (n;Y (h))h=1;:::;m !

where the vector (V0 ; : : : ; Vm ) is jointly 1 =2-stable in Rm+1 . (2) If 1 2 (2; 4) and for h = 0; : : : ; m,

! nX ?h ? 2=1 Yt Yt+h IfjYtYt+h ja2ng = 0 ; lim lim sup var n !0 n!1 t=1

(3.9) then

(3.10) (3.11)



n1?2= ( n;Y (h) ? Y (h))



1

n1?2= (n;X (h) ? X (h) 1

 

h=0;:::;m

!d (Vh)h

h=1;:::;m

!d X? (0) (Vh ? X (h) V )h

where (V0 ; : : : ; Vm ) is jointly 1 =2-stable in Rm+1 .

11

;:::;m ;

=0

1

0

;:::;m

=1

;

(3) If 1 > 4 then (3:10) and (3:11) hold with normalization n1=2 , where (V1 ; : : : ; Vm ) is mulP cov(Y Y ; Y Y )] tivariate normal with mean zero and covariance matrix [ 1 0 i k k+j i;j =1;:::;m k=?1 2 and V0 = E (Y0 ).

Remark 3.5 The limit random vectors in parts (1) and (2) of the theorem can be expressed in terms of the Pi 's and Qij 's de ned in Theorem 3.3. For more details, see Davis and Mikosch [13]

where the proof of (1) and (2) is provided. Part (3) follows from a standard central limit theorem for strongly mixing sequences; see for example Doukhan [17].

Remark 3.6 The conclusions of Theorem 3.4 are also valid for other functions of Xt including

linear combinations of powers of the components. Indeed, the constructed process inherits strong mixing as well as joint regular variation from the (Xt ) process and point process convergence follows from the continuous mapping theorem.

4 Application to GARCH processes One of the major applications of SRE's is to the class of GARCH processes. Recall that a generalized autoregressive conditionally heteroscedastic process (Xt ) of orders p  0 and q  0 (GARCH(p; q)) is given by the equations

Xt = t Zt t2 = 0 +

p X i=1

i Xt2?i +

q X j =1

j t2?j ;

where (Zt ) is an iid sequence of random variables, and the i 's and j 's are non-negative constants. This class of processes was introduced by Bollerslev [7] and has since then found a multitude of applications for modelling nancial time series. For q = 0 the process is called an ARCH(p) process. The squared processes (Xt2 ) and (t2 ) satisfy the following SRE:

Xt = AtXt? + Bt ;

(4.1) where

1

?

Xt = Xt ; : : : ; Xt?p ; t ; : : : ; t?q 2

2

+1

2

2

12

+1


0 ;

0Z BB 1 t BB BB ... BB BB 0 BB  BB BB 0 BB .. @ .

2

1

(4.2)

At =

1

0




p? Zt pZt Zt


q? Zt q Zt 1 C


0 0 0


0 0 C C C 2

1

.. . 1

...

2

.. . 0







p?


0

p






0

1

.. .

.. . 0

0 .. .

1

2

.. . 0

1 1 .. .

0

2

1

.. .

.. . 0







q ?


0

1

...






.. . 1

2

C C C C C C; q C C C 0 C C C .. C . C A .. . 0

0

Bt = ( Zt ; 0; : : : ; 0;  ; 0 : : : ; 0)0 : 0

2

0

In the following proposition we collect some of the basic properties of the process (Xt ).

Proposition 4.1 Consider the SRE (4:1). (A) (Existence of stationary solution) Assume that the following conditon holds:

(4.3)

E ln+ jZ j < 1 and the Lyapunov exponent < 0.

Then there exists a unique causal stationary solution of the SRE (4:1). (B ) (Strong mixing) Assume there is a  > 0 such that

(4.4)

E jZ j < 1 and < 0.

Then there exists a unique causal stationary solution of the SRE (4:1) and the Markov chain (Xn ) is strongly mixing with geometric rate. (C) (Regular variation of the nite-dimensional distributions) Let j
j denote the Euclidean norm and k
k the corresponding operator norm. In addition to the Lyapunov exponent being less than 0, assume the following conditions: 1. Z has a positive density on R such that either E jZ jh < 1 for all h > 0 or E jZ jh = 1 for some h0 > 0 and E jZ jh < 1 for 0  h < h0 . 0

2. Not all of the parameters i and i vanish. Then there exists a 1 > 0 such that X is regularly varying with index 1 .

Remark 4.2 Necessary and sucient conditions for (4.3) in terms of the parameters i and i and the distribution of Z are known only in a few cases. This includes the ARCH(1) (see Goldie [22]; 13

cf. Embrechts et al. [18], Section 8.4) and the GARCH(1,1) cases (see Nelson [33]). The latter case can be reduced to a one-dimensional SRE for (t2 ); see for example Mikosch and Starica [32]. The general case can be found in Bougerol and Picard [9], where, to the best of our knowledge, the most general sucient conditions are given. We formulate here some of their results. Throughout we assume that the conditions 0 > 0 and E ln+ jZ j < 1 hold.

 < 0 is necessary and sucient for the existence of a unique strictly stationary causal solution to (4.1).

 Pqi i < 1 is necessary for < 0. =1

 Ppi i + Pqj j < 1 implies < 0. =1

=1

 If Z has in nite support and no atom at zero, i > 0 and j > 0 for all i and j then

Pp  + Pq = 1 implies < 0. i j i j =1

=1

An alternative way to check whether < 0 is via Monte-Carlo simulations by using (2.3).

Proof. Part (A) is an immediate consequence of Theorem 2.1. For parts (B) and (C), we apply Proposition 2.5 and Kesten's Theorem 2.7 to an iterate of (4.1). That is, choose m  1 such that Xt = Ae t Xt?m + Be t ;

e t = At


At?m+1 has positive entries with probability one. Increase m if necessary so where A e t k < 0 and each entry of Ae t has a term consisting of at least one even power of Zt2?j for that E ln kA some j = 0; : : : ; m ? 1. The former holds for m large since the top Lyapunov exponent is assumed e 1 k < 1. to be negative. By Remark 2.6 this implies that there exists an  > 0 such that E kA Note that the skeleton process (Xtm ; t  1) satis es this new SRE. These conditions, speci cally E kAe 1 k < 1, imply that (Xtm ) is strongly mixing with geometric rate (see Proposition 2.5). Using the monotonicity of the mixing function, it follows that (Xt ) is strongly mixing with geometric rate. e t . We start with (2.10). Let C be any d  d matrix We now check conditions (2.10){(2.12) for A with non-negative entries. By the Perron{Frobenius theorem (cf. Chapter 9 in Lancaster [27]), the spectral radius (C) is equal to one of the real eigenvalues of the matrix C. Let 0 d d 1= XX A cij kCk = @

1 2

2

2

i=1 j =1

denote the Frobenius norm of C. Since 2 (C) = (C0 C) it is easy to check that (4.5)

d?1 kC0 Ck22  2 (C0 C) = 4 (C)  kC0 Ck22  kCk42 : 14

e n


Ae 1. In view of (4.5), it suces to show that the set of values ln k0nnk2 , n  1, Set n = A are dense in R a.s. Since the entries of n consist of linear combinations of products of even powers of 1; Z1 ; : : : ; Zn with positive coecients, we can write k0n n k22 = c0 + c1 , where c0  0 is a constant depending on n and c1 is a linear combination of products of even powers of Z1 ; : : : ; Zn . e k < 1 for some  > 0, It follows that the support of k0n n k22 is [c0 ; 1). Moreover, since E kA k0n nk22 a!:s: 0 as n ! 1, and therefore c0 (n) ! 0. Hence k0n nk22 ; n  1 is dense in [0; 1) with probability one. This proves (2.10). By the choice of m above, it follows that P (Qd > 1) > 0 and hence (2.11) holds by Remark 2.8. e t consist of linear combinations of products of 1; Zt2; : : : ; Zt2?m+1 , the moment Since the entries of A e. condition of (2.12) holds for A e t . Since 1  0 , we From Theorem 2.7, there exists a 1 > 0 such that (2.13) holds for the A also have E jBe j < 1. We conclude that X is regularly varying with index 1 . 2 1

Corollary 4.3 Let (Xt ) be a stationary GARCH(p; q) process. Assume the conditions of part C of Proposition 4.1 hold. Then there exists a  > 0 such that the nite-dimensional distributions of the process ((t ; Xt )) are regularly varying with index .

Proof. As for Corollary 2.12, it follows that the nite-dimensional distributions of ((t ; Xt )) are regularly varying with index  . It suces to show that for all k  1, the random vector Yk = ( ; X ; : : : ; k ; Xk )0 is regularly varying with index  = 2 . This is proved by induction 2

2

1

1

1

1

= 1, (1 ; X1 )0 = (Z1 ; 1)0 1 and since

on k. For k 1 is regularly varying, so is the vector by Corollary 5.12. Now suppose Yk is regularly varying with k  max(p; q). Using the representation k2+1 = 0 + 1 Xk2 +


+ pXk2+1?p + 1k2 +


+ q k2+1?q , it follows that (Yk0; k+1 )0 is regularly varying with exponent . Now writing

Yk = +1

I

k

2 +1

0

0

Zk+1

 Y  k ; k +1

we conclude again from Corollary 5.12 that Yk+1 is regularly varying with exponent  which completes the induction argument. 2 In what follows we want to apply the theory of Section 3.3 to derive the distributional limits of the sample autocovariances and autocorrelations of certain GARCH(p; q) processes. The case of an ARCH(1) process, its absolute values and squares has been treated in Davis and Mikosch [13]. The case of a GARCH(1; 1) was dealt with in Mikosch and Starica [32]. From these two examples we learnt that it is dicult to prove the limit behaviour of the sample ACVF and ACF of a general GARCH(p; q) in terms of the limiting point processes, especially for the case when the process has a nite rst but an in nite second moment. In what follows we restrict ourselves to a particular case, the GARCH(2; 1) model, to describe the method. 15

Consider a GARCH(2; 1) process (Xt ) with positive parameters i and i given by the SRE Xt = t Zt , where

t2 = 0 + 1 Xt2?1 + 1 t2?1 + 2 t2?2 = 0 + At?1 t2?1 + 2 t2?2 and At = 1 Zt2 + 1 . We assume that the conditions of Proposition 4.1 are satis ed for the parameters i , i and the noise sequence (Zt ). By Corollary 4.3, the nite-dimensional distributions of (Xt ) are regularly varying with index  > 0.

The sample ACF of (Xt ) Theorem 3.4 is not directly applicable to the sample ACF and ACVF of (Xt ) since only the squares of the process satisfy a SRE of type (2.2). However, another application of the results in Davis and Mikosch [13] also ensures that the statement of Theorem 3.4 remains valid for the X -sequence; see also Remark 3.6. This result is then directly applicable to the sample ACF and ACVF in the case  2 (0; 2) [ (4; 1). Now assume that  2 (2; 4) and for simplicity that Zt has a symmetric distribution. For h  1, the condition (3.9) is easy to verify since (Xt Xt+h IfjXt Xt h jan g ) is a sequence of uncorrelated random variables. For the case h = 0, we have 2

+

a?n 2

n X t=1

(Xt2 ? EX 2 )

= a?n 2 = a?n 2

n X t=1 n X t=1

t2 (Zt2 ? 1) + a?n 2

n X t=1

(t2 ? E2 )

t2 (Zt2 ? 1)Ift an g + a?n 2

= I + II + III :

n X t=1

t2 (Zt2 ? 1)Ift >an g + na?n 2 ( n; (0) ? E2 )

Using Karamata's theorem on regular variation, it follows that (4.6)

lim lim sup var (I ) = lim lim sup na?n 2 var(Z 2 )Et2 Ift an g = 0 : !0

!0 n!1

As for the third term, we have

n!1

X X III = a?n 2 t2?1 (At ? (1 + 1 )) + (1 + 1 + 2 ) (t2 ? E2 ) + oP (1) = a?n 2

n

n

t=1 n X

t=1

t=1

t (At ? (1 + 1 ))Ift ang 2

+(1 + 1 + 2 )III + oP (1) = IV + V + V I : 16

X + a?2 2 (At ? (1 + 1 ))I n

n

t=1

t

ft >an g

Relation (4.6) with I replaced by IV holds. Hence the distributional limit behaviour of the sequence (na?n 2 ( n;X (0) ? EX 2 )) is determined by the limit of

II + (1 ? (1 + 1 + 2 ))?1 V =

n 1 ? ( 1 + 2 ) a?2 X 2 2 1 ? (1 + 1 + 2 ) n t=1 t (Zt ? 1))Ift an g :

As in Davis and Mikosch [13], Section 4, one can now apply the point process convergence of Theorem 3.1 to conclude that na?n 2 ( n;X (0) ? EX 2 ) has a =2-stable limit. Moreover, since the convergence for the sample ACVF at lags h  1 is based on the same point process result, one has joint convergence to a =2-stable limit for any nite vector of sample autocovariances. This fact together with the continuous mapping theorem implies that the statement of part 2 of Theorem 3.4 holds for (Xt ), both for the sample ACF and ACVF.

The sample ACF of (Xt ) 2

In the analysis of nancial returns it is common use to study the autocorrelations of the absolute values and their powers in order to detect the non-linearity in the dependence structure. In what follows, we restrict ourselves to the second powers (Xt2 ) of a GARCH(2; 1) process. The absolute values (and any powers) can be treated in a similar way by applying the same kind of argument; see for example Davis and Mikosch [13] for the ARCH(1) case and Mikosch and Starica [32] for the GARCH(1; 1) case. In the cases  2 (0; 4) [ (8; 1) one can apply parts 1 and 3 of Theorem 3.4 directly to the sample ACF and ACVF of (Xt2 ) to obtain weak convergence to =4-stable limits. In the case  2 (4; 8) one would have to verify condition (3.9). This turns out to be dicult and, therefore, as in Davis and Mikosch [13] for the ARCH(1) case, we will apply point process convergence results to derive the weak limits of the sample autocovariances and autocorrelation. Since we are only interested in showing that the joint limit of the centered and normalised sample autocovariances is =4-stable, we restrict ourselves to one particular case, namely the sample autocovariance n;X (1). Observe that 2

a?n 4

n X t=1

(Xt2 Xt2+1 ? E (X0 X1 )2 )

= a?n 4 = a?n 4

n X t=1 n X

Xt2 t2+1 (Zt2+1 ? 1) + a?n 4 Xt t (Zt

t=1 n X

+a?n 4

t=1

2

2 +1

2 +1

n  X t=1

Xt2 t2+1 ? EX02 12



X ? 1)IfjXtjt+1a2n g + a?n 4 Xt2 t2+1 (Zt2+1 ? 1)IfjXt jt+1>a2n g n

t=1

(Xt2 [At t2 + 2 t2?1 ] ? [E4 EAZ 2 + 2 E02 12 ]) + oP (1) 17

= I + II +a?n 4

n X t=1

t4 [At Zt2 ? E (AZ 2 )] + E (AZ 2 )a?n 4

X + 2 a?n 4 [Xt2 t2?1 ? E (X12 02 )] = I + II

n

t=1

[t4 ? E4 ]

t=1

X +a?4 4 [At Z 2 ? E (AZ 2 )]I n

n X

n

t=1

t

t

+E (AZ 2 )a?n 4 + 2 a?n 4

n X

n X t=1

ft an g

n X

= I + II + III + IV + V + 2 a?n 4 + 2 a?n 4

n X

t=1 n X t=1

n

n

t=1

t

t

ft >an g

[t4 ? E4 ]

t2 t2?1 (Zt2 ? 1) + 2 a?n 4

t=1

X + a?4 4 [At Z 2 ? E (AZ 2 )]I

t=1

[t2 t2?1 ? E (02 12 )]

t2 t2?1 (Zt2 ? 1)Ift t? an g + 2 a?n 4 1

2

n X t=1

t2 t2?1 (Zt2 ? 1)Ift t? >an g 1

2

[t2 t2?1 ? E (02 12 )]

= I + II + III + IV + V + V I + V II + V III : The terms I,III,VI can be dealt with as in (4.6). As in Section 4 of Davis and Mikosch [13] one can use point process convergence and a continuous mapping argument to show that II+IV+VII has a =4-stable limit. The same is true for V+VIII, and these terms also converge jointly with the sum of the others. Thus it remains to show convergence of V+VIII. Notice that

X a?4 [2 n

n

= a?n 4 = a?n 4

t=1 n X

t+1 t

? E  ] 2 0

2 1

[(t4 At ? E4 EA) + 2 (t2 t2?1 ? E02 12 )] + oP (1)

t=1 n X t=1

2

t4 (At ? (1 + 1 )) + 2 a?n 4

+oP (1) :

n X t=1

[t2 t2+1 ? E02 12 ] + (1 + 1 )a?n 4

Thus

V + V III 18

n X t=1

(t4 ? E4 )

 X n n X  1 + 1 4 4 ? 1 ?4 ? 4 t4 (At ? (1 + 1 )) + oP (1) : [t ? E ] + (1 ? 2 ) an = E (AZ ) + 1 ? an 2 t=1 t=1 

2

Using the same point process approach as mentioned above, the second term can be shown to have a =4-stable limit. Thus it remains to show the weak convergence of n; (0). Notice that 2

X a?4 [4 ? E4 ] n

n

= a?n 4 = a?n 4

t=1 n  X t=1 n X t=1

t



t4 (A2t + 22 ) + 2t2 t2?1 2 At ? (E4 (EA2 + 22 ) + 2 2 E02 12 (1 + 1 )) + oP (1)

t (At + ? (EA + )) + (EA +

+2 2 a?n 4

4

n X t=1

2

2 2

2

2 2

2

2 2

X )a?4 [4 ? E4 ] n

n

t=1

t2 t2?1 (At ? (1 + 1 )) + 2 2 a?n 4 (1 + 1 )

t

n X t=1

(t2 t2+1 ? E02 12 ) + oP (1) :

The right-hand side can be re-expressed in terms of n; (0) and other terms which can be dealt with by point process techniques. This nally shows that the sample autocovariance n;X (1) has a =4-stable limit. Since for the distributional convergence only the point process convergence and the continuous mapping theorem have been used, it is immediate that the same kind of argument yields the joint convergence of the sample autocovariances to a =4-stable limit as described in part 2 of Theorem 3.4. 2

2

5 Appendix

5.1 Multivariate regular variation In what follows we compare dierent notions of multivariate regular variation. Recall that a onedimensional random variable X (or its distribution) is said to be regularly varying if the tail probabilities can be written as (5.1) P (X > x) = x? L(x) ; x > 0 ; where L is a slowly varying function, i.e. L(cx)=L(x) ! 1 for any choice of c > 0. Relation (5.1) can be extended to the case of multivariate vectors X with values in Rd in dierent ways. In what follows, we write a  b; a < b; a > b, etc., where ; ; : : : refer to the natural (componentwise) partial ordering in Rd . We say that the vector x is positive (x > 0) if each of its components is positive. We also use the following notation:

1 = (1; : : : ; 1) ; 0 = (0; : : : ; 0) ; [a; b] = fx : a  x  bg : Unless stated otherwise, we assume in what follows that X is positive with probability 1. However, in such cases we will choose the state space D = [0; 1]d nf0g. 19

Condition R1. The following limit exists and is nite for all x > 0: P (X 2 t [0; x]c ) ; lim t!1 P (X 2 t [0; 1]c )

where the complements are taken in the state space D. Condition R2. There exist   0 and a random vector  with values in Sd+?1 = Sd?1 \ [0; 1)d

(Sd?1 is the unit sphere in Rd ) such that the following limit exists for all x > 0: P (jXj > t x ; X=jXj 2
) !v x? P (
) ; t ! 1 ; (5.2)  P (jXj > t)

v where ! denotes vague convergence on Sd+?1 and P is the distribution of . Condition R3. There exist a non-zero Radon measure  on D and a relatively compact Borel set E  D such that X 2 t
) !v (
) ; t ! 1 ; t(
) := PP ((X 2 t E)

v denotes vague convergence on D. where ! Condition R4. The following limit exists and is nite for all x > 0: P ((x; X) > t) : lim t!1 P ((1; X) > t) Remark 5.1 Conditions R1 and R2 are frequently used in extreme value theory in order to characterize maximum domains of attractions of extreme value distributions; see for example Resnick [35]. They are also used for the characterization of domains of attractions of stable laws in the theory of sums for independent random vectors; see for example Araujo and Gine [1]. Condition R3 has been introduced by Meerschaert [38] in the context of regularly varying measures. Condition R4 was used by Kesten [25] in order to characterize the tail behaviour of solutions to SRE's.

Remark 5.2 If R3 holds, then by a standard regular variation argument, there exists an   0 such that (vS ) = v? (S ) for every Borel set S  D and v > 0. If one uses polar coordinates (r;  ) = (jxj; x=jxj) then it is convenient to describe the measure  as c r?? dr  P (d), where P is a probability measure on the Borel - eld of Sd? and c is a positive constant. The case  = 0 corresponds to the measure "f1g(dr)  P (d), where "x denotes Dirac measure at x. 1

+

1

Remark 5.3 R1{R4 can be formally extended to the case  = 1. For example, R2 can be formulated as follows: For a given P -continuity set S , the left-hand side probability in (5.2) converges to 0 or 1 according as x 2 (0; 1) or x > 1. For d = 1, the case  = 1 corresponds to rapid variation; see Bingham et al. [5].

20

Remark 5.4 The limits in R1{R4 can be compared by choosing in R3 the particular sets:  for R1: E = [0; 1]c ,  for R2: E = fx > 0 : x=jxj 2 Sd?1; jxj > 1g, +

 for R4: E = fy : (y; 1) > 1g

Remark 5.5 If R3 holds for some set E , it holds for any bounded Borel set E  D with (@E ) = 0.

It is also straightforward to show that this condition is equivalent to the existence of a Radon measure  on D and a sequence (an ); an ! 1 such that (5.3)

n P (a?n 1 X 2
) !v  (
) ;

where  is a measure with the property that  (E ) > 0 for at least one relatively compact set E  D. The other conditions can be represented similarly.

Theorem 5.6 Conditions R1{R4 are equivalent. Remark 5.7 Following the lines of the proofs below, one can show that R2 and R3 are equivalent on the enlarged state space R d nf0g provided one replaces Sd? with Sd? in R2. +

1

1

Proof. R3 =) R1. De ne Uy = [0; y]c ; y > 0 : From Remark 5.2 we conclude that (@Uy ) = 0. Now take E = U1 . R3 =) R2. Write Vu;S = fx : x=jxj 2 S and jxj > ug ; where u > 0 and S is any Borel set of Sd+?1. Put E = V1;Sd? . It follows from the above argument in combination with Remark 5.2 that for every continuity set V1;S of , P (jXj > tx ; X=jXj 2 S ) ! (V ) = x? (V ) ; x;S 1;S P (jXj > t) for some non-negative number . Since (V1;S ) corresponds to a probability measure on the Borel sets S of Sd+?1, there exists a random vector  with this distribution. This establishes R2. R3 =) R4. De ne for x > 0, Wx = fy : (x; y) > 1g +

1

and E = W1 . The same arguments as above prove the implication. R1 =) R3. Choose E = U1. The set of measures ft g in R3 is relatively compact in the vague 21

topology. That is, for any bounded Borel set B  D, supt t (B ) < 1 : Any such set B is contained in some Uy . To establish vague convergence of (t ) suppose there exist two sequences (ti ); (si ) ! 1 v such that ti ! I and si !v II . It then suces to show that I = II . By R1, I and II agree on the sets Uy . Any rectangle in D can be represented by applying a nite number of dierences of sets Uy . Hence I and II must agree on all Borel sets of D, i.e. I = II . R2 =) R3. The sets

o

n

Vu;S : u > 0; S is a Borel subset of Sd+?1

constitute a -system that generates the Borel - eld on D; see Billingsley [4] for its de nition. We can use the same arguments as above to show that R2 implies R3. R4 =) R3. Take E = W1. The relative compactness of ft g follows from R4. Then, arguing as above, suppose (t ) has vague limits I and II along two dierent subsequences. De ne

W = fWx ; x > 0g : It is not a -system. However, by R4, I and II agree on sets W 2 W . Since for every xed x > 0 the functions expf?(x; t)g are measurable with respect to the - eld (W ) generated by W , we conclude that for any bounded Borel set B 2 (W ), (5.4)

Z

B

f (t) dI (t) =

Z

B

f (t) dII (t) ;

for every function f in the algebra of functions generated by expf?(x;
)g. By the Stone{Weierstrass theorem (cf. Lang [28], p.150) and Lebesgue dominated convergence we conclude that (5.4) holds for all continuous functions f with compact support on D. Letting B " D through subsequences B 2 (W ), the equality (5.4) holds for B = D and all continuous functions f with compact support on D. This implies that I = II . 2 In view of Theorem 5.6 and Remark 5.7 we adopt the following de nition for regular variation:

De nition 5.8 The random vector X with values in Rd is said to be regularly varying with index  and spectral measure P on Sd?1 if condition R2 holds.

5.2 Functions of regularly varying vectors In what follows we consider a regularly varying random vector X with index   0 and spectral measure P . We consider suitable transformations of X such that the transformed vector is again regularly varying. Our rst result tells us that regular variation is preserved under power transformations. De ne for any vector x > 0 and p > 0, xp = (xp1 ; : : : ; xpd ) : 22

Proposition 5.9 Assume X > 0 is regularly varying in the sense of R2 where j
j denotes the max-norm. For every p > 0, Xp is regularly varying with index =p and spectral measure Pp . After noting that jxp j = jxjp , the proof is immediate from condition R2.

Our next theorem extends a well-known one-dimensional result of Breiman [11] to d > 1. This result says that, for any independent non-negative random variables  and  such that  is regularly varying with index  and E < 1 for some > ,

P (  > x)  E  P ( > x) :

(5.5)

In view of (5.3) and Remark 5.5 the regular variation condition can be formulated as follows: there exist a sequence (an ) and a measure  on D such that




?

P a?n 1 X 2
v P (jXj > an) ! (
) ; v denotes vague convergence on D. where !

The multivariate version of Breiman's result reads as follows.

Proposition 5.10 Let A be a random q  d matrix, independent of the vector X which satis es (5:3). Also assume that 0 < E kAk < 1 for some > . Then

?
n P a?n 1 AX 2
!v e(
) := E [  A?1 (
)] ;

v denotes vague convergence on D and A?1 is the inverse image of A. where !

Remark 5.11 Assume that B is a d-dimensional random vector such that P (jBj > x) = o(P (jXj > x)) as x ! 1 and the conditions of Proposition 5.10 hold. Then AX + B is regularly varying with the same limit measure e. Moreover, if B itself is regularly varying with index  and independent of AX, then AX + B is regularly varying with index . This follows from the fact that (AX; B) is regularly varying.

Proof. For a xed bounded e-continuity set B de ne

 An (B ) = a?n 1 AX 2 B :

Then for every 0 < " < M < 1,

P (An(B )) = P (An (B ) \ fkAk  "g) + P (An (B ) \ f" < kAk  M g) + P (An (B ) \ fkAk > M g) =: p1 + p2 + p3 : 23

Note that by (5.5) for some t > 0 (one can t to be the distance of the set B from 0),

?




P jXj kAkI(M;1) (kAk) > ant = t? E [kAk I(M;1) (kAk)] : lim sup P (jXpj 3> a )  nlim !1 P (jXj > an) n!1 n Since E kAk < 1 we conclude by Lebesgue dominated convergence that p3 lim lim sup (5.6) = 0: M !1 n!1 P (jXj > an ) Now consider p2 : p2 (5.7) lim n!1 P (jXj > an ) Z P (An(B )j A) P (dA) = E I (kAk)(A?1 B ) : = nlim (";M ] !1 " an ) In the limit relation we made use of a Pratt's lemma; cf. Pratt [34]. The right-hand side of (5.7) converges to the desired E(A?1 B ) if we rst let M ! 1 and then " ! 0. The so obtained limit is nite since E kAk < 1 and







E I(";M ](kAk)(A?1 B ) = E I(";M ](kAk)fx : Ax 2 B g



 E [kAk ] fx : jxj > tg < 1 : Finally, we consider p1 . Then

P ("jXj > an t) = (t?1 ") : lim sup P (jXpj 1> a )  nlim !1 P (jXj > an ) n!1 n

We conclude that (5.8)

p1 lim lim sup = 0: "!0 n!1 P (jXj > an ) Combining the limit results for p1 ; p2 ; p3 , we conclude that the proof is nished.

2

Corollary 5.12 Let X be regularly varying with index , independent of the vector (Y ; : : : ; Yd ) which has independent components. Assume that E jYi j  < 1 for some  > 0, i = 1; : : : ; d. Then 1

+

(Y1 X1 ; : : : ; Yd Xd ) is regularly varying with index . A particular consequence is the following.

Proposition 5.13 Let " = (" ; : : : ; "d ) be a vector of iid Bernoulli random variables such that P (" = 1) = 0:5. Also assume that " and X are independent. Then Y = (" X ; : : : ; "d Xd ) is 1

1

regularly varying with index  and spectral measure P("  ;:::;"dd ) . 1 1

24

1

1

Proof. Without loss of generality assume that the coordinates of X are non-negative. Observe that jYj = jXj and that the event fY=jYj 2 S g can be rewritten as fX=jXj 2 S  (")g for some S  (")  Sd? . Then, assuming that S  (") is a P -continuity set for every ", the proof is complete +

1

2

after taking the expectation with respect to ".

Acknowledgement. This research has been conducted during mutual visits of each of the authors

at the other's home institution. The authors would like to thank Colorado State Unversity and the University of Groningen for their hospitality.

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Thomas Mikosch Bojan Basrak Department of Mathematics P.O. Box 800 University of Groningen NL-9700 AV Groningen THE NETHERLANDS