S a is ics & Ope a ions Resea ch T ansac ions
SORT 27 (2) July-Decembe 2003, 165-174
S a is ics &
Ope a ions Resea ch
T ansac ions
Asymp o ic s udy o canonical co ela ion analysis:
om ma ix and analy ic app oach o ope a o
and enso app oach
Jeanne Fine∗
Uni e si ´e Paul Saba ie , F ance
Abs ac
Asymp o ic s udy o canonical co ela ion analysis gi es he oppo uni y o p esen he di e en s eps
o an asymp o ic s udy and o show he in e es o an ope a o and enso app oach o mul idimensional
asymp o ic s a is ics a he han he classical, ma ix and analy ic app oach. Using he las app oach,
Ande son (1999) assumes he andom ec o s o ha e a no mal dis ibu ion and he non ze o canonical
co ela ion coe icien s o be dis inc . The new app oach we use, Fine (2000), is coo dina e- ee,
dis ibu ion- ee and pe mi s o ha e no es ic ion on he canonical co ela ion coe icien s mul iplici y
o de . O cou se, when ec o s ha e a no mal dis ibu ion and when he non ze o canonical co ela ion
coe icien s a e dis inc , i is possible o ind again Ande son’s esul s bu we di e ge on wo o hem.
In his me hodological p esen a ion, we insis on he analysis ame (Dauxois and Pousse, 1976), he
sampling model (Dauxois, Fine and Pousse, 1979) and he di e en ma hema ical ools (Fine, 1987,
Dauxois, Romain and Viguie , 1994) which pe mi o sol e p oblems encoun e ed in his ype o s udy,
and e en o ob ain asymp o ic beha io o he analyses andom elemen s such as p incipal componen s
and canonical a iables.)
MSC:
62E20, 62H20, 62H25, 47N30
Keywo ds:
mul i a ia e analysis, canonical co ela ion analysis, asymp o ic s udy, ope a o , coo dina e-
ee, dis ibu ion- ee
1 Classical app oach
1.1 Popula ion canonical co ela ion analysis
Le Xand Ybe wo andom ec o s, pand qdimensional espec i ely (p≤q) de ined
on a same p obabili y space (Ω,A,P), cen e ed and admi ing o de 4 momen s. We
∗Add ess o co espondence: Labo a oi e de S a is ique e P obabili ´
es, Uni e si ´
e Paul Saba ie , 118, ou e de
Na bonne, 31062 Toulouse cedex, F ance. e-mail: [email p o ec ed].
Recei ed: Oc obe 2003
Accep ed: Decembe 2003
166
Asymp o ic s udy o canonical co ela ion analysis: om ma ix and analy ic...
assume he ma ix co a iance VXo X o be non-singula and we deno e by HX he ec o
space o eal- alued andom a iables ( . . .) linea combina ions o Xcomponen s.
We in oduce simila ly VYand HYand we deno e by VXY he c oss co a iance he
componen s o Xwi h he ones o Y.
The aim o canonical co ela ion analysis (CCA) o (X,Y) is o measu e he
ela ionships be ween Xand Y. CCA may be de ined as he sea ch o 1and g1,
. . . o HXand HYwi h uni a iance and maximal co ela ion ρ1 hen, i e a i ely, o
j=2, ..., , ( ≤p), as he sea ch o jand gj, . . . o HXand HYwi h uni a iance,
unco ela ed wi h he ( k)k<jand (gk)k<jand wi h maximal co ela ion ρj. The . . . j
and gja e called j h canonical a iables and he eal ρjo [0,1] is called j h canonical
co ela ion coe icien .
Le RX=V−1
2
XVXY V−1
YVYXV−1
2
Xand he same o RYpe mu ing Xand Y oles. I is
easy o e i y ha RXand RYha e he same non-ze o eigen alues deno ed by (ρ2
j)j=1,...,
when w i en in a dec easing o de . We se : λj=ρ2
j, o j=1, ..., ,and, excep in he
pa icula case =p=q, we se λj=ρj=0 o j> . Fo j> we de ine jand gjas
. . . o HXand HY espec i ely wi h uni a iance and unco ela ed wi h he ( k)k<jand
(gk)k<j.
CCA o (X,Y) is hen :
((ρj)j=1,..., +1,( j)j=1,...,p,(gj)j=1,...,q).(1)
CCA o (X,Y) depends only on HXand HY, which a e also gene a ed by componen s
o X0:=V−1
2
XXand Y0:=V−1
2
YY espec i ely. We show ha , i (uj)j=1,...,pand ( j)j=1,...,q
deno e uni eigen ec o s bases o RXand RYassocia ed wi h (λj)j=1,...,pand (λj)j=1,...,q
espec i ely, we can ob ain canonical a iables jand gjas linea combina ions o X0
and Y0componen s, using ujand jcomponen s as coe icien s, ha is, by se ing:
j=huj,X0ipe gj=h j,Y0iq, whe e h., .ipand h., .iqdeno e Rpand Rqusual scala
p oduc s.
Decomposi ion (1) is no unique because each canonical a iable associa ed wi h a
simple eigen alue may be eplaced by i s opposi e and he se o canonical a iables
associa ed wi h a mul iple eigen alue may be eplaced by an o he se acco ding o he
choice o RXand RYeigen ec o s associa ed wi h his eigen alue.
1.2 Sample canonical co ela ion analysis
Le (Xl,Yl)l=1,...,nbe a n-sample i.i.d. as (X,Y). We index by n he elemen s de ined
p e iously and calcula ed on he sample : µn
X, µn
Y,Vn
X,Vn
Y,Vn
XY ,Rn
X,Rn
Y.
Le (λn
j)j=1,...,pbe he dec easing sequence o he peigen alues o Rn
X(and o he p
la ges eigen alues o Rn
Y, he o he ones, i q>p, being null), (un
j, n
j)j=1,...,pa sequence
o associa ed uni eigen ec o s o Rn
Xand o Rn
Yand ( n
j,gn
j)j=1,...,p he canonical a iables
sequence, ec o s o Rn, ob ained by :
Jeanne Fine
167
∀l∈ {1, ..., n}( n
j)l=hun
j,(Vn
X)−1
2(Xl−µn
X)ipand (gn
j)l=h n
j,(Vn
Y)−1
2(Yl−µn
Y)iq.
I he case a ises (q>p), we le λn
p+1=0, we comple e ( n
j)j=1,...,pwi h Rn
Y
eigen ec o s in o de o ob ain an o hono mal basis ( n
j)j=1,...,qo Rqand we de ine
he canonical a iables associa ed.
A las , o all jin {1, ..., p+1}le ρn
j=qλn
j. Sample CCA o (X,Y) is hen:
((ρn
j)j=1,...,p+1,( n
j)j=1,...,p,(gn
j)j=1,...,q).(2)
1.3 Asymp o ic s udy
Asymp o ic s udy o CCA consis s in es ablishing a.s. con e gence o he canonical
elemen s sequences o he sample CCA (2) o he co esponding canonical elemen s
o he popula ion CCA (1) and in es ablishing con e gence in dis ibu ion o he
s anda dized canonical elemen s sequences.
Di icul ies a e nume ous : canonical a iables a e es ima ed (“p edic ed”) by Rn
ec o s, he space dimension inc easing wi h sample size. Then he use is o es ic
asymp o ic s udy o Rn
Xand Rn
Yeigen ec o s : (un
j)j=1,...,pand ( n
j)j=1,...,q espec i ely,
called canonical ec o s and also o he Rpand Rq ec o s de ined by: xn
j=(Vn
X)−1
2un
j
and yn
j=(Vn
Y)−1
2 n
j espec i ely, called canonical ac o s; hese ec o s pe mi o ob ain
di ec ly canonical a iables by:
∀l∈ {1,...,n}( n
j)l=hxn
j,Xl−µn
Xipe (gn
j)l=hyn
j,Yl−µn
Yiq.
Mul iple eigen alues case is di icul o p ocess because he eigen ec o s associa ed
wi h a e no uniquely de ined. Then he use is o es ic asymp o ic s udy o he case
whe e all eigen alues a e simple. Uniqueness is hen e i ied by choosing sys ema ically
he uni ec o (be ween he wo ones) which has he i s non null coo dina e in espec
wi h he canonical basis posi i e.
As o all mul idimensional analyses, co a iance ma ices o sample andom
ma ices Vn
X,Vn
XY ,Rn
X, ... a e “supe -ma ices” ( ha is, ma ices o ma ices). Tools such
as he “ ec” ope a o which ans o ms ma ix in o ec o , ha e been in oduced in o de
o handle heses supe -ma ices; he di icul y comes om he necessi y o ixing he
o de o lines and columns elemen s.
In o he espec s, we know ha he sequence ( √n(Vn
X−VX)) con e ges in dis ibu ion
o a cen e ed no mal a iable, he co a iance supe -ma ix o which is known in some
special cases, when Xhas a no mal o ellip ical dis ibu ion o example.
In he CCA ame wo k, we need o s udy con e gence in dis ibu ion o he sequence
(√n(Vn
Z−VZ)) wi h Z=(X,Y), hen o s udy con e gence in dis ibu ion o he sequence
(√n(Rn
Z−RZ)) wi h RZ=(RX,RY) and Rn
Z=(Rn
X,Rn
Y),be o e s udying con e gence
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Asymp o ic s udy o canonical co ela ion analysis: om ma ix and analy ic...
o he Rn
Zeigenelemen s sequences and con e gence o he sample canonical elemen s
sequences. This asymp o ic s udy is much mo e complex han he one o P incipal
Componen Analysis (PCA) because p incipal alues and p incipal ec o s o he X
PCA a e eigenelemen s o he Xco a iance ma ix.
I is only in 1999 ha Ande son publishes a CCA asymp o ic s udy when (X,Y)
has a no mal dis ibu ion and when all non ze o eigen alues a e simple. Canonical
ac o s componen s and canonical co ela ion coe icien s o he popula ion CCA a e
di e en iable unc ions o VZ. Resul s a e hen ob ained om Taylo expansions. So,
his classical app oach may be quali ied as ma icial and analy ic.
In o de o simpli y calcula ions, we p opose o change a iables om (X,Y) o
(X0,Y0), which is equi alen o changing he basis in Rpand Rq. We hen ha e: VX0=Ip,
and RX=RX0=VX0Y0VY0X0, and simila ly o VY0and RY.
2 Ope a o and enso app oach
2.1 In oduc ion
Di icul ies p e iously desc ibed a e ensued only om he ac ha ma icial ool
is no con enien . Wo king di ec ly on linea ope a o s in Euclidean spaces a oid
indices p oblems and can be easily ex ended o an Hilbe ian ame. Mo eo e , ins ead
o s udying eigen ec o s associa ed wi h simple eigen alues, i is possible o s udy
eigenp ojec o s associa ed wi h mul iple eigen alues. Ea on (1983) also ad ices a
“ ec o space app oach” o he mul idimensional s a is ics.
Dauxois and Pousse (1976) enla ge he PCA de ini ion o a Rp andom ec o o a
Hilbe andom a iable and e en o a Hilbe andom unc ion, ha is, a Hilbe andom
a iable depending on a pa ame e in o de o p ocess empo al o spa ial da a. They
ede ine each ac o ial analysis (PCA, CCA, Co espondence Analysis, Disc iminan
Analysis, ...) in an ope a o ial and s ochas ic ame, ha leads hem o de ine, be ween
o he s, nonlinea analyses.
The i s asymp o ic s udy in his ame has been ealized by Romain (1979) o
a Hilbe andom unc ion PCA (see also Dauxois, Pousse and Romain, 1982), s udy
comple ed by A con e (1980) who also s a ed on CCA asymp o ic s udy bu all ools
we e no a ailable o con inue he s udy. Dauxois, Romain and Viguie (1994) p opose
o use some enso p oduc s and es ablish a dic iona y be ween ma icial and ope a o ial
o mula. This wo k pe mi s o compa e common esul s ob ained in bo h ames, bu
also o ob ain mo e easily complex esul s; w i ings in espec wi h eigen ec o s basis
a e es ablished a e concise o mula ions wi h ope a o s.
These new ools pe mi o ealize in Fine (2000) he CCA asymp o ic s udy
wi hou es ic ion, ha is, wi hou assump ion on he (X,Y) dis ibu ion, in he gene al
case whe e eigen alues may be mul iple and wi hou excluding canonical a iables
Jeanne Fine
169
asymp o ic s udy (CCA andom elemen s). The e o e, ou app oach may be quali ied
as an ope a o and enso app oach. We gi e below he di e en s eps o he CCA
asymp o ic analysis, some ools used and some examples o esul s.
2.2 Di e en s eps o he CCA asymp o ic s udy, ools, esul s
1) Popula ion CCA
Fi s , he ma e is o de ine CCA o a pai o Euclidean andom a iables (popula ion
CCA). Again, we use classical app oach no a ions subs i u ing (Rp,h., .ip) and (Rq,h., .iq
o pand qdimensional Euclidean spaces (X,h., .iX) and (Y,h., .iY) espec i ely.
Ob iously, we wo k wi hou e e ence o any basis ( ee coo dina e).
Le L2(P) he Hilbe space o . . . de ined on (Ω,A,P) and admi ing o de 2
momen s, scala p oduc o which associa es E( g) o ( ,g).
The ope a o ΦX om X o L2(P) which associa es hx,XiX o xplays an essen ial
ole in he ope a o app oach o mul idimensional s a is ics. In pa icula , Xis a no mal
Euclidean andom a iable i , and only i , ∀x∈ X,hx,XiXis a no mal . . ..
The expec ed alue o Xis he unique elemen o X(Riesz heo em), deno ed by
E(X), e i ying: ∀x∈ X,hx,E(X)iX=E(hx,XiX).
Fo all (x,y)∈ X×Y, we deno e by x⊗y he ope a o om X o Ywhich associa es
hx0,xiXy o x0; i is an elemen o he Hilbe space σ2(X,Y) o ope a o s om X o
Ywi h he scala p oduc : hA,Bi2= (AB∗).Due o he Riesz heo em, we may hen
de ine co a iance ope a o s VXo X,VYo Y, and c ossed co a iance ope a o s VXY and
VYX o Xand Y:VX=E((X−µX)⊗(X−µX)),...
As in he CCA classical app oach (§1.1), Xand Ya e assumed o be cen e ed. The
adjoin ope a o Φ∗
Xo ΦXis he ope a o om L2(P) o Xwhich associa es E( X) o
and hen we ha e : Φ∗
X◦ΦX=VX,Φ∗
X◦ΦY=VXY , ...
I is con enien o ep esen ope a o s ela ionships in he ollowing commu a i e
diag am, also called a duali y scheme; he e, each space is iden i ied wi h i s dual space.
The HXand HYspaces a e image spaces o ΦXand ΦY espec i ely and he o hogonal
p ojec o s o L2(P) on hese subspaces a e: ΠX= ΦX◦V−1
X◦Φ∗
Xand ΠY= ΦY◦V−1
Y◦Φ∗
Y.
X
Φ∗
X
←− L2(P)
Φ∗
Y
−→ Y
V−1
X↓↑ VX↑I VY↑↓ V−1
Y
X−→
ΦXL2(P)←−
ΦYY
Ope a o s RXand RY, and also CCA o (X,Y), a e de ined as p e iously (symbols ◦
a e dele ed in o de o educe no a ion).

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Asymp o ic s udy o canonical co ela ion analysis: om ma ix and analy ic...
As in he classical app oach, in o de o acili a e calcula ions, we change he scala
p oduc on Xso ha he co a iance ope a o o Xis he iden i y o X, and simila ly o
Y. We hen ha e: RX=VXY VYX and RY=VYXVXY .
2) Sample model and sample CCA
We use a sample model (Dauxois, Fine and Pousse, 1979) es ablishing a link be ween
he sample used in Da a Analysis and he i.i.d. sample o S a is ics. A sample (Xl,Yl)l∈N∗
i.i.d. as (X,Y) is buil om an elemen ωo ΩN∗se ing, o all lo N∗(πldeno ing he
l h p ojec ion o ΩN∗on o Ω) : Xl=X◦πland Yl=Y◦πl ha is Xl(ω)=X(ωl) and
Yl(ω)=Y(ωl).
We hen p o ide L2(P) wi h he scala p oduc ( andom scala p oduc as i depends
on ω):
∀( ,g)∈L2(P)×L2(P),En( g)=1
n
n
X
l=1
(ωl)g(ωl).
Then we ha e:
En(X)=µn
X,Φn
X=h., X−µn
XiX,Vn
X=1
n
n
X
l=1
(Xl−µn
X)⊗(Xl−µn
X), . . .
This sample model is he clue o dis inguish andomness implied by he model (L2(P)
elemen s) om andomness implied by sampling. I pe mi s o ob ain he canonical
a iables asymp o ic dis ibu ion.
The duali y scheme o sample CCA is he same as he popula ion one a e
subs i u ing L2(P) o (L2(P),En) and indexing ope a o s by n.
Sample ope a o s Rn
Xand Rn
Yand sample CCA a e de ined as p e iously.
3) Con e gence o sample ope a o s sequence
Limi heo ems in Euclidean o Hilbe spaces pe mi o ob ain a.s. con e gence and
con e gence in dis ibu ion o he co a iance ope a o s sequence wi hou assump ion on
he dis ibu ion o (X,Y) excep he exis ence o o de 4 momen s. Fo CCA, we ob ain
( emind we le Z=(X,Y)):
Wn
Z:=√n(Vn
Z−VZ)D
−→ WZ∼N(0 ; KZ),
whe e KZis he co a iance ope a o o Z⊗Z.
In wha conce ns he sample ope a o s Rn
Z=(Rn
X,Rn
Y), elemen s o σ2(Z) (wi h
Z=X×Y), a.s. con e gence de i es om he ac ha i is possible o w i e Rn
Xand Rn
Y
as con inuous unc ion o Vn
Z.
Le Un
Z=√n(Rn
Z−RZ) (:=(Un
X,Un
Y)).
We w i e Un
X= Ψn
X(Wn
Z) whe e (Ψn
X) is a sequence o andom ope a o s om
σ2(Z) o σ2(X) a.s. con e ging o ΨX. We hen deduce he con e gence in dis ibu ion
Jeanne Fine
171
o (Un
X) o UX= ΨX(WZ), cen e ed no mal a iable, co a iance ope a o o which
being LX= ΨX◦KZ◦Ψ∗
X, and he same esul o (Un
Y) pe mu ing Xand Y oles.
The p oposi ion used he e, is easy o p o e om classical esul s in me ic spaces
(Billingsley, 1968). We ob ain o example:
UX=−1
2(WXRX+RXWX)+WXY VYX +VXY WYX −VXY WYVYX ∼N(0 ; LX)
4) Con e gence o eigenelemen s and CCA elemen s sequences
Wha e e may be he “ ac o ial” me hod, which is an analysis o a model ob ained
om a spec al (o singula - alue) decomposi ion, all esul s conce ning eigenelemen s
(eigen alues, eigenp ojec o s, eigen ec o s associa ed wi h simple eigen alues, ...) a e
easily ob ained hanks o pe u ba ion heo y o linea ope a o s (Ka o, 1980). In Fine
(1987), his heo y has been adap ed o bounded pe u ba ions ha pe mi s o use i , due
o he i e a ed loga i hm law, in he asymp o ic s udy ame. So we ob ain a.s. expansions
o eigenelemen s o a symme ic posi i e ope a o s sequence.
We may also consul Dossou-Gbe e and Pousse (1991) o limi esul s bu , o he
con e gence in dis ibu ion o some CCA elemen s, limi esul s a e no su icien when
pe u ba ion expansions pe mi o conclude.
Fo example, o he canonical ac o s associa ed o a simple eigen alue λi, we ha e:
xi=uibecause VX=IXand xn
i=(Vn
X)−1
2un
iso:
√n(xn
i−xi)=−(Vn
X)−1
2((Vn
X)1
2+IX)−1[√n(Vn
X−IX)]un
i+[√n(un
i−ui)].
We know ha ( √n(Vn
X−IX)) con e ges in dis ibu ion o WXand ( √n(un
i−ui)) o
SXiUXxi(wi h SXi=(RX−λiIX)−) bu , hanks o pe u ba ion expansions, i is possible
o es ablish: √n(xn
i−xi)D
−→ 1
2WXxi+SXiUXxi∼N(0 ; LXi)
5) Asymp o ic co a iance ope a o s in he ellip ical case
We ha e al eady seen ha he asymp o ic co a iance ope a o o ( √n(Rn
X−RX)) is
LX= ΨX◦KZ◦Ψ∗
Xwhe e KZis he asymp o ic co a iance ope a o o ( √n(Vn
Z−VZ))
and whe e he ope a o ΨX om σ2(Z) o σ2(X) can be w i en explici ly. All he
dis ibu ion limi s o eigenelemen s o CCA elemen s sequences a e cen e ed no mal
a iables (o unc ion o cen e ed no mal a iables), co a iance ope a o o which being
w i en as unc ion o KZin he same way.
Now, we may w i e explici ly hese asymp o ic co a iance ope a o s in he case
whe e Zhas an ellip ical dis ibu ion wi h mean µZ, co a iance ope a o VZand ku osis
κ( eal pa ame e , which, when i is null, leads o a N(µZ,VZ) dis ibu ion). We hen
know ha KZis he ope a o om σ2(Z) o i sel which associa es o T:
KZ(T)=(1 +κ)VZ(T+T∗)VZ+κhVZ,Ti2VZ.
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Asymp o ic s udy o canonical co ela ion analysis: om ma ix and analy ic...
A his s ep, we need mo e algeb aic ools. The enso p oduc in spaces o ype σ2
is deno ed by ˜
⊗.Fo example:
∀(A,B)∈σ2(Z)×σ2(Z),∀T∈σ2(Z),A˜
⊗B(T)=hT,Ai2B.
We de ine also p oduc `
⊗in spaces o ype σ2.Fo example:
∀(A,B)∈σ2(Z)×σ2(Z),∀T∈σ2(Z),A`
⊗B(T)=BT A∗.
We de ine he commu a ion ope a o Cwhich associa es o an ope a o Ti s adjoin
T∗. A las , we subs i u e he space p oduc Z=X×Y o he Hilbe ian sum Z=X⊕Y;
his pe mi s o plunge all he ope a o s in o σ2(Z) in o de o simpli y no a ion. P ojec o
PX om σ2(Z) on o σ2(X) becomes in his ame a symme ic ope a o o σ2(Z).
The ope a o KZ om σ2(Z) o i sel may be w i en as:
KZ=(1 +κ)VZ
`
⊗VZ(I+C)+κVZ˜
⊗VZ.
Le (xi)i=1,...,pbe an o hono mal basis o med by canonical ac o s o X, hen
(xi⊗xj)i,j=1,...,pis an o hono mal basis o σ2(X) and ((xi⊗xj)˜
⊗((xk⊗xl))i,j,k,l=1,...,p
is an o hono mal basis o σ2(σ2(X)).
A e calcula ions ob ained in a concise way, i is easy o decompose ope a o s in
espec wi h his ype o basis. Fo example, we ob ain o he asymp o ic co a iance
ope a o o ( √n(Rn
X−RX)):
LX=(1 +κ)(I+C)[−3
4R2
X
`
⊗IX+R2
X
`
⊗RX+RX
`
⊗IX−5
4RX
`
⊗RX](I+C)
and, in espec wi h he basis o canonical ac o s ( emembe ha (λj)j=1,...,pis he
dec easing sequence o eigen alues o RX):
LX=1
2(1 +κ)
p
X
j=1
p
X
k=1Ã−3
4λ2
j−3
4λ2
k+λ2
jλk+λjλ2
k+λj+λk−5
2λjλk!
(xj⊗xk+xk⊗xj)˜
⊗(xj⊗xk+xk⊗xj)
When (X,Y) has a no mal dis ibu ion and when all eigen alues a e simple, i is
possible o edisco e Ande son’s esul s bu we di e ge on wo o hem.
6) Con e gence o CCA andom elemen s sequences
As p e iously announced (§2.1.2) he sample model pe mi s o ob ain a.s.
con e gence and con e gence in dis ibu ion o canonical a iables sequences. We ha e
o example, o he canonical a iable associa ed wi h a simple eigen alue λi:
√n( n
i− i)) D
−→ h1
2WXxi+SXiUXxi,XiX∼N(0 ; MXi)
Jeanne Fine
173
wi h, in he pa icula case whe e (X,Y) has an ellip ical dis ibu ion:
MXi =1
4(2 +3κ) i⊗ i+(1 +κ)X
j,i
(1 −λi)(λi+λj−2λiλj)(λi−λj)−2 j⊗ j
7) In e en ial applica ions and conclusion
These esul s on CCA asymp o ic s udy pe mi o ackle easily in e en ial
applica ions (con idence in e al es ima ion, s a is ical es s, ...) which imply CCA
elemen s, pa icula ly he p oximi y measu es buil on canonical co ela ion coe icien s.
See Ande son (1999) and Dauxois and Nkie (2002).
Fu he aspec s and esul s may be consul ed in Fine (2000). This me hodological
p esen a ion shows ha he ope a o app oach pe o ms qui e well in sol ing asymp o ic
p oblems in mul i a ia e s a is ics.
3 Re e ences
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Analysis, 70, 1-29.
A con e, A. (1980). ´
E ude asymp o ique de l’analyse en composan es p incipales e de l’analyse ca-
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eme cycle, Uni e si ´
e de Pau e des Pays de l’Adou .
Billingsley, P. (1968). Con e gence o p obabili y measu es. Wiley, New Yo k.
Dauxois, J., Fine, J. and Pousse A. (1979). ´
Echan illonnage en segmen a ion, ´
e ude de la con e gence. S a-
is ique e Analyse des Donn´ees, 3, 45-53.
Dauxois, J. and Nkie , G. M. (2002). Measu es o Associa ion o Hilbe ian subspaces and some ap-
plica ions. Jou nal o Mul i a ia e Analysis, 82, 263-298.
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essai d’´e ude syn h´e ique. Th`
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o a ec o andom unc ion; some applica ions o s a is ical in e ence. Jou nal o Mul i a ia e
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Dauxois, J., Romain, Y. and Viguie , S. (1994). Tenso p oduc s and s a is ics. Linea Algeb a and i s
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Ea on, M. L. (1983). Mul i a ia e s a is ics. A ec o space app oach. Wiley, New Yo k.
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—(2000). ´
E ude Asymp o ique de l’Analyse Canonique. Pub. Ins . S a . Uni . Pa is, 44, 2-3, 21-72.
Ka o, T. (1980). Pe u ba ion heo y o linea ope a o s. Sp inge -Ve lag, New Yo k.
Romain, Y. (1979). ´
E ude Asymp o ique des app oxima ions pa ´echan illonnage de l’analyse en compo-
san es p incipales d’une onc ion al´ea oi e. Quelques applica ions. Th`
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