In this section, we provide a proof that the multivariable tail estimator of Section 8 is still valid for certain sequences of dependent heavy tailed random vectors. We say that a sequence(Bn)of invertible linear operators is regularly varying with index−Eif for any λ >0 we have
B[λn]Bn−1→λ−E asn→ ∞.
For further information about regular variation of linear operators see Meerschaert and Scheffler (2001a, Chapter 4).
In view of Theorem 2.1.14 of Meerschaert and Scheffler (2001a) we can writeRd= V1⊕ ã ã ã ⊕Vp andE=E1⊕ ã ã ã ⊕Epfor some 1pd where eachVi isEinvariant, Ei:Vi→Viand Re(λ)=ai for all real parts of the eigenvalues ofEi and somea1<ã ã ã<
ap. By Definition 2.1.15 of Meerschaert and Scheffler (2001a) this is called the spectral decomposition ofRdwith respect toE. By Definition 4.3.13 of Meerschaert and Scheffler (2001a) we say that(Bn)is spectrally compatible with−Eif everyVi isBn-invariant for alln. Note that in this case we can writeBn=B1n⊕ ã ã ã ⊕Bpnand eachBin:Vi →Vi
is regularly varying with index−Ei. [See Proposition 4.3.14 of Meerschaert and Scheffler (2001a).] For the proofs in this section we will always assume that the subspacesVi in the spectral decomposition ofRdwith respect toEare mutually orthogonal. We will also assume that(Bn)is spectrally compatible with−E. Letπidenote the orthogonal projection operator ontoVi. If we letPi=πi+ ã ã ã +πpandLi=Vi⊕ ã ã ã ⊕VpthenPi:Rd→Li is a orthogonal projection. Furthermore,Pi=π1+ ã ã ã +πi is the orthogonal projection onto L¯i=V1⊕ ã ã ã ⊕Vi.
Now assume 0< a1<ã ã ã< ap. Since(Bn)is spectrally compatible with−E, Propo- sition 4.3.14 of Meerschaert and Scheffler (2001a) shows that the conclusions of Theo- rem 4.3.1 of Meerschaert and Scheffler (2001a) hold withLi =Vi ⊕ ã ã ã ⊕Vp for each i=1, . . . , p. Then for anyε >0 and anyx∈Li\Li+1we have
n−ai−ε Bnx n−ai+ε (9.1)
for all largen. Then log Bnx
logn → −ai asn→ ∞ (9.2)
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 627
and since this convergence is uniform on compact subsets ofLi\Li+1we also have log πiBn
logn → −ai asn→ ∞. (9.3)
It follows that log Bn
logn → −a1 asn→ ∞. (9.4)
Since (Bn)−1is regularly varying with indexE, a similar argument shows that for any x∈ ¯Li\ ¯Li−1we have
nai−ε(((Bn)−1x((nai+ε (9.5)
for all largen. Then log (Bn)−1x
logn →ai asn→ ∞ (9.6)
and since this convergence is uniform on compact subsets ofL¯i\ ¯Li−1we also have log πi(Bn)−1
logn →ai asn→ ∞. (9.7)
Hence
log (Bn)−1
logn →ap asn→ ∞. (9.8)
Suppose thatXt,t=1,2, . . . ,areRd-valued random vectors and letMnbe the sample covariance matrix of(Xt)defined by (6.3). Note thatMnis symmetric and positive semi- definite. Let 0λ1nã ã ãλdndenote the eigenvalues ofMnand letθ1n, . . . , θdnbe the corresponding orthonormal basis of eigenvectors.
Basic Assumptions. Assume that for some exponentE with real spectrum 1/2< a1<
ã ã ã< apthe subspacesVi in the spectral decomposition ofRd with respect toEare mu- tually orthogonal, and there exists a sequence(Bn)regularly varying with index−Eand spectrally compatible with−Esuch that:
(A1) The set{n(BnMnBn): n1}is weakly relatively compact.
(A2) For any limit pointMof this set we have:
(a) Mis almost surely positive definite.
(b) For all unit vectorsθthe random variableθMθ has no atom at zero.
628 M.M. Meerschaert and H.-P. Scheffler
Now letRd=V1⊕ ã ã ã ⊕Vpbe the spectral decomposition ofRdwith respect toE. Put di=dimViand fori=1, . . . , pletbi=di+ ã ã ã +dpandb¯i=d1+ ã ã ã +di. Our goal is now to estimate the real spectruma1<ã ã ã< apofEas well as the spectral decomposition V1, . . . , Vp. In various situation, these quantities completely describe the moment behavior of theXt.
Theorem 9.1. Under our basic assumptions, fori=1, . . . , pandb¯i−1< jb¯i we have log(nλj n)
2 logn →ai in probability asn→ ∞.
The proof of Theorem 9.1 is in parts quite similar to the Theorem 2 in Meerschaert and Scheffler (1999b). See also Section 10.4 in Meerschaert and Scheffler (2001a), and Scheffler (1998). We include it here for sake of completeness.
Proposition 9.2. Under our basic assumptions we have log(nλdn)
2 logn →ap in probability.
Proof: Forδ >0 arbitrary we have P!!
!!log(nλdn) 2 logn −ap
!!!!> δ
P
λdn> n2(ap+δ)−1 +P
λdn< n2(ap−δ)−1 . Now choose 0< ε < δ and note that by (9.8) we have (Bn)−1 nap+ε for all largen.
Using assumption (A1) we obtain for all largen P
λdn> n2(ap+δ)−1
=P
Mn > n2(ap+δ)−1
P(((Bn)−1((2 nBnMnBn > n2(ap+δ)
P
nBnMnBn > n2(δ−ε) and the last probability tends to zero asn→ ∞.
Now fix anyθ0∈ ¯Lp\ ¯Lp−1and write (Bn)−1θ0=rnθn for some unit vectorθn and rn>0. Theorem 4.3.14 of Meerschaert and Scheffler (2001a) shows that every limit point of(θn)lies in the unit sphere inVp. Then since (9.5) holds uniformly on compact sets we have for any 0< ε < δthatnap−εrnnap+εfor all largen. Then for all largenwe get
P
λdn< n2(ap−δ)−1
=P )
maxθ =1Mnθãθ < n2(ap−δ)−1
*
P
Mnθ0ãθ0< n2(ap−δ)−1
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 629
=P
nBnMnBnθnãθn< rn−2n2(ap−δ)−1
P
nBnMnBnθnãθn< n2(ε−δ) .
Given any subsequence (n)there exists a further subsequence(n)⊂(n)along which θn→θ. Furthermore, by assumption (A1) there exists another subsequence(n)⊂(n) such thatnBnMnBn ⇒Malong(n). Hence by continuous mapping [see Theorem 1.2.8 in Meerschaert and Scheffler (2001a)] we have
nBnMnBnθnãθn⇒Mθãθ along(n).
Now, given anyε1>0 by assumption (A2)(b) there exists aρ >0 such thatP{Mθãθ < ρ}
< ε1/2. Hence for all largen=nwe have P
nBnMnBnθnãθn< n2(ε−δ)
P{nBnMnBnθnãθn< ρ} P{Mθãθ < ρ} +ε1
2
< ε1.
Since for any subsequence there exists a further subsequence along which P
nBnMnBnθnθn< n2(ε−δ)
→0,
this convergence holds along the entire sequence which concludes the proof.
Proposition 9.3. Under the basic assumptions we have log(nλ1n)
2 logn →a1 in probability.
Proof: Since the set GL(Rd)of invertible matrices is an open subset of the vector space ofd×d real matrices, it follows from (A1) and (A2)(a) together with the Portmanteau Theorem [cf., Theorem 1.2.2 in Meerschaert and Scheffler (2001a)] that limn→∞P{Mn∈ GL(Rd)} =1 holds. Hence we can assume without loss of generality thatMnis invertible for all largen.
Given anyδ >0 write P!!
!!log(nλ1n) 2 logn −a1
!!!!> δ
P
λ1n> n2(a1+δ)−1 +P
λ1n< n2(a1−δ)−1 .
To estimate the first probability on the right-hand side of the inequality above choose a unit vectorθ0∈ ¯L1and write(Bn)−1θ0=rnθnas above. Then, since (9.5) holds uniformly
630 M.M. Meerschaert and H.-P. Scheffler
on the unit sphere inL¯1=V1, for 0< ε < δ we havena1−εrnna1+εfor all largen.
Therefore for all largen P
λ1n> n2(a1+δ)−1
P
) min
θ =1Mnθãθ > n2(a1+δ)−1
*
P
Mnθ0ãθ0> n2(a1+δ)−1
P
nBnMnBnθnãθn> n2(δ−ε) .
It follows from assumption (A1) together with the compactness of the unit sphere inRd and continuous mapping that the sequence(nBnMnBnθnãθn)is weakly relatively compact and hence by Prohorov’s Theorem this sequence is uniformly tight. Sinceδ > εit follows thatP{λ1n> n2(a1+δ)−1} →0 asn→ ∞.
Since the smallest eigenvalue ofMnis the reciprocal of the largest eigenvalue ofMn−1 we have
P
λ1n< n2(a1−δ)−1
=P 1
λ1n
> n2(δ−a1)+1
=P )
maxθ =1Mn−1θãθ > n2(δ−a1)+1
*
=P((Mn−1((> n2(δ−a1)+1
P((
((1
n(Bn)−1Mn−1Bn−1((
((> Bn −2n2(δ−a1)
.
It follows from (9.4) that for any 0< ε < δthere exists a constantC >0 such that Bn Cn−a1+ε for allnand hence for some constantK >0 we get Bn −2Kn2(a1−ε)for alln. Note that by assumptions (A1) and (A2)(a) together with continuous mapping the sequence
1
n(Bn)−1Mn−1Bn−1
is weakly relatively compact and hence by Prohorov’s theorem this sequence is uniformly tight. Hence
P((
((1
n(Bn)−1Mn−1Bn−1((
((> Bn −2n2(δ−a1)
P((
((1
n(Bn)−1Mn−1Bn−1((
((> Kn2(δ−ε)
→0 asn→ ∞. This concludes the proof.
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 631
Proof of Theorem 9.1: LetCj denote the collection of all orthogonal projections onto subspaces ofRd with dimensionj. The Courant–Fischer Max–Min Theorem [see, Rao (1965, p. 51)] implies that
λj n= min
P∈Cj
max
θ =1P MnP θãθ
= max
P∈Cd−j+1
minθ =1P MnP θãθ . (9.9)
Note thatPi2=Piand thatBnandPicommute for alln, i. Furthermore(PiBn)is regularly varying with indexEi⊕ ã ã ã ⊕Ep. Since
n(PiBn)PiMnPi(BnPi)=nPi(BnMnBn)Pi
it follows by projection from our basic assumptions that the sample covariance matrix formed from theLi valued random variablesPiXt satisfies again those basic assumptions withE=Ei⊕ ã ã ã ⊕EponLi. Hence ifλndenotes the smallest eigenvalue of the matrix PiMnPi it follows from Proposition 9.3 that
log(nλn)
2 logn →ai in probability.
Similarly, the sample covariance matrix formed in terms of theL¯i-valued random vec- torsL¯iXt again satisfies the basic assumptions withE=E1⊕ ã ã ã ⊕Ei as above. Then, ifλ¯ndenotes the largest eigenvalue of the matrixPiMnPi it follows from Proposition 9.2 above that
log(n¯λn)
2 logn →ai in probability.
Now apply (9.9) to see that λnλj nλ¯n
wheneverb¯i−1< jb¯i. The result now follows easily.
After dealing with the asymptotics of the eigenvalues of the sample covariance in The- orem 9.1 above we now investigate the convergence of the unit eigenvectors ofMn. Recall thatπi:Rd→Vi denotes the orthogonal projection ontoVi fori=1, . . . , p. Define the random projection
πin(x)=
b¯i
j= ¯bi−1+1
(xãθj n)θj n.
632 M.M. Meerschaert and H.-P. Scheffler
Theorem 9.4. Under the basic assumptions we haveπ1n→π1andπpn→πpin proba- bility asn→ ∞.
Again the proof is quite similar to the proof of Theorem 3 in Meerschaert and Schef- fler (1999b) and Theorem 10.4.8 in Meerschaert and Scheffler (2001a). See also Scheffler (1998). We include here a sketch of the arguments.
Proposition 9.5. Under our basic assumptions we have:Ifj >b¯p−1andr < pthen πrθj n→0 in probability.
Proof: Sinceπrθj n=(πrMn/λj n)θj nwe get πrθj n ((
((πrMn λj n
(((( πrBn−1 nBnMnBn (Bn)−1 nλj n
.
By assumption (A1) together with continuous mapping it follows from Prohorov’s theorem that(n BnMnBn )is uniformly tight. Also, by (9.7), (9.8) and Theorem 9.1 we get
log( πrBn−1 nBnMnBn (Bn)−1 )/(nλj n) logn
=log πrBn−1
logn +log (Bn)−1
logn −log(nλj n) logn
→ar+ap−2ap<0 in probability.
Hence the assertion follows.
Proposition 9.6. Under our basic assumptions we have:Ifjb¯1andr >1then πrθj n→0 in probability.
Proof: Sinceπrθj n=(πrMn−1λj n)θj nwe get πrθj n ((πrMn−1λj n(( πrBn ((
((1
n(Bn)−1Mn−1Bn−1((
(( Bn (nλj n).
As in the proof of Proposition 9.3 the sequence(1n(Bn)−1Mn−1Bn−1 )is uniformly tight and now the assertion follows as in the proof of Proposition 9.5.
Proof of Theorem 9.4: The proof is almost identical to the proof of Theorem 3 in Meer- schaert and Scheffler (1999b) or Theorem 10.4.8 in Meerschaert and Scheffler (2001a) and therefore omitted.
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 633
Corollary 9.7. Under our basic assumptions, ifp3 thenπin→πi in probability for i=1, . . . , p.
Proof: Obvious.
Example 9.8. Suppose thatZ,Z1,Z2, . . .is a sequence of independent and identically distributed (IID) random vectors with common distributionà. We assume thatàis regu- larly varying with exponentE. That means that there exists a regularly varying sequence (An)of linear operators with index−Esuch that
n(Anà)→φ asn→ ∞. (9.10)
For more information on regularly varying measures see Meerschaert and Scheffler (2001a, Chapter 6).
Regularly varying measures are closely related to the generalized central limit theorem discussed in Section 3. Recall that if
An(Z1+ ã ã ã +Zn−nbn)⇒Y asn→ ∞ (9.11) for some nonrandombn∈Rd, we say thatZ belongs to the generalized domain of at- traction of Y and we writeZ∈GDOA(Y). Corollary 8.2.12 in Meerschaert and Schef- fler (2001a) shows that Z ∈GDOA(Y) and (9.11) holds if and only if à varies regu- larly with exponentEand (9.10) holds, where the real parts of the eigenvalues ofEare greater than 1/2. In this case, Y has an operator stable distribution and the measure φ in (9.10) is the Lévy measure of the distribution ofY. Operator stable distributions and Lévy measures were discussed in Section 4, where (9.10) is written in the equivalent form nP (AnZ∈ dx)→φ(dx). The spectral decomposition was discussed in Section 5. Theo- rem 8.3.24 in Meerschaert and Scheffler (2001a) shows that we can always choose norm- ing operatorsAnand limitY in (9.11) so thatYis spectrally compatible withZ, meaning thatAnvaries regularly with some exponent−E, the subspacesVi in the spectral decom- position ofRd with respect toE are mutually orthogonal, and these subspaces are also An-invariant for everyn. In this case, we writeZ∈GDOAc(Y).
Recall from Section 6 that, since the real parts of the eigenvalues of E are greater than 1/2,
nAnMnAn⇒W asn→ ∞, (9.12)
whereMnis the uncentered sample covariance matrix Mn=1
n n i=1
ZiZi
andW is a randomd×d matrix whose distribution is operator stable. Theorem 10.2.9 in Meerschaert and Scheffler (2001a) shows thatW is invertible with probability one, and
634 M.M. Meerschaert and H.-P. Scheffler
Theorem 10.4.2 in Meerschaert and Scheffler (2001a) shows that for all unit vectorsθ∈Rd the random variableθãW θ has a Lebesgue density. Then the basic assumptions of this section hold, and hence the results of this section apply.
The tail estimator proven in this section approximates the spectral index functionα(x) defined in (5.2). This index function provides sharp bounds on the tails and radial projec- tion moments ofZ. Given ad-dimensional data setZ1, . . . ,Znwith uncentered covariance matrixMn, let 0λ1nã ã ãλdndenote the eigenvalues ofMnandθ1n, . . . , θdnthe cor- responding orthonormal basis of eigenvectors. Writing xj =xãθj we can estimate the spectral indexα(x)by
ˆ
α(x)=min{ ˆαj: xj=0}, whereαˆj= 2 logn log(nλj n)
using the results of this section. Hence the eigenvalues are used to approximate the tail behavior, and the eigenvectors determine the coordinate system to which these estimates pertain. A practical application of this tail estimator appears in Example 8.1.
Example 9.9. The same tail estimation methods used in the previous example also apply to the moving averages considered in Section 7. This result is apparently new. Given a sequence of IID random vectorsZ,Zjwhose common distributionàvaries regularly with exponentE, so that (9.10) holds, we define the moving average process
Xt= ∞
j=−∞
CjZt−j, (9.13)
where we assume that thed ×d matricesCj fulfill for each j eitherCj =0 or Cj is invertible andAnCj=CjAnfor alln. Moreover ifap denotes the largest real part of the eigenvalues ofEwe assume further
∞ j=−∞
Cj δ<∞ (9.14)
for someδ <1/ap withδ1. Recall from Section 7 that under those conditionsXt is almost surely well defined, and that if the real parts of the eigenvalues ofE are greater than 1/2 we have that
nAnΓ%n(0)An⇒M= ∞ j=−∞
CjW Cj asn→ ∞, (9.15)
where the sample covariance matrix Γ%n(h) is defined by (7.6) and W is a random d×dmatrix whose probability distribution is operator stable. Suppose that the norming
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 635
operatorsAn are chosen so that (9.11) holds andZ∈GDOAc(Y). Then in view of our basic assumptions (A1) and (A2) it remains to show:
Lemma 9.10. Under the assumptions of the paragraph above the limiting matrix M in (9.15)is a.s. positive definite and for any unit vectorθthe random variableMθãθhas no atom at zero.
Proof: SinceWin (9.12) is a.s. positive definite we have for anyθ=0 thatCjW Cjθãθ= W CjθãCjθ0 for allj and strictly greater that zero for thosejwithCj=0. Hence
Mθãθ= ∞
j=−∞
CjW Cjθãθ >0
for anyθ=0 soMis positive definite.
Moreover if for a given unit vectorθ we setzj=Cjθ thenzj0 =0 for at least onej0. SinceW is almost surely positive definite we have
P{Mθãθ < t} =P ∞
j=−∞
Wzjãzj< t P{Wzj0ãzj0< t} →0
ast→0 using the fact thatWzj0ãzj0 has a Lebesgue density as above. HenceMθãθhas no atom at zero.
It follows from (9.15) together with Lemma 9.10 that the Xt defined above fulfill the basic assumptions of this section. Hence it follows from Theorems 9.1 and 9.4 that the tail estimator used in Example 9.8 also applies to time-dependent data that can be modeled as a multivariate moving average. We can also utilize the uncentered sample covariance matrix (6.3), which has the same asymptotics as long asEZ=0 [cf. Theorem 10.6.7 and Corollary 10.2.6 in Meerschaert and Scheffler (2001a)]. In either case, the eigenvalues can be used to approximate the tail behavior, and the eigenvectors determine the coordinate system in which these estimates apply.
Example 9.11. Suppose now thatZ1,Z2, . . .are IIDRd-valued random vectors with com- mon distributionà. We assume thatàis ROV∞(E, c), meaning that there exist(An)reg- ularly varying with index−E, a sequence(kn)of natural numbers tending to infinity with kn+1/kn→c >1 such that
kn(Aknà)→φ asn→ ∞. (9.16)
See Meerschaert and Scheffler (2001a, Section 6.2) for more information on R–O varying measures.
636 M.M. Meerschaert and H.-P. Scheffler
R–O varying measures are closely related to a generalized central limit theorem. In fact, ifàis ROV∞(E, c)and the real parts of the eigenvalues ofEare greater than 1/2 then (9.16) is equivalent to
Akn(Z1+ ã ã ã +Zkn−knbn)⇒Y asn→ ∞,
whereY has a so called (cE, c) operator semistable distribution. See Meerschaert and Scheffler (2001a), Sections 7.1 and 8.2 for details. Once again, a judicious choice of norm- ing operators and limits guarantees thatYis spectrally compatible withZ, so thatAnvaries regularly with some exponent−E, the subspacesVi in the spectral decomposition ofRd with respect toEare mutually orthogonal, and these subspaces are alsoAn-invariant for everyn. It follows from Theorem 8.2.5 of Meerschaert and Scheffler (2001a) thatZ has the same moment and tail behavior as for the generalized domain of attraction case consid- ered in Section 5. In particular, there is a spectral index functionα(x)taking values in the set{a1−1, . . . , ap−1}wherea1<ã ã ã< apare the real parts of the eigenvalues ofE. Given x=0, for any smallδ >0 we have
r−α(x)−δ< P
|Zãx|> r
< r−α(x)+δ
for allr >0 sufficiently large. ThenE(|Zãx|β)exists for 0< β < α(x)and diverges for β > α(x).
Now let Mn=1
n n i=1
ZiZi
denote the sample covariance matrix of(Zi). Then it follows from Theorem 10.2.3, Corol- laries 10.2.4 and 10.2.6, Theorem 10.2.9, and Lemma 10.4.2 in Meerschaert and Schef- fler (2001a) thatMnfulfills the basic assumptions (A1) and (A2) of this section. Hence, by Theorems 9.1 and 9.4 we rediscover Theorems 10.4.5 and 10.4.8 of Meerschaert and Scheffler (2001a). See also Scheffler (1998). In other words, the approximationα(x)ˆ from Example 9.8 still functions in this more general case, which represents the most general setting in which sums of IID random vectors can approximated in distribution via a central limit theorem.