Journal: LAA
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Contents lists available atScienceDirect
Linear
Algebra
and
its
Applications
www.elsevier.com/locate/laa 1Q1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14Q2 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31Q4 31 32 32 33 33 34 34 35 35 36 36 37 37 38Q3 38 39 39 40 40 41 41 42 42Eigenvalue-eigenvector
structure
of
Schoenmakers–Coffey
matrices
via
Toeplitz
technology
and
applications
E. Salinellia, S. Serra-Capizzanob,c,∗, D.Sesanab
a
DipartimentodiStudiperl’Economiael’Impresa,UniversitàdelPiemonte Orientale“AmedeoAvogadro”Alessandria,Novara,Vercelli,Via Perrone 18, 28100Novara, Italy
bDipartimentodiScienzaeAltaTecnologia,Universitàdell’Insubria,
Via Valleggio 11,22100Como, Italy
cDepartmentofInformationTechnology,UppsalaUniversity,Box337,SE-75105
Uppsala, Sweden a r t i c l e i n f o a b s t r a c t Articlehistory: Received12November2014 Accepted11March2015 Availableonlinexxxx Submittedby L.Cvetkovic MSC: 15B05 15A18 47B36 62H25 Keywords: Schoenmakers–Coffeymatrices Greenmatrices Tridiagonalmatrices Spectraldistribution GLTsequence Eigenvectoroscillation
The paper is concerned with the spectral properties of Green matrices and of a special subclass of the latter, known as Schoenmakers–Coffey matrices, which have a role in financial applications. The main results are related to the eigenvalue distribution of sequences of Green matrices of increasing size, while for the subclass of interest mentioned above, we also study the eigenvector oscillation structure: interestingly enough, even if these matrices are not shift invariant (Toeplitz), the results are obtained by using tools coming from Toeplitz technology. Indeed, for the asymptotic spectral distribution analysis, we use the theory of Generalized Locally Toeplitz sequences, while techniques taken from the study of Kac–Murdoch–Szegö matrices (again connected to Toeplitz matrices) are employed for the eigenvector oscillation structure results of the Schoenmakers–Coffey
* Corresponding author at: Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio11,22100Como,Italy.
E-mailaddresses:[email protected](E. Salinelli), [email protected],
[email protected](S. Serra-Capizzano), [email protected](D. Sesana).
http://dx.doi.org/10.1016/j.laa.2015.03.017
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33Q5 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
matrices. Few numerical tests are reported in order to illustrate the theoretical findings.
© 2015 Published by Elsevier Inc.
1. Introduction
ThepaperisconcernedwiththespectralpropertiesofGreenmatricesandofaspecial subclassofthelatter,knownasSchoenmakers–Coffeymatrices.Themainresultsconcern theeigenvaluedistributionofsequencesofGreenmatricesofincreasingsizeinthesense ofWeyl[5]:inparticular,despiteitsapparentcomplicateexpression,weprovethatthese matricescanbeseenastheproductofdiagonalmatricesandinverseofToeplitzmatrices generated bylineartrigonometricpolynomials.Undertheassumption thatthediagonal matrices can be seen as samplings of Riemann integrable functions over the interval [0,1], we provethatthe limit distribution ofthe spectra of the consideredsequence is independent ofsuchfunctionsand indeedequals theconstantzero.Inother words,the Green matrixsequence {Gn}showsaclusteratzerooftheeigenvalues,i.e.,forevery, for thesizen largeenough, mostoftheeigenvaluesofGn havemodulusboundedby, except o(n) of them.In practice,as shown inthe numericalexperiments, the quantity indicated as o(n) behaves just as O(√n), so itseems thatthere isroom for improving thetheoretical result.
ConcerningthesubclassoftheSchoenmakers–Coffeymatriceswealsostudythe eigen-vector oscillation behavior.We recall thatthese structures appearas special instances of correlationmatricesand theoscillationsofthefirstthree eigenvectorsprovide useful information intrendyfinancialproblems,associated withinterestratesmodelsand risk management/valuation(seeSection4forabriefaccountand[12,14,17]formoredetails). Even ifthesematrices arenotshift invariant,thatis theydonotenjoytheToeplitz structure,itisworthstressingthattheresultsareobtainedbyusingtoolscomingfrom Toeplitztechnology.Indeed,concerningtheasymptoticspectralresults,weusethetheory of Generalized Locally Toeplitz (GLT) sequences [23,24], while, for thesecond typeof results, tools used in the study of Kac–Murdoch–Szegö matrices (again connected to Toeplitz matrices)areemployed[10].
The paper is organized as follows. In Section2 we report the definition of spectral distribution in the sense of Weyl and we briefly introduce Toeplitz matrices and GLT matrix sequences. In Section3 we introduce the Green matrices and we use the tools of the previous section in order to identify the distribution symbol of Green matrix sequences: selectednumericaltestsarereportedandcommented.InSection4wedefine thesubclassoftheSchoenmakers–Coffeymatricesandwestudytheoscillatorybehavior of the eigenvectors, by including also some examples. Finally Section 5 is devoted to conclusionsandto stressfewopenproblems.
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
2. AsymptoticanalysisofGreenmatrixsequencesviatheGLTtheory
The section contains the mathematical tools that we use for studying the global behavior ofthe spectra ofGreen matrixsequences. In Subsection2.1 we introducethe notionofspectraldistributionintheWeylsense [5,8]andinSubsection2.2wegivethe definitionof Toeplitzmatrices generated byasymbol (see[5])and of theclass ofGLT sequences(see[23,24]),whose mainpropertiesarereportedand discussed.
2.1. Definitions anddistributionresults
Before starting, let us introduce some notations. We denote by C0(C) and C0(R+0)
theset ofcontinuousfunctionswith boundedsupportdefinedover Cand R+0 = [0,∞), respectively. Given afunction F : C→ C and amatrix A of size m, with eigenvalues
λj(A) andsingularvaluesσj(A), j= 1,. . . ,m,weset
Σλ(F, A) := 1 m m j=1 F(λj(A)), Σσ(F, A) := 1 m m j=1 F(σj(A)).
Weuse the notation Ap for the Schatten p-norm of A, defined as the p-norm of the
vector formed by the singular values of A. In symbols, Ap = ( m
j=1σ
p
j(A))1/p for
1≤p<∞andA∞= maxj=1,...,mσj(A)=Aistheusualspectralnorm[4]. Definition 2.1. Let f :G → Cbe acomplex-valued measurable function, defined ona measurable set G⊂R with finite andpositive Lebesguemeasure, 0< μ(G)<∞. Let
{An}be amatrix-sequence,withAn ofsizedn,dn < dn+1.Wesaythat:
• {An} is distributed as the pair (f,G) in the sense of the eigenvalues, in symbols
{An}∼λ(f,G),ifforallF∈ C0(C) wehave
lim n→∞Σλ(F, An) = 1 μ(G) G F(f(t))dt. (1)
• {An}isdistributedasthepair(f,G)inthesense of thesingularvalues,insymbols {An}∼σ(f,G),ifforallF ∈ C0(R+0) wehave
lim n→∞Σσ(F, An) = 1 μ(G) G F(|f(t)|)dt. (2)
Finallywesaythattwosequences{An}and{Bn}areequallydistributed inthesense of theeigenvalues[30] if,∀F ∈ C0(C),wehave
lim
n→∞[Σλ(F, An)−Σλ(F, Bn)] = 0.
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
Noticethatiftwosequencesareequallydistributedandoneofthemhasadistribution function,then theother necessarilyhasthesamedistributionfunction.
Amatrixsequence{An}isdistributedintheeigenvaluesenseasthepair(f,G) ifand
onlyifthesequenceoflinearfunctionals {φn}definedbyφn(F)= Σλ(F,An) converges
weak-∗to thefunctional
φ(F) = 1
μ(G)
G
F(f(t))dt,
asin(1),thesameistrueforthesingularvalues.Ausefultoolforthestudyofthespectral distributionofamatrixsequenceisthenotionof approximatingclassofsequences.
Definition2.2.Supposeasequenceofmatrices{An},An ofsizedn,dn< dn+1,isgiven.
Wesaythat{{Bn,m}:m∈N}misanapproximatingclassofsequences(a.c.s.)for{An}
if,forallsufficientlylargem∈N,thefollowing splittingshold:
An =Bn,m+Rn,m+Nn,m,
with
rank(Rn,m)≤dnc(m), Nn,m ≤ω(m), ∀n > nm,
where nm,c(m) andω(m) dependonlyonmand
lim
m→∞c(m) = 0, mlim→∞ω(m) = 0.
Proposition 2.1. (See [22].) Suppose a sequence of matrices {An}, An of size dn, dn < dn+1, is givenand let {{Bn,m}:m ∈N}m be an a.c.s. for{An} in the sense of
Definition 2.2.Supposethat,forallsufficientlylargem∈Nwehave{Bn,m}∼σ(fm,G),
and limm→∞fm=f.Thenitholds{An}∼σ(f,G).
Similarly, if the matrices An and Bn,m are eventually Hermitian, and for all
suffi-cientlylargem∈Nwehave{Bn,m}∼λ(gm,G),withlimm→∞gm=g,thenitalsoholds {An}∼λ(g,G).
Finallywe introducethedefinitionofsparselyvanishing sequence:asequence of ma-trices{An},An ofsizedn,issaidtobesparselyvanishingif,foreachM >0 thereexists annM suchthatforn≥nM wehave
#{i:σi(An)< M−1} ≤r(M)dn, lim
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
2.2. Toeplitz, LocallyToeplitzand GeneralizedLocallyToeplitzsequences
Given an integrable complex-valued function f, f ∈ L1(Q), Q = (−π,π), from the Fouriercoefficientsoff aj = 1 2π Q f(x)e−ijxdx, j∈Z, i2=−1,
wecanbuild thesequenceofToeplitzmatrices{Tn(f)}asfollows:
Tn(f) := [ai−j]ni,j=1.
The eigen/singular values asymptotic distribution of a sequence of Toeplitz matrices, startedinafamoustheorem bySzegö[8],hasbeenstudiedbymanyauthors(see[5,26, 31]andthereferencesreportedtherein).Theresultsarereportedbelow.
Theorem2.1. (See[31].) Iff ∈L1(Q) and{Tn(f)}isthesequenceof Toeplitzmatrices
generatedby f,then
{Tn(f)} ∼σ (f, Q).
Moreover,if f isalso real-valued,theneach matrixTn(f)isHermitianand
{Tn(f)} ∼λ(f, Q). (3)
Nowweintroducethenotionof(unilevel)LocallyToeplitzmatrix-sequencethatleads toageneralizationof(unilevel)Toeplitzsequences.Wementionthat,withrespecttothe originalpaperbyTilli[27],thedefinitionswilltakeintoaccountveryminorimprovements (asdiscussed inRemark 1.1of [23]).We recallthat,giventwo matrices A∈Cn×n and B∈Cm×m,theirdirectsumisdefinedas
A⊕B= A O O B ∈C(n+m)×(n+m),
where O isthenullmatrix,and thetensorproduct A⊗B ∈Cnm×nm isdefinedas the n×n block matrixwith m×m blocks,whose block (i,j), i,j = 1,. . . ,m, is given by
ai,jB.
Definition2.3.Asequenceofmatrices{An},An ofsizen,issaidto beLocallyToeplitz with respect to a pair of functions (a,f), with a : [0,1] → C and f : Q → C, if f is Lebesgueintegrableand, forallsufficientlylargem∈N,thefollowingsplittingholds:
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 with rank(Rn,m)≤c(m), Nn,m1≤ω(m)n, ∀n > nm,
where nm, c(m) and ω(m) are functions of m and limm→∞ω(m) = 0. The matrix LTnm(a,f) in (4)isdefinedas
LTnm(a, f) =Dm(a)⊗Tn/m(f)⊕Onmodm,
where n/misthe integerpartofm/nandnmodm=n−mn/m(itisunderstood thatthezeroblockOnmodmisnotpresentifnisamultipleofm).MoreoverDm(a) isthe m×mdiagonalmatrixwhoseentriesare givenbya(j/m),j= 1,. . . ,m,Tk(f) denotes
the Toeplitz matrixof order kgenerated by f and Oq isthe nullmatrixof order q. In
this casewe writeinshort {An}∼LT(a,f).
The topologicalclosure ofthe spaceof LocallyToeplitz sequencesis thatformed by GeneralizedLocallyToeplitzsequences.
Definition 2.4. A sequenceof matrices {An}, An ofsize n, isapproximated byunilevel
LocallyToeplitzsequences withrespecttoameasurablefunctionκif,forevery>0,
• there exist pairs of functions {(ai,,fi,)}N
i=1 with fi, polynomial and ai, defined
overΩ= [0,1] suchthatN
i=1ai,fi,−κconvergeinmeasuretozerooverΩ×Qas
tendstozero,
• thereexist matrixsequences{{A(ni,)}}Ni=1 suchthat{A(ni,)}∼LT(ai,,fi,)
• {{N i=1A
(i,)
n }:= (m+ 1)−1,m∈N}misana.c.s.for{An}.
In this case the sequence {An} is said to be a Generalized Locally Toeplitz sequence
with respecttoκandwewrite inshort {An}∼GLTκ.
Someremarksareinorder.When{An}∼LT(a,f),wecallatheweightfunction,and
f thegeneratingfunction.Furthermore,inthesplitting(4),thematricesRn,marecalled
rankcorrections,whilethematricesNn,m arecallednormcorrections.
If {An}∼GLT κ, itisevidentthatthe uniquefunctionκhassimultaneouslytherole
of weightfunctionand ofgeneratingfunction:wecall κthekernel functionorsymbol.
FortheclassofGeneralizedLocallyToeplitzsequencesthefollowingSzegö-likeresult holds.
Theorem 2.2. (See [23].) Assume that {An}, An of size n, is a sequence of complex
matrices.Letκbe measurableover Ω×Q.Then
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
holds whenever {An}is aGLT sequencewith respect toκ asin Definition 2.4 and the
functionsai,involvedinDefinition 2.4areRiemannintegrableoverΩ.Ifinadditionthe
matricesAn are eventuallyHermitian,then therelationistruealso fortheeigenvalues.
TheGLT sequencesform a∗-algebra. Moreprecisely, the GLTsequences are stable underlinearcombinations,product,pseudo-inversion,and adjoint.
Theorem2.3. (See[23,24].)If {An}∼GLTκA and{Bn}∼GLTκB,thenwehave
• {αAn+βBn}∼GLT ακA+βκB;
• if the functions a(i,A) and a(i,B) involved in Definition 2.4 are Riemann integrable over Ω,then {AnBn}∼GLTκAκB;
• if thefunction a(i,A) involved in Definition 2.4 is Riemann integrable over Ω andif
{An} is sparsely vanishing then {A+n} ∼GLT κ−A1, where A+n is the pseudo inverse
of An (we canreplacethesuperscript+with −1if An isalsoinvertible);
• {A∗n}∼GLTκ∗A,where A∗n denotetheconjugate transpose ofAn.
TheclassofGLTsequencescontainsallToeplitzsequences,generatedbyL1symbols,
anddiagonalsequences,obtainedasauniformsamplingofRiemannintegrablefunctions.
Theorem 2.4. (See [23].) If {Tn(f)} is a sequence of Toeplitz matrices generated by f ∈L1(Q),then{Tn(f)}isaGLTsequencewith respectto thefunctionf.
Theorem 2.5. (See [23].) If {Dn(a)} is a sequence of diagonal matrices [Dn(a)]i,i = a(i/n), i = 1,. . . ,n, generated by a Riemann integrable function a, then {Dn(a)} is a
GLTsequencewithrespecttothefunctiona.
3. Greenmatrices
AGreenmatrixGn= [gij]ni,j=1 [16]isdefinedby
gij =
aicj if i≤j,
ajci if i > j, ai, cj∈R\ {0}. (5)
Byconstruction,Gnissymmetric.Furthermore,itcanbeprovedthatGnisanonsingular GreenmatrixifandonlyifitsinverseG−1
n isasymmetrictridiagonalmatrixwithnonzero
superdiagonalelements, whose explicitform is(see [3,2,11] and [15] for astudy of the conditioning)
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 G−n1 ij= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 aici+1−ai+1ci i=j−1, 1 ai−1ci−aici−1 i=j+ 1, ai+1ci−1−ai−1ci+1
(aici−1−ai−1ci)(ai+1ci−aici+1) i=j = 1, n, a2 a1(a2c1−a1c2) i=j = 1, cn−1 cn(ancn−1−an−1cn) i=j =n, 0 otherwise. (6)
Hence, thepair (λ,h),λ∈R and h∈Rn, h= [h
1,. . . ,hn]T, isan eigenpair ofG−n1 if
and onlyifitsatisfies thediscrete Sturm–LiouvilleProblem (SLP)
−Δ (pi−1Δhi−1) = (μ−qi)hi i= 1, . . . , n, h0= 0; hn+1= 0 (7) with μ= 1/λ, and Δhi=hi+1−hi, pi= 1 ai+1ci−aici+1 , i= 0, . . . , n, qi=ai−1(ci−ci+1) +ai(ci+1−ci−1)−ai+1(ci−ci−1) (aici−1−ai−1ci)(ai+1ci−aici+1) , i= 1, . . . , n,
where,byconvention,wehaveseta0=cn+1= 0 andan+1=c0= 1.
RegardingthespectralpropertiesofGreenmatrices,werecallaresultontheordering oftheeigenvaluesandontheoscillatorypropertiesoftheeigenvectorsthatpassesthrough the theory of totally positivematrices [11,16]. We first introduce adefinition and two theorems.
Definition 3.1. Ann×nmatrix An iscalled: (strictly) totally positive, denotedbyTP (STP),ifallitsminorsarenonnegative(positive);oscillatoryifitisTPandthereexists aq∈N\{0}suchthatAqisSTP.
Of course,anSTPmatrixisalsooscillatorywithq= 1.
Theorem 3.1. (See [11], Theorem 3.1 of Chapter 3.) A Greenmatrix Gn is TP if and
only if {ai} and {ci} have the same strict sign and {ai/ci} is increasing. Moreover,
Gn−1
n isSTPifandonlyif{ai}and{ci}havethesamestrictsignand{ai/ci}isstrictly
increasing.
Theimportanceofbeing TPiswell-illustratedbythefollowingwell-knownresult.
Theorem 3.2. An n×n oscillatorymatrix An has eigenvalues λ1 > λ2 >· · ·> λn >0
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
Regardingthe previousresult, werecall thatthesymbol S(vk) denotesthe number
ofsignvariationsof vk thatcoincideswiththecommonvalueS−(vk) =S+(vk),where S−(vk) isthenumberofsignchangesof vkwhenzerotermsarediscarded,andS+(vk)
is the maximum number of sign changes of vk when values +1 or −1 are arbitrarily
assigned to zeroterms. Notice thatS+(v
k) and S−(vk) can coincide only if the first
andthelastcomponentsof vkarenotzeroandifforeveryzerocomponentthepreceding
andthefollowingcomponentsarenotzeroandof differentsign.
Inthe following,wewill be interested incharacterizing the asymptoticspectral dis-tribution of specific sequences of Green matrices and in analyzing, when possible, the monotonicity propertiesof the components of the eigenvectors; the latter analysis has animpactfromaneconomicviewpoint(see[12,14,17]).
3.1. Spectral distributionof sequencesof Greenmatrices
InthissectionweusethetheoryreportedinSection2inordertoshowthatasequence ofGreenmatricesisaGLTsequence:asanoteworthyresultweobtainthatthespectral distributionisindependentofthepossibleparametersandinfactitisequaltozero.
Theorem 3.3. Suppose that aj and cj in (5) are equally spaced sampling of Riemann
integrable real valued functions a(x) and c(x), respectively, on the interval [0,1], i.e.
aj=a(j/n) andcj =c(j/n),j= 1,. . . ,n,then{Gn}∼σ0and{Gn}∼λ0. Proof. ThematrixGn in(5)canbewrittenas
Gn=Dn(a)Dn(c) +Dn(a)Tn(1)Dn(c) +Dn(c)Tn(2)Dn(a), (8) where Dn(a) = diag{a1, a2, . . . , an}, Dn(c) = diag{c1, c2, . . . , cn}, and Tn(1)= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 · · · 1 .. . . .. ... ... .. . . .. 1 0 · · · · 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦, T (2) n = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 · · · · 0 1 . .. ... .. . . .. ... ... 1 · · · 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦.
Westudyseparatelythefour sequences{Dn(a)},{Dn(c)},{Tn(1)}and{Tn(2)}.
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
ConsidertheToeplitzmatrixTn(1),wecanwrite
Tn(1) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 · · · 1 .. . . .. ... ... .. . . .. 1 0 · · · · 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 −1 0 0 . .. ... .. . . .. ... −1 0 · · · 0 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ −1 −I=Tn(f1)−1−I,
with f1(θ) = 1 −e−iθ ∈ L1(Q). By Theorem 2.4 {Tn(f1)} ∼GLT f1. Since the
se-quence{Tn(f1)}issparselyvanishing(Tn(f1) isinvertible),fromTheorem 2.3weobtain {Tn(f1)−1}∼GLTf1−1.
Thesamereasoningcanbe repeatedverbatimforTn(2):
Tn(2) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 · · · · 0 1 . .. ... .. . . .. ... ... 1 · · · 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 · · · 0 −1 . .. ... ... . .. ... 0 0 −1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ −1 −I=Tn(f2)−1−I,
with f2(θ)= 1−eiθ∈L1(Q).Againwecanconcludethat{Tn(f2)−1}∼GLT f2−1.
Using Theorem 2.3,putting togetherthepreviousresultswehave
{Gn} ∼GLTa(x)c(x) +a(x) 1 1−e−iθ −1 c(x) +c(x) 1 1−eiθ−1 a(x) =a(x)c(x) 1 + 1 1−e−iθ −1 + 1 1−eiθ −1 = 0,
thatis {Gn}∼GLT0.Theorem 2.2ensuresthat{Gn}∼σ 0 and{Gn}∼λ0 since Gn is
also Hermitian. 2
Remark 3.1. We canprove Theorem 3.3 using a result on the spectral distribution of nonscaledsamplingmatricesobtainedin[1],butagainwiththesupportofGLTtheorems. The Green matrix Gn in(5) can be obtainedas anonscaled sampling of theGreen function(kernel)definedas
K(x, y) = a(x)c(y) 0< x≤y≤1, a(y)c(x) 0< y≤x≤1, =a(x)c(y) + 0 0< x≤y≤1, a(y)c(x)−a(x)c(y) 0< y≤x≤1, (9)
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
thatis,K(x,y) isthesumofaseparablefunctiona(x)c(y) andafunctionφ(x,y) whose associatednonscaled samplingmatrix(usingthenotationin(8))isgiven by
Φn:=Dn(c)Tn(2)Dn(a)−Dn(a)Tn(2)Dn(c).
Ifa(x) and c(y) areRiemannintegrablerealvaluedfunctions,using theGLTtheorems as in theproof of Theorem 3.3, it follows that{Φn}∼σ 0, so,from Theorem 5in [1], {Gn}∼σ0,and{Gn}∼λ 0 sinceGn isalsoHermitian.
Remark 3.2. If Bn−Gn = En,m, where Gn is the Green matrix defined in (5) and, forall sufficientlylargem∈N, En,m=Rn,m+Nn,m(where Rn,m and Nn,m areas in
Definition 2.2), thatis,if Bn isaperturbation ofGn, then{Bn}∼σ 0 and{Bn}∼λ 0
ifEn,m areHermitian(seeProposition 2.1);thatis, {Gn}and{Bn}havethesamenull distribution.
Remark3.3.Thedecomposition(8)ofGn isinterestingfromthecomputationalpointof
view.Starting fromthetwosets{aj}and{cj},theproductGnx,x∈Cn,requires only nmultiplicationsforeachdiagonalmatrixandn−2 sumsforeachToeplitzmatrix.The totalcostis6n+ 2(n−2)= 8n−4 operations.
3.2. Fewnumerical experiments
We start bygiving the notion of proper (or strong) cluster and the notionof weak cluster.
Definition 3.2. A matrix sequence{An} ofsize dn,dm < dm+1 for each m, is properly
(orstrongly)clustered at s∈C(intheeigenvaluesense), ifforany >0 thenumberof outliers,thatistheeigenvaluesof An offthedisk
B(s, ) :={z:|z−s|< },
canbe boundedby apureconstantq possiblydepending on,butnot onn.In other words
q(n, s) := #{i:λi(An)∈/B(s, )}=O(1), n→ ∞.
If every An has only real eigenvalues (at least for all n large enough), then s is real andthediskB(s,) reducestotheinterval(s−,s+).Finally,theterm“properly(or strongly)”isreplacedby“weakly”,if
q(n, s) =o(dn), n→ ∞,
inthecaseofapoints(aclosedset S),respectively.
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 Table 1
NumberofeigenvaluesofGn,definedin (5)withai= (i/n+ 1)−1,cj= ln(j/n+ 2),greaterthan1= 10−1, 2= 10−2,and3= 10−3. n q1(n,0) q2(n,0) q3(n,0) q1(n,0) n q2(n,0) n q3(n,0) n 20 4 16 20 0.2000 0.8000 1.0000 40 6 19 40 0.1500 0.4750 1.0000 80 8 25 80 0.1000 0.3125 1.0000 160 11 35 142 0.0688 0.2188 0.8875 320 16 49 174 0.0500 0.1531 0.5437 640 22 68 226 0.0344 0.1063 0.3531 1280 31 96 311 0.0242 0.0750 0.2430 2560 43 136 433 0.0168 0.0531 0.1691 5120 61 192 609 0.0119 0.0375 0.1189
Itisimmediatetoseethatasequenceofmatricesisweaklyclusteredatsifandonlyif itadmitsadistributioninthesenseofDefinition 2.1andthedistributionfunctionisequal totheconstants(almosteverywhereonareferencedomain[0,1]).ThereforeTheorem 3.3
tells us thatanyGreen matrix sequence, withRiemannintegrable coefficientsa(·) and
c(·),isweaklyclusteredatzero.
Inthefollowingwe reportselectedexperimentsandwealsodiscuss thefactthatthe cluster isoftenstrong,dependingonthenatureoftheconsideredfunctions.This shows thatthereisroomforimprovingthetheoreticalfindingsinTheorem 3.3.
Wechoosethefunctionsa(x) andc(x),x∈[0,1],computethenumberofeigenvalues ofthematrixGnin(5)greater,inmodulus,ofacertaintolerance∈ {10−1,10−2,10−3},
forvarious sizesn.
Firstofall,weobservethat,ifwesetc(x)=ka(x),withk∈C,weobtainarank-one matrix.Indeed,ifweusethedecompositionin(8)we have
Gn=k2Dn(a)(In+Tn(1)+Tn(2))Dn(a) =k2Dn(a)1nDn(a),
where 1n isthematrixofsizenwithallelementsequaltoone.Now,sincerank(1n)= 1
and rank(AB)≤min{rank(A),rank(B)}, where A,B are matrices ofsize n, the result follows. Inthiscase,obviously,theclusterisstrong.
InTables 1–3we fixthefunctiona(x)= (x+ 1)−1 and wechoosethefunctionsc(x)
with differentgrowth:logarithmic: c(x)= ln(x+ 2) (Table 1);polynomialc(x)=x+ 1 (Table 2); and exponential: c(x) = ex (Table 3). The experiments seem to suggest a deterioration of the cluster when the slope of c(x) increases and a(x) is a decreas-ing function, moreover, in all three cases, we cannote that q2(n,0) ≈ 3q1(n,0) and q3(n,0)≈10q1(n,0).Finally,wecanobservethatthegrowthoftheoutliersis
propor-tionaltothesquareroot ofthedimensionnofthematrix.
In Table 4 we consider a combination of a sinusoidal function a(x) with a linear function c(x) with positive and negative values, in detail a(x) = sinπ2x+π4 and
c(x) = x−1/3. Also in this case the same considerations of the previous cases on theproportionality ofq(n,0) andonthegrowthoftheoutliers as thesquarerootof n
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 Table 2
NumberofeigenvaluesofGn,definedin (5)withai= (i/n+ 1)−1,cj =j/n+ 1,greaterthan1= 10−1, 2= 10−2,and3= 10−3. n q1(n,0) q2(n,0) q3(n,0) q1(n,0) n q2(n,0) n q3(n,0) n 20 6 20 20 0.3000 1.0000 1.0000 40 8 31 40 0.2000 0.7750 1.0000 80 11 37 80 0.1375 0.4625 1.0000 160 15 50 160 0.0938 0.3125 1.0000 320 22 68 290 0.0688 0.2125 0.9063 640 30 96 338 0.0469 0.1500 0.5281 1280 43 135 445 0.0336 0.1055 0.3477 2560 60 190 612 0.0234 0.0742 0.2391 5120 85 268 854 0.0166 0.0523 0.1668 Table 3
NumberofeigenvaluesofGn, definedin (5)withai = (i/n+ 1)−1,cj =ej/n,greaterthan1 = 10−1, 2= 10−2,and3= 10−3. n q1(n,0) q2(n,0) q3(n,0) q1(n,0) n q2(n,0) n q3(n,0) n 20 7 20 20 0.3500 1.0000 1.0000 40 9 40 40 0.2250 1.0000 1.0000 80 13 45 80 0.1625 0.5625 1.0000 160 18 59 160 0.1125 0.3688 1.0000 320 25 81 320 0.0781 0.2531 1.0000 640 36 113 422 0.0563 0.1766 0.6594 1280 50 158 533 0.0391 0.1234 0.4164 2560 71 223 724 0.0277 0.0871 0.2828 5120 100 314 1007 0.0195 0.0613 0.1967 Table 4
NumberofeigenvaluesofGn,definedin (5)withai= sin(πi/(2n)+π/4), cj =j/n −1/3,greaterthan 1= 10−1,2= 10−2,and3= 10−3. n q1(n,0) q2(n,0) q3(n,0) q1(n,0) n q2(n,0) n q3(n,0) n 20 5 20 20 0.2500 1.0000 1.0000 40 7 25 40 0.1750 0.6250 1.0000 80 10 32 80 0.1250 0.4000 1.0000 160 14 44 160 0.0875 0.2750 1.0000 320 19 61 246 0.0594 0.1906 0.7688 640 27 85 291 0.0422 0.1328 0.4547 1280 38 120 391 0.0297 0.0938 0.3055 2560 54 169 542 0.0211 0.0660 0.2117 5120 76 238 759 0.0148 0.0465 0.1482
4. EigenvectorsofSchoenmakers–Coffey matrices
Inthissectionweprovearesultonthe“shape”oftheeigenvectorsofageneralsubclass ofGreenmatrices,theso-calledSchoenmakers–Coffeymatrices,motivatedbyafinancial problemthatwenowbrieflyillustrate.
Werecallfirst thatΣn = [σij]i,jn =1 isacovariance matrix ifit issymmetricand (for
simplicity) positive definite. Its corresponding correlation matrix Rn = [ρij]ni,j=1 has
elements ρij = σij/√σii√σjj that is Rn = DΣnD where D = diag (Σn)−1/2. Thus,
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
Covarianceand/orcorrelationmatricesplayacrucialroleinmultifactormodelsof
in-terest rates wherechangesintheshapeoftheyield curvearelargelyattributedtosome
unobservable factors. Their estimation on real data through the multivariate statistic technique ofprincipal component analysis highlightedtheimportance ofthe firstthree factors, formally captured by the first three eigenvectors of the covariance (or corre-lation) matrix of yields: for details refer to [12,14,17]. These three eigenvectors were respectivelycalledshift,slope andcurvature(oftheyieldcurve),hereafterSSC,because of thebehavior oftheirelements.Approximately:
• a shift has constant sign and an “humped shape”: when it is positive, it is first increasingthendecreasing(seee.g.[7]);
• aslopeismonotone, withachangeofsign;
• acurvature hasaone-peakedshapewithtwo changesofsign.
These features areformally capturedinthefollowing definition(that resumestheones in [18] and [20]) in termsof changes of signof vectors v ∈Rn, v = [v1,. . . ,vn]T, and
Δv∈Rn−1 definedby(Δv)i=vi+1−vi fori= 1,. . . ,n−1.
Definition 4.1.LetΓn be ann×n,n≥3,correlation(or covariance)matrixhaving its
first threeeigenvaluessimple,whosecorresponding eigenvectors v1, v2 and v3 have,by
convention, nonnegativefirstelement.Wedefine:
v1 weakshift ifS−(v1) = 0, shift ifit isweakshift andS−(Δv1) = 1 wherethefirst
nozeroelement ofΔv1ispositive,pure shift ifitisconstant;
v2 weakslope ifS−(v2) = 1,slopeifitisweakslopeandS−(Δv2) = 0;
v3 weakcurvature ifS−(v3) = 2,curvature ifitisweakcurvatureandS−(Δv3) = 1.
In theempirical literatureboth casesof SSCand SSCinaweakform canbe found (see Figures3.16and3.17in[14],Exhibit5in[7],Figures 1and2in[12,13]).Anyway, inalltheweakcases,ineachintervalwhere theelementsof thefirstthree eigenvectors are of constant sign, there is at most one hump, that is the conjecture S(Δv1) ≤ 1,
S(Δv2)≤2 andS(Δv3)≤3 appearsreasonable.
WewillshowthatthisisthecaseforaclassofGreenmatrices,thoseofSchoenmakers– Coffey,denotedbySCn= [rij]ni,j=1 where[21]:
rij=
min{bi, bj}
max{bi, bj}, i, j= 1, . . . , n, (10)
and
H1) {bi}isrealandstrictly positive; H2) {bi}isstrictlyincreasing;
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
We provenow thatSCn is a correlation matrix thatpossesses the following standard
propertiesof thecorrelationsρij ofinterestrates:
a) ρij >0 foralli,j;
b) {ρij}isincreasinginiand decreasinginj fori< j; c) {ρi,i+1}isincreasing.
Infact,assumptionsH1)andH2)ensurethatSCn satisfiesa)andb);furthermore,H1)
andH3)implyc)inastrictsense.Moreover,SCn isGreen’swithai=bi andcj= 1/bj
fori< j,andsince{ai ci}={b
2
i}ispositiveandstrictly increasing,byTheorem 3.1SCn
is oscillatory (see also Corollary 4 of [13]), so it is a correlation matrix. Theorem 3.2
ensuresthatthefirstthree eigenvectorsofSCn are SSCinaweakform (obviously,the
existenceofaweakshifteigenvectorcanbejustifiedviathePerron–FrobeniusTheorem too).
FromwhatwehavejustshownitfollowsthatSCn isanonsingularsymmetricGreen
matrixandthenhastridiagonalinverseSCn−1 asin(6).ThecorrespondingSLP(7)has coefficients pi= bibi+1 b2 i+1−b2i , i= 0, . . . , n, qi= bi(bi+1−bi−1) (bi+bi−1) (bi+bi+1) , i= 1, . . . , n, withb0= 0 andbn+1=∞.
Since{bi}ispositiveand increasing,both {pi}ni=1−1 and{qi}ni=1−1 arepositive.Further insightaregiveninthefollowing.
Lemma 4.1. The sequence {pi}ni=1−1 is (strictly) increasing if and only if {bi/bi+1} is
(strictly) increasing.
Thesequence{qi}ni=2−1isboundedfromaboveby1and(strictly)decreasingif{bi/bi+1}
is(strictly) increasing.
Proof. Themonotonicitystatementsstraightforwardlyfollowsby
pi−pi−1= b2ibi+1(bi−1+bi+1) bi bi+1 − bi−1 bi b2 i −b2i−1 b2 i+1−b2i , i= 2, . . . , n−1, qi−qi−1= bi−1bi+1 bi−2 bi−1 − bi bi+1 (bi−1+bi−2) (bi+bi+1) , i= 3, . . . , n−1,
andpropertiesH1)–H3)of{bi}.Furthermore,sincebi>0 foranyi= 1,. . . ,n,wehave:
qi <bi+1−bi−1 bi+bi+1
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
Werecallnow adefinitionandatheorem dueto Hartman[9].
Definition 4.2. A solution h = [h1, . . . , hn]T of Eq. (7) has a generalized zero at i0
providedhi0 = 0 ifi0= 1 andifi0>1 eitherhi0 = 0 orhi0−1hi0 <0.
Theorem4.1.(Rolle’s)Assumethesequenceofrealnumbersv1,. . . ,vnhasNj generalized zeros andΔv1,. . . ,Δvn−1 has Mj generalizedzeros.Then Mj ≥Nj−1.
Given avector h∈Rn, set
Ωi=−Δ
pi−1(Δh)i−1 =pi−1(hi−hi−1)−pi(hi+1−hi).
Obviously if hi−1 = hi = hi+1 then Ωi = 0. Furthermore, if in i there is a strict
maximum, thatis(Δh)i−1>0 and(Δh)i<0,thenΩi>0;analogously,ifinithereis
astrictminimumthen Ωi <0.
Wearenow ableto provethemainresultofthis section.
Theorem 4.2.The k-theigenvector hk of aSchoenmakers–CoffeymatrixSCn of sizen,
n≥4,has exactlyk−1changesofsign;betweentwoconsecutive changesof signof hk
there isexactlyonechange of monotonicity andk−2≤S−(Δhk)≤k.
Proof. Thefirststatementfollows byTheorem 3.2.
Assume now hk has two consecutive generalized zeroes in i∗,i∗∗ ∈ {2, . . . , n} with i∗ < i∗∗. Since SCn is oscillatory, if hk,i∗ = hk,i∗∗ = 0 there exists an index i∗ such
that i∗ < i∗ < i∗∗ and hi∗ = 0. By Rolle Theorem, Δhk has (at least) a generalized
zero betweeni∗ andi∗∗. Leti+ betweeni
∗ and i∗∗ the minimumindex for whichΔhk
has a generalized zero, so (Δhk)i+ = 0 or (Δhk)i+−1(Δhk)i+ < 0. If, for example,
(Δhk)i+−1>0 thenhk,i+ >0 andinboth thepreviouscasesweobtainΩi+>0.Since
{μ−qi}ni=2−1isstrictlyincreasing,by(7)itfollowsΩi>0 foralli> i+ forwhichhi>0
and this prevent the existence of other generalized zero of Δhk between i∗ and i∗∗.
Assumingalternatively(Δhk)i+−1<0 (andthenhk,i+<0)wegetthesameconclusion.
With thesameargument, itis possible to showthatbefore thefirst (theminimum) andafterthelast(themaximum)generalizedzeroesof hk thereexistsatmostachange
of signof Δhk andthelastconclusionfollows. 2
Ifoneremovestheassumptionthat{bi}islog-concave,thenthesequence{μ−qi}ni=2−1
is no longer strictly increasing and the statement of Theorem 4.2 on the changes of monotonicity does not necessarily hold.In this last case, by theproof of the previous theoremitemergesthatthefirsteigenvectorpresenting“internal”humpsisthefirstone, as illustratedbythefollowing example.
Example4.1.Thesequence{bi}definedbybi= exp65i+ sin65i+1 isstrictlyincreasing butnotlog-concave.Onecanverifythatthefirsteigenvectorofthecorrespondingmatrix
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
Theorem 4.2tellsusthatthefirstthreeeigenvectorsofaSchoenmakers–Coffeymatrix
SCn areSSC inastrict sense if and onlyif theydo notpresent an humpbefore their
firstandaftertheirlast zero.Thefollowing corollaryformalizesthisidea.
Corollary 4.1. Let (λ,h), with h ∈ Rn, h = [h
1,. . . ,hn]T, be an eigenpair of a
Schoenmakers–Coffeymatrix SCn with,forexample, h1 >0.Then, hdoesnot present
anhumpbefore(after) itsfirst(last)zero ifan onlyif λ<1+b1
b2 (λ<1+ bn−1
bn ).
Proof. hdoes notpresentahump beforeits firstzeroif andonly ifh1 > h2.The first
rowofI−λ(SCn)−1 h=0gives: h2= λ b 2 2 b2 1−b22 + 1 λb1b2 b2 1−b22 −1 h1, therefore h1−h2= h1 λb1b2 (b2−b1) (b1+b2−λb2),
and the claim follows by 0< b1 < b2, h1 > 0 and λ > 0, as SCn is oscillatory. The
secondstatementisobtainedinthesameway. 2
GivenaSchoenmakers–CoffeymatrixSCn,twoconclusionsimmediatelyfollowbythe
previouscorollary.First,ifaneigenvector hofSCndoesnothaveahumpbeforeitsfirst
zero, then it does not have anyone after its last zero. Second, since SCn has positive
elementsand{bi/bi+1}isincreasing,bytheclassicalinequalities
min j i rij≤λ1≤max j i rij,
we obtain λ1 > 1+ b1/b2, namely, in financial terms, the first eigenvector of a
Schoenmakers–Coffeymatrixisalwaysshift.
Forwhatconcernsthesecondand thirdeigenvectorofSCn,wemustnoticethatthe
conditionsforthe non-existenceof aninitial and/orfinal hump foragiven eigenvector
hofCorollary 4.1arebasedonknowledge(atleastintermsofestimation)ofthe corre-spondingeigenvalueλ.Intheabsence (as usual)ofsuch information,it isstill possible toobtainsomepartial conclusions,as illustratedbythefollowing example.
Example4.2.Ifbs=sα withα >0,then:
rij= i j α i≤j, j i α i > j.
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
Since limα→0+rij = 1 andlimα→+∞rij = 0,weobtain:
lim
α→0+λ2= 0; α→lim+∞λ2= 1.
Ontheother hand
lim α→0+1 + b1 b2 = lim α→0+1 + bn−1 bn = 2, lim α→+∞1 + b1 b2 = lim α→+∞1 + bn−1 bn = 1.
Thereforeforαinasuitablerightneighborhoodof0 therearenoinitialandfinalhumps in h2.Similarconsiderationsapplyto λ3 and h3.
A furtherconsideration.Asit iswell-known,the eigenvectorsofacovariancematrix are different from the ones of the corresponding correlation matrix. However, in [13]
it has been shown thatan invertible covariance matrix is oscillatory if and only ifits correlationmatrixisoscillatory.ThismeansthatallGreen(covariance)matriceshavinga correspondingcorrelationmatrixofSchoenmakers–Coffeytype,areoscillatoryandtheir firstthreeeigenvectorsareSSCinaweaksense.Thisraisesthequestionofwhetherthe results obtained hereon the numberof monotonicity changes of Schoenmakers–Coffey matrices extendinanaturalway tothe correspondingcovarianceGreen matrices. The negative answerisgiven bythefollowing example.
Example 4.3. Consider the Green matrix Gn defined by the sequences {ai} and {cj}
with: tk = π 2 1 + k−1 n−1 , k= 1, . . . , n, ai=ti(sin(4ti) + 2), i= 1, . . . , n, cj = sin(4tj) + 2, j= 1, . . . , n.
The sequences {ai} and {cj} are positive but not monotone. However we have that {ai/ci} = {ti} is strictly increasing, therefore Gn is oscillatory and in particular is a
covariancematrix.Thesequence{bi}={
ai/ci}={√ti}ispositive,strictlyincreasing
and suchthat{bi/bi+1}isstrictlyincreasingtoo:
bi bi+1 − bi−1 bi = 1− 1 n−1 +i− 1− 1 n−2 +i >0.
Therefore,thecorrespondingcorrelationmatrixSCnisofSchoenmakers–Coffeytypeand
itsfirstthreeeigenvectorsareSSC(Fig. 1(a)).NeverthelessthefirsteigenvectorofGnis
notshiftandpresentstwohumps(Fig. 1(b)).Ifwesubstitutesin(4ti) withsin(10ti) also thesecondandthethirdeigenvectorofGn arenotmoreslopeandcurvature(Fig. 1(c)).
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26Q6 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
Fig. 1.Firstthreeeigenvectors,h1(+),h2(×)andh3(∗),ofSC50forsin(4ti) (a),ofG50forsin(4ti) (b),
andofG50forsin(10ti) (c).
In thenext example the estimation of theeigenvalues of the considered correlation matrixallowsto obtainacompleteresultontheexistenceof slopeandcurvature.
Example 4.4. Consider the well-known Kac–Murdock–Szegö (KMS) matrices R =
ρ|i−j|ni,j=1, ρ∈(0,1),[6,8,10,29].TheyareGreen’s(see(6))with
ai =ρ1−i, cj=ρj−1, (11)
and represent a limit case of the Schoenmakers–Coffey ones, as {bi/bi+1} is constant.
Indeed,bytheproofofLemma 4.1,{pi}in=1−1isaconstantsequenceifandonlyif{bi/bi+1}
isconstant.Itis possibleto show(seeTheorem 3.1in[21])thatunderH1)–H3)(10)is equivalentto (R)ij = exp − n l=i+1 min{l−i, j−i}Δl , i < j, (12) where Δl= lnbl−lnbl−1−(lnbl+1−lnbl), l= 2, . . . , n−1, Δn= lnbn−lnbn−1= lnb2− n−1 l=2 Δl.
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42
Hence, {bi/bi+1} is constant if and only if one chooses Δl = 0 in (12) for l =
2,. . . ,n − 1, obtaining (without any restrictions we assume b1 = 1) (R)ij =
exp{−min{n−i, j−i}Δn}=b2i−j for i< j. Therefore this is theunique casewhere
the main diagonalof SC−n1 (except for thefirst and the last elements) and the super-diagonalsareconstant.
KMSmatricesareoscillatory [18],sotheirfirstthree eigenvectorsareSSCinaweak sense. Theauthors in[19] haveshownthatthese matricesadmit shift forallρ∈(0,1) and slope and curvature if and only if ρ is greater than a threshold value. We briefly show how to extend and precise these results for n ≥ 4 (if n = 3 all is obvious, see Example 15 in[19]).
As it can be easily verified, the conclusions of Theorem 4.2 apply despite being in a borderline case. The same result can be obtained recalling that KMS matrices are Toeplitz’s, with the n2! symmetric and "n2# skew-symmetric eigenvectors hk whose
expressions (see[28]): htk= cos t−n+ 1 2 θk , kodd, (13) htk= sin t−n+ 1 2 θk , keven, (14)
areobtainedbysolvingproblem(7)withpi= 1−ρρ2 andqi =11+−ρρ.Asaconsequence,the
factthatbetweentwoconsecutivezerosthereisoneandonlyonechangeof monotonic-ity follows bynoting thateach eigenvector isan equispacedsampling ofsine or cosine functions.
For whatconcerns thepresence ofinitial (and bysymmetry, final)humps, we recall thattheeigenvaluesof Rare(interlaced,withλ1even and)givenby
λk =
1−ρ2
1 +ρ2−2ρcosθk, (15)
where,asprovedin[25]improvingapreviousresultof[8],
(k−1)π
n < θk< kπ
n+ 1, k= 1, . . . , n. (16)
As Corollary 4.1 holds with 1+b1 b2 = 1+
bn−1
bn = 1+ρ, themonotonicity of the“first
positivepart”(withh1k>0)ofanyeigenvectorofRdependsonthesignofλk−(1 +ρ).
Apartfrom theobvious conclusionsaboutλ1(>1 +ρ) (andλn<1+ρ),we shownow
that:
i) forn= 4,5 wehave(λ3<)λ2<1+ρforallρ∈(0,1);forn≥6 wehaveλ21+ρ
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41Q7 41 42 42
ii) forn= 6,7,8 wehaveλ3 <1+ρforall ρ∈(0,1);for n≥9 we haveλ31+ρif
andonlyifρ2cosn2−π2−1. Infact,observe preliminarilythat
lim
ρ→1−λk(ρ) =λk(1) = 0<2 = limρ→1−1 +ρ, k= 2,3, (17)
hence,bycontinuity,λk <1+ρfor“sufficientlylarge”ρ.
i)By(15)itfollows λ2 1+ρifand onlyif2cosθ2 1+ρ,whichis equivalentto
h12h22.
From(14),h12=h22 ifand onlyif(see(16))
θ2∈ π n, 2π n+ 1 : sin θ2 n−1 2 = sin θ2 n−3 2 , (18)
thatisifandonlyifθ∗2 =nrπ−2 wheretheintegerrsatisfies
1− 2
n < r <2−
6
n+ 1.
Then for (3 ≤)n ≤ 5 there are no solutions, whereas for n ≥ 6 Eq. (18) admits the uniquesolutionθ2= nπ−2.These conclusionsjointlywith(17)provethestatementwith
ρ= 2cosn−π2−1.
ii) Operating as inthe proof of i), forn ≥9 we find aunique valueθ$3 = n2−π2 such
thath13=h23 andρ= 2cosn2−π2−1.
The“translation”inthefinanciallanguageoftheseresultsisimmediate.
5. Conclusionsandremarks
Wehaveidentified somespectral propertiesofGreenmatrices,whose eigenvalue dis-tribution isalwaysequalto zero:this resultis notsurprisinggivingthe factthatthese matricescome from integraloperators(see [1]andreferences therein),butwehave ob-tainedit inawider generality,by using thetheory of GLTsequences. Concerning this firstpart weobserve that thezerospectral distribution isequivalent to the weak clus-tering:however,inournumericaltests,oftenweobserveda√n-clusteringintheaverage caseandhenceitseemsthatthereisroomforimprovingthetheoreticalresults.
As a second step, we have considered a subclass of the Green matrices, called Schoenmakers–Coffey matrices, which have a role in financial context. For this class, in view of its importance in applications such as in the portfolio estimation, we have analyzedtheeigenvectoroscillationstructure.
Asalreadyobservedintheintroduction,evenifthese matricesarenotshiftinvariant (Toeplitz), the results are obtained by using tools coming from Toeplitz technology, related to GLTsequences and to Kac–Murdoch–Szegö matrices, connected to Toeplitz matrices.
1 1 2 2 3 3 4Q9 4 5 5 6Q10 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23Q8 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 Acknowledgements
The work of the second and third author has been partly financed by the Italian national group GNCS-INDAM. The work of the second author has been also partly supportedby agrantassociated to theProject“BecomingtheNumber One–2014” of theKnutandAliceWallenberg Foundation,Sweden.
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