Copulas Related to Manneville-Pomeau Processes

0.73 MB·English

by  Sílvia R. C. Lopes

Copulas Related to Manneville-Pomeau
Processes
S´ılvia R.C. Lopes and Guilherme Pumi
Mathematics Institute
Federal University of Rio Grande do Sul
This version: October, 10th, 2011
Abstract
In this work we derive the copulas related to Manneville-Pomeau processes. We examine
both bidimensional and multidimensional cases and derive some properties for the related
2
1 copulas. Computationalissues,approximationsandrandomvariategenerationproblemsare
0 also addressed and simple numerical experiments to test the approximations developed are
2 also perform. In particular, we propose an approximation to the copula which we show to

n converge uniformly to the true copula. To illustrate the usefulness of the theory, we derive
a a fast procedure to estimate the underlying parameter in Manneville-Pomeau processes.
J
Keywords. Copulas; Manneville-Pomeau Processes; Invariant Measures; Parametric Esti-
8
1 mation.

]
T
S 1 Introduction
.
h
t
a Thestatisticsofstochasticprocessesderivedfromdynamicalsystemshasseenagrownattention
m
in the last decade or so (see Chazottes et al. (2005) and references therein). The relationship
[ between copulas and areas such ergodic theory and dynamical systems also have seen some

2 development, especially in the last few years (see, for instance, Koles´arov´a et al. (2008)). In
v this work our aim is to contribute with the area by identifying and studying the copulas related
8
to random vectors coming from the so-called Manneville-Pomeau processes, which are obtained
2
6 as iterations of the Manneville-Pomeau transformation to a specific chosen random variable (see
2 Definitions 2.1 and 2.2). We cover both, bidimensional and n-dimensional cases, which share a
.
9 lot more in common than one could expect.
0
1 The copulas derived here depend on a probability measure which has no closed formula. In
1
order to minimize this deficiency, we propose an approximation to the copula which we show
:
v to converge uniformly to the true copula. The copula also depend on several functions which
i
X have to be approximated as well, so the approximation depends on several intermediate steps.
r The results related to the convergence of the proposed approximation presented here are far
a
more general than we need and actually allows one to change these intermediate approximations
and still obtain the uniform convergence result for the approximated copula. We also address
problems related to random variate generation of the copula and present the results of some
simple numerical experiments in order to assess the stability and precision of the intermediate
approximations. Theusefulnessofthetheoryisillustratedbyasimpleapplicationtotheproblem
of estimating the underlying parameter in Manneville-Pomeau processes.
The paper is organized as follows: in the next section, we briefly review some concepts
and results on Manneville-Pomeau transformations and processes and on copulas. Section 3
is devoted to determine the copulas related to any pair (X ,X ) from a Manneville-Pomeau
t t+h
process and to explore some consequences. In Section 4, the multidimensional extensions are
shown. In Section 5 an approximation to the copulas derived in Section 3 is proposed. This
2 Copulas Related to Manneville-Pomeau Processes
approximation,whichisshowntoconvergeuniformlytothetruecopula,isthenappliedtoexploit
some characteristics of the copulas related to Manneville-Pomeau process through statistical
and graphical analysis. Some computational and random variate generation problems are also
addressed. In Section 6 we illustrate the usefulness of the theory by deriving a fast procedure to
estimate the underlying parameter in Manneville-Pomeau processes. Conclusions are reserved
to Section 7.
2 Some Background
In this section we shall briefly review some basic results on Manneville-Pomeau transformations
and related processes as well as some concepts on copulas needed later. We start with the
definition of the Manneville-Pomeau transformation.
Definition 2.1. The map T : [0,1] −→ [0,1], given by
s
T (x)=x+x1+s(mod1),
s
for s > 0, is called the Manneville-Pomeau transformation (MP transformation, for short).
In what follows, λ shall denote the Lebesgue measure in I := [0,1] and the k-fold compo-
sition will be denoted, as usual, by Tk = T ◦ ··· ◦ T . Figure 1 shows the plot of the MP
s s s
transformation for the values of s ∈ {0.5,1,10,100}. The plots show the usual behavior of the
MP transformations: for any s, they are increasing and differentiable functions by parts in I.
Furthermore, for any s > 0, the function Tk will have exactly 2k parts.
s
Figure 2.1: Plot of the Manneville-Pomeau transformation for different values of s∈{0.5,1,10,100}.
Pianigiani (1980) shows the existence of a T -invariant and absolutely continuous measure
s
with respect to the Lebesgue measure in I which will be denoted henceforth by µ . However,
s
the proof uses Perron-Frobenius operator theory and is, for practical purposes, non-constructive
so that an explicit form for a T -invariant measure is unknown. However, this measure will be
s
a Sinai-Bowen-Ruelle (SBR) measure in the sense that the weak convergence
n−1
1 (cid:88)
δ (A)−→µ (A) (2.1)
n Tsk(x) s
k=0
holds for almost all x ∈ I and all µ -continuity sets1 A, where δ (·) is the Dirac measure at a.
s a
As a dynamical system, the triple (I,µ ,T ) is exact (that is, lim (µ ◦Tk)(A) = 1, for
s s k→∞ s s
all positive µ -measurable sets A) which implies ergodicity and strong-mixing. When s < 1,
s
1Recall that a set A is a µ-continuity set if µ(∂A) = 0, where ∂A denotes the boundary of A. The measure
theoreticalresultsappliedherecanbefound,forinstance,inRoyden(1988). Agoodreferenceinweakconvergence
of probability measures is Billingsley (1999) and for ergodic theoretical related results, see Pollicott and Yuri
(1998).
S.R.C. Lopes and G. Pumi 3
µ is a probability measure, while if s ≥ 1, µ is no longer finite, but σ-finite (see Fisher
s s
and Lopes (2001)). Furthermore, it can be shown that µ has a positive, bounded continuous
s
Radon-Nikodym derivative dµ = h (x)dx, fact that will be useful later. For further details
s s
in the theory of MP transformations and related results, we refer to Pianigiani (1980), Young
(1999), Maesetal. (2000)andFisherandLopes(2001). Forapplications, seeZebrowsky(2001),
Olbermann et al. (2007) and Lopes and Lopes (1998).
Definition 2.2. Let s ∈ (0,1) and let U be a random variable distributed according to (the
0
probability measure) µ . Let ϕ : [0,1] −→ R be a function in L1(µ ). The stochastic process
s s
given by
X =(ϕ◦Tt)(U ), for all t∈N,
t s 0
is called a Manneville-Pomeau process (or MP process, for short).
The MP process, as defined above, is stationary since µ is a T -invariant measure and
s s
µ (cid:28) λ. It is also ergodic since µ is ergodic for T . By its turn, copulas are distribution
s s s
functions whose marginals are uniformly distributed on I. The copula literature has grown
enormously in the last decade, especially in terms of empirical applications and have become
standard tools in financial data analysis (see Nelsen (2006) and references therein). The next
theorem, known as Sklar’s theorem, is the key result for copulas and elucidates the role played
by them. See Schweizer and Sklar (2005) for a proof.
Theorem 2.1 (Sklar). Let X ,··· ,X be random variables with marginals F ,··· ,F , respec-
1 n 1 n
tively, and joint distribution function H. Then, there exists a copula C such that,
H(x ,··· ,x ) = C(cid:0)F (x ),··· ,F (x )(cid:1), for all (x ,··· ,x ) ∈ Rn.
1 n 1 1 n n 1 n
If the F ’s are continuous, then C is unique. Otherwise, C is uniquely determined on Ran(F )×
i 1
···×Ran(F ). The converse also holds. Furthermore,
n
C(u ,··· ,u ) = H(cid:0)F(−1)(u ),··· ,F(−1)(u )(cid:1), for all (u ,··· ,u ) ∈ In,
1 n 1 1 n n 1 n
where for a function F, F(−1) denotes its pseudo-inverse given by F(−1)(x) := inf(cid:8)u ∈ Ran(F) :
(cid:9)
F(u) ≥ x .
The next theorem, whose proof can be found, for instance, throughout Nelsen (2006), shall
prove very useful in what follows. Except stated otherwise, the measure implicit to phrases like
“almost sure”, “almost everywhere” and so on will be the (appropriate) Lebesgue measure.
Theorem 2.2. Let X and Y be continuous random variables with copula C. If f is an almost
everywhere decreasing function then C (u,v) = u−C (u,1−v). Furthermore, if f and
f(X),Y X,Y 1
f are functions increasing almost everywhere, then C (u,v) = C (u,v).
2 f1(X),f2(Y) X,Y
For an introduction to copulas, we refer the reader to Nelsen (2006). For more details and
extensions to the multivariate case with emphasis in modeling and dependence concepts, see
Joe (1997). The theory of copulas is also intimately related to the theory of probabilistic metric
spaces, see Schweizer and Sklar (2005) for more details in this matter.
3 Copulas and MP Processes: Bidimensional Case
In this section we shall investigate the bidimensional copulas associated to pairs of random
variables coming from MP processes which we shall call MP copulas. As we will see later, the
4 Copulas Related to Manneville-Pomeau Processes
multidimensionalcaseisverysimilartothebidimensionalcase, soweshallgivespecialattention
to the latter.
First, let {Xn}n∈N be an MP process with parameter s ∈ (0,1) and ϕ ∈ L1(µs) be an
increasing almost everywhere function. Throughout this section and in the rest of the paper, we
shall treat s ∈ (0,1) as a given fixed number. Let
F (x):=P(U ≤x)=µ (cid:0)[0,x](cid:1).
0 0 s
Since µ (cid:28) λ, µ is non-atomic and, therefore, F will be (uniformly) continuous. The existence
s s 0
of a positive Radon-Nikodym density for µ also shows that F will be increasing and its inverse
s 0
will be well defined. Let F be the distribution function of Tt(U ), for all t ∈ N. For x ∈ I,
t s 0
notice that
F (x):=P(cid:0)Tt(U )≤x(cid:1)=µ (cid:0)(Tt)−1(cid:0)[0,x](cid:1)(cid:1)=µ (cid:0)[0,x](cid:1)=F (x), (3.1)
t s 0 s s s 0
since µ is a T -invariant measure.
s s
In what follows, we shall need the solution for the inequality Tt(X) ≤ y, y ∈ (0,1), in X,
s
for X a random variable taking values in I. Now, since each of the 2t parts of Tt is one-to-one
s
in its domain, the inverse of Tt will also be continuous by parts and each part will also be a
s
one-to-one function in its domain. Let 0 = a ,··· ,a = 1 be the end points of each part of
t,0 t,2t
Tt. We shall call each interval [a ,a ) a node of Tt, for k = 0,··· ,2t −1 and t > 0. The
s t,k t,k+1 s
(piecewise) inverse of Tt can be conveniently written as
s
(Tt)−1 :I −→ I2t
s
(cid:0) (cid:1)
y (cid:55)−→ T (y),··· ,T (y) , (3.2)
t,0 t,2t−1
where T (y) denotes the inverse of Tt restricted to its k-th node, for all k ∈ {0,··· ,2t −1}.
t,k s
Notice that both T and a depend on s for each k, but since no confusion will arise, and for
t,k t,k
the sake of simplicity, we shall omit this dependence from the notation as we shall do in several
other occasions. Now, the solution of the inequality Tt(X) ≤ y in X can be determined and is
s
given by X ∈ A (y)(cid:83) ···(cid:83)A (y), where
t,0 t,2t−1
(cid:2) (cid:3)
A (y)= a ,T (y) , (3.3)
t,k t,k t,k
which will be a proper closed subinterval of [a ,a ), for each k = 0,··· ,2t−1. Notice that
t,k t,k+1
A (y) (whose dependence on s was omitted from the notation) is just the inverse image of
t,k
[0,y] by the transformation Tt restricted to the node [a ,a ). We can now use this result
s t,k t,k+1
to prove the following useful lemma.
Lemma 3.1. Let X be a random variable taking values in I and let T be the MP transformation
s
with parameter s > 0. Then, for any t ∈ N and x ∈ I,
2t−1
P(cid:0)Tt(X)≤x(cid:1)=P(cid:0)X ∈(cid:83)2t−1A (x)(cid:1)= (cid:88) P(cid:0)X ∈A (x)(cid:1),
s k=0 t,k t,k
k=0
where A ’s are given by (3.3).
t,k
Proof: The result follows easily from what was just discussed and from the fact that the
intervals A ’s are (pairwise) disjoints. (cid:4)
t,k
As for the copulas related to MP processes, in view of the stationarity of the MP process,
the following result follows easily.
Proposition 3.1. Let {Xn}n∈N be an MP process with parameter s ∈ (0,1) and ϕ ∈ L1(µs) be
an almost everywhere increasing function. Then, for any t,h ∈ N,
C (u,v) = C (u,v),
Xt,Xt+h X0,Xh
everywhere in I2.
S.R.C. Lopes and G. Pumi 5
Proof: As consequence of the stationarity of {Xt}t∈N, if we let the joint distribution of the
pair (Xp,Xq) for any p,q ∈ N, p (cid:54)= q, be denoted by H(cid:101)p,q(·,·), it follows that for all x,y ∈ (0,1),
t ∈ N and h ∈ N∗ := N\{0}, H(cid:101)t,t+h(x,y) = H(cid:101)0,h(x,y). Now, upon applying Sklar’s Theorem
and (3.1), it follows that
CXt,Xt+h(u,v)=H(cid:101)t,t+h(cid:0)Ft−1(u),Ft−+1h(v)(cid:1)=H(cid:101)0,h(cid:0)F0−1(u),Fh−1(v)(cid:1)=CX0,Xh(u,v),
for all (u,v) ∈ I2. (cid:4)
Corollary 3.1. Let T be the MP transformation for some s ∈ (0,1), µ be a T -invariant
s s s
probability measure and let U be distributed as µ . Then, for any t,h ∈ N, h (cid:54)= 0,
0 s
C (u,v) = C (u,v)
Tst(U0),Tst+h(U0) U0,Tsh(U0)
everywhere in I2.
Proof: Immediate from Theorem 2.2 applied to Proposition 3.1. (cid:4)
Nowweturnourattentiontodeterminethecopulaassociatedtoanypairofrandomvariables
(X ,X ), p,q ∈ N, obtained from an MP process with ϕ increasing almost everywhere. For the
p q
sake of simplicity, let us introduce the following functions: let h be a positive integer and for
k = 0,··· ,2h−1, let F :I →(cid:2)F (a ),F (a )(cid:3)be given by
h,k 0 h,k 0 h,k+1
F (x):=F (cid:0)T (cid:0)F−1(x)(cid:1)(cid:1).
h,k 0 h,k 0
Notice that for each k, F (0) = F (a ) and F (1) = F (a ) and F is a one to one,
h,k 0 h,k h,k 0 h,k+1 h,k
increasing and uniformly continuous function.
Proposition 3.2. Let {Xn}n∈N be an MP process with parameter s ∈ (0,1), ϕ ∈ L1(µs) be an
increasing almost everywhere function and let F be the distribution function of U . Then, for
0 0
any t,h ∈ N, h (cid:54)= 0 and (u,v) ∈ I2,
(cid:32)n(cid:88)0−1 (cid:33) (cid:8) (cid:9)
CXt,Xt+h(u,v)= Fh,k(v)−F0(ah,k) δN∗(n0)+min u,Fh,n0(v) −F0(ah,n0), (3.4)
k=0
where δN∗(x) equals 1, if x ∈ N∗ and 0, otherwise, {ah,k}2kh=0 are the end points of the nodes of
Th and n := n (u;h) = (cid:8)k : u ∈ (cid:2)F (a ),F (a )(cid:1)(cid:9) ∈ {0,··· ,2h−1}.
s 0 0 0 h,k 0 h,k+1
Proof: By Propositions 3.1 and 2.2, it suffices to derive the copula of the pair (cid:0)U ,Th(U )(cid:1).
0 s 0
So let again {Xn}n∈N be an MP process with parameter s ∈ (0,1) and ϕ ∈ L1(µs) be an
increasing almost everywhere function and let H (·,·) denote the distribution function of the
0,h
pair (cid:0)U ,Th(U )(cid:1). Notice that
0 s 0
H (x,y)=P(U ≤x,Th(U )≤y)=P(cid:0)U ≤x,U ∈(cid:83)2h−1A (y)(cid:1)
0,h 0 s 0 0 0 k=0 h,k
=P(cid:0)U ∈[0,x](cid:84)(cid:83)2h−1A (y)(cid:1)=P(cid:0)U ∈(cid:83)2h−1(cid:2)[0,x](cid:84)A (y)(cid:3)(cid:1)
0 k=0 h,k 0 k=0 h,k
2h−1
= (cid:88) P(cid:0)U ∈[0,x](cid:84)A (y)(cid:1),
0 h,k
k=0
for any x,y ∈ (0,1). Now let n := n (x;h) = (cid:8)k : x ∈ [a ,a )(cid:9) ∈ {0,··· ,2h −1} and
1 1 h,k h,k+1
(cid:2) (cid:3)
assume for the moment that n ≥ 1. Since A (y) = a ,T (y) , it follows
1 h,k h,k h,k
H (x,y)=n(cid:88)1−1P(cid:0)U ∈A (y)(cid:1)+P(cid:0)U ∈A (y)(cid:84)[a ,x](cid:1)
0,h 0 h,k 0 h,n1 h,n1
k=0
6 Copulas Related to Manneville-Pomeau Processes
=n(cid:88)1−1µ (cid:0)A (y)(cid:1)+µ (cid:0)(cid:2)a ,T (y)(cid:3)(cid:84)[a ,x](cid:1)
s h,k s h,n1 h,n1 h,n1
k=0
n(cid:88)1−1 (cid:0)(cid:2) (cid:3)(cid:1) (cid:0)(cid:2) (cid:3)(cid:1)
= µ a ,T (y) +µ a ,min{x,T (y)} ,
s h,k h,k s h,n1 h,n1
k=0
which can be written, since F (x) = µ ([0,x]) is increasing, as
0 s
n(cid:88)1−1(cid:2) (cid:0) (cid:1) (cid:3) (cid:8) (cid:0) (cid:1)(cid:9)
H (x,y)= F T (y) −F (a ) +min F (x),F T (y) −F (a ).
0,h 0 h,k 0 h,k 0 0 h,n1 0 h,n1
k=0
If n = 0, the summation is absent of the formula and we have
1
(cid:8) (cid:0) (cid:1)(cid:9)
H (x,y)=min F (x),F T (y) −F (a ),
0,h 0 0 h,0 0 h,0
so that, in any case, we have
(cid:32)n(cid:88)1−1(cid:2) (cid:0) (cid:1) (cid:3)(cid:33) (cid:8) (cid:0) (cid:1)(cid:9)
H0,h(x,y)= F0 Th,k(y) −F0(ah,k) δN∗(n1)+min F0(x),F0 Th,n1(y) −F0(ah,n1).
k=0
Now upon applying Sklar’s Theorem, it follows that
C (u,v)=H (cid:0)F−1(u),F−1(v)(cid:1)=H (cid:0)F−1(u),F−1(v)(cid:1)
U0,Tsh(U0) 0,h 0 h 0,h 0 0
(cid:32)n(cid:88)0−1 (cid:33) (cid:8) (cid:9)
= Fh,k(v)−F0(ah,k) δN∗(n0)+min u,Fh,n0(v) −F0(ah,n0),
k=0
where n := n (u;h) = n (cid:0)F−1(u);h(cid:1) = (cid:8)k : u ∈ (cid:2)F (a ),F (a )(cid:1)(cid:9). The result now
0 0 1 0 0 h,k 0 h,k+1
follows from Proposition 3.1. (cid:4)
Remark 3.1. Notice that the copula (3.4) can be expressed in terms of µ as
s
CXt,Xt+h(u,v) = (cid:32)n(cid:88)0−1µs(cid:16)(cid:2)ah,k,Th,k(cid:0)F0−1(v)(cid:1)(cid:3)(cid:17)(cid:33)δN∗(n0)+
k=0
+µ (cid:16)(cid:2)a ,min(cid:8)F−1(u),T (cid:0)F−1(v)(cid:1)(cid:9)(cid:3)(cid:17), (3.5)
s h,n0 0 h,n0 0
which will prove useful in Section 5. Also, expression (3.5) is helpful if one desires to verify
directly that the marginals of (3.4) are indeed uniform.
In the next proposition we address the case where ϕ is an almost everywhere decreasing
function. In view of Theorem 2.2, one could, at first glance, think that a result like C =
X0,Xh
C would not hold anymore, but in fact it still does, as it is shown in the next proposition.
Xt,Xt+h
Proposition 3.3. Let {Xn}n∈N be an MP process with parameter s ∈ (0,1), ϕ ∈ L1(µs) be
an almost everywhere decreasing function and let F be the distribution function of U . Then,
0 0
C (u,v) = C (u,v) everywhere in I2 and, for any t,h ∈ N and h (cid:54)= 0,
X0,Xh Xt,Xt+h
(cid:32)(cid:88)n0 (cid:33)
CXt,Xt+h(u,v) = u+v−1+ [Fh,k(1−v)−F0(ah,k)] δN∗(n0)+
k=0
(cid:8) (cid:9)
+ min 1−u,F (1−v) −F (a ), (3.6)
h,n0 0 h,n0
for all (u,v) ∈ I2, where {a }2h are the end points of the nodes of Th and n := n (u;h) =
h,k k=0 s 0 0
(cid:8) (cid:0) (cid:3)(cid:9)
k : u ∈ 1−F (a ),1−F (a ) .
0 h,k+1 0 h,k
S.R.C. Lopes and G. Pumi 7
Proof: Since the inverse of an almost everywhere decreasing function is still decreasing almost
everywhere and X = ϕ(cid:0)Tt(U )(cid:1), upon applying Theorem 2.2 twice, it follows that
t s 0
C (u,v) = C (u,v)=u−C (u,1−v)
Tst(U0),Tst+h(U0) ϕ−1(Xt),ϕ−1(Xt+h) Xt,ϕ−1(Xt+h)
(cid:0) (cid:1)
= u− 1−v−C (1−u,1−v) ,
Xt,Xt+h
or, equivalently (changing u by 1−u and v by 1−v),
C (u,v)=u+v−1+C (1−u,1−v). (3.7)
Xt,Xt+h Tst(U0),Tst+h(U0)
Now(3.6)followsuponapplyingProposition3.2withtheidentitymapandsubstitutingequation
(3.4) into (3.7). As for the equality C (u,v) = C (u,v), Corollary 3.1 and Theorem
X0,Xh Xt,Xt+h
2.2 applied to (3.7) yield
C (u,v) = u+v−1+C (1−u,1−v)
Xt,Xt+h U0,Tsh(U0)
= u+v−1+C (1−u,1−v)
ϕ−1(ϕ(U0)),ϕ−1(ϕ(Tsh(U0)))
= C (u,v) = C (u,v),
ϕ(U0),ϕ(Tsh(U0)) X0,Xh
everywhere in I2, as desired. (cid:4)
Remark 3.2. In view of the “stationarity” results of Theorems 3.1 and 3.3, a copula associated
to a pair (X ,X ) from an MP process will be referred as lag h MP copula.
t t+h
Thecopulasin(3.4)and(3.6)are bothsingular, asitcan bereadilyverified. Sothequestion
that naturally arises is, for each h, what is the support of C ? The question is addressed
Xt,Xt+h
in the next proposition, which will be useful in Sections 5 and 6. For simplicity, for a given MP
process and h > 0, let (cid:96)+ ,(cid:96)− : (cid:2)F (a ),F (a )(cid:1) → I be functions defined by
h,k h,k 0 h,k 0 h,k+1
x−F (a ) F (a )−x
(cid:96)+ (x) = 0 h,k and (cid:96)− (x) = 0 h,k+1 ,
h,k F (a )−F (a ) h,k F (a )−F (a )
0 h,k+1 0 h,k 0 h,k+1 0 h,k
for all k = 0,··· ,2h−1. Notice that, for each k, (cid:96)+ is the linear function connecting the points
h,k
(cid:0)F (a ),0(cid:1)and(cid:0)F (a ),1(cid:1), while(cid:96)− connectsthepoints(cid:0)F (a ),1(cid:1)and(cid:0)F (a ),0(cid:1).
0 h,k 0 h,k+1 h,k 0 h,k 0 h,k+1
Proposition 3.4. Let {Xn}n∈N be an MP process with parameter s ∈ (0,1), for ϕ1 ∈ L1(µs)
an almost everywhere increasing function and let {Yn}n∈N be an MP process with parameter s ∈
(0,1), for ϕ ∈ L1(µ ) an almost everywhere decreasing function. Also let F be the distribution
2 s 0
function of U . Then, for any t,h ∈ N, h > 0,
0
supp{C }=(cid:83)2h−1(cid:8)(cid:0)u,(cid:96)+ (u)(cid:1):u∈(cid:2)F (a ),F (a )(cid:1)(cid:9) (3.8)
Xt,Xt+h k=0 h,k 0 h,k 0 h,k+1
and
supp{C }=(cid:83)2h−1(cid:8)(cid:0)u,(cid:96)− (u)(cid:1):u∈(cid:2)F (a ),F (a )(cid:1)(cid:9). (3.9)
Yt,Yt+h k=0 h,k 0 h,k 0 h,k+1
Proof: Let R = [u ,u ]×[v ,v ] be a rectangle in I2 and let its C -volume be denoted
by V (R). Let k ∈1{0,2··· ,2h1−21} be fixed and suppose that u ∈ (cid:2)XFt,X(at+h),F (a )(cid:3). This
CX i 0 h,k 0 h,k+1
implies that n = k for all four terms in V (R), hence the summands and constants on the
0 CX
copula cancel out so that we have
(cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)
V (R)=min u ,F (v ) +min u ,F (v ) −min u ,F (v ) −min u ,F (v )
CX 1 h,k 1 2 h,k 2 1 h,k 2 2 h,k 1
(cid:0) (cid:1)
=V [u ,u ]×[F (v ),F (v )] ,
M 1 2 h,k 1 h,k 1
where M(u,v) = min{u,v} is the Frech`et upper bound copula whose support is the main
diagonal in I2. Since [u ,u ]×[F (v ),F (v )] ⊂ [F (a ),F (a )]2, V (R) > 0 if, and
only if, R(cid:84)(cid:8)(cid:0)u,(cid:96)+ (u)(cid:1)1: u2∈ (cid:2)F h(a,k 1),F h(a,k 1 )(cid:1)(cid:9) (cid:54)=0∅.h,k 0 h,k+1 CX
h,k 0 h,k 0 h,k+1
8 Copulas Related to Manneville-Pomeau Processes
(cid:2)
Analogously, denoting the C -volume of R by V (R), if u ∈ 1 − F (a ),1 −
(cid:3) Yt,Yt+h CY i 0 h,k
F (a ) , we have
0 h,k+1
(cid:8) (cid:9) (cid:8) (cid:9)
V (R)=min 1−u ,F (1−v ) +min 1−u ,F (1−v ) −
CY 1 h,k 1 2 h,k 2
(cid:8) (cid:9) (cid:8) (cid:9)
− min 1−u ,F (1−v ) −min 1−u ,F (1−v )
1 h,k 2 2 h,k 1
(cid:0) (cid:1)
=V [1−u ,1−u ]×[F (1−v ),F (1−v )] . (3.10)
M 1 2 h,k 2 h,k 1
Since [1 − u ,1 − u ] × [F (1 − v ),F (1 − v )] ⊂ [F (a ),F (a )]2, V (R) is pos-
itive if, and1only if2, R(cid:84)(cid:8)h(cid:0),ku,(cid:96)− (u1)(cid:1) :h,uk ∈ (cid:2)F2(a ),F0 (ah,k )0(cid:1)(cid:9)h,(cid:54)=k+1∅ (notCicYe the terms
h,k 0 h,k 0 h,k+1
1 − v in expression (3.10), for i = 1,2). Now (3.8) and (3.9) follow by observing that I =
i
(cid:83)2h−1(cid:2)F (a ),F (a )(cid:3) = (cid:83)2h−1(cid:2)1−F (a ),1−F (a )(cid:3). (cid:4)
k=0 0 h,k 0 h,k+1 k=0 0 h,k+1 0 h,k
Remark 3.3. We end up this section by noticing that as an application of Propositions 3.1
and 3.3, together with the so-called copula version of Hoeffding’s lemma (see Nelsen (2006)),
we can show in a rather different way that an MP process is weakly stationary. Let F be the
Xt
distribution function of X and notice that F (x) = F (x), for all t ∈ N, by the stationarity
t Xt X0
of {Xt}t∈N and since CXt,Xt+h(u,v) = CX0,Xh(u,v), the result follows immediately.
4 Multidimensional Case
In this section we are interested in extending the results from the previous section to the mul-
tidimensional case, that is, in this section we are interested in deriving the copulas associated
to n-dimensional vectors (X ,··· ,X ), t ,··· ,t ∈ N, coming from an MP process with ϕ
t1 tn 1 n
an increasing almost everywhere function. In view of Theorem 2.2, it suffices to derive the
copula associated to the vector (cid:0)Tst1(U0),··· ,Tstn(U0)(cid:1). It turns out that there are more simi-
larities between the bidimensional and multidimensional cases than one could expect. In fact,
an expression very similar in form to (3.4) holds for the multidimensional case as well.
Let {Xn}n∈N be an MP process with parameter s ∈ (0,1) and ϕ ∈ L1(µs) be an al-
most everywhere increasing function. For the sake of simplicity, we shall use the following
notation: let a,b ∈ N, a < b, we shall write x := (x ,··· ,x ) and for a function f,
a:b a b
(cid:0) (cid:1)
f(x ) := f(x ),··· ,f(x ) . Again we shall denote the distribution function of U by F .
a:b a b 0 0
Theorem 4.1. Let {Xn}n∈N be an MP process with parameter s ∈ (0,1), with ϕ ∈ L1(µs) an
almost everywhere increasing function. Let t ,··· ,t ∈ N and set h := t −t . Then, for all
1 n i i 1
(u ,··· ,u ) ∈ In,
1 n
CXt1,···,Xtn(u1,··· ,un)=(cid:32)n(cid:88)0−1F0(cid:16)bhn,k(cid:0)F0−1(u2:n)(cid:1)(cid:17)−F0(ahn,k)(cid:33)δN∗(n0)+
k=0
+ min(cid:8)u ,F (cid:0)b (cid:0)F−1(u )(cid:1)(cid:1)(cid:9)−F (a ), (4.1)
1 0 hn,n0 0 2:n 0 hn,n0
where n := n (cid:0)u ,n) = (cid:8)k : u ∈ (cid:2)F (a ),F (a )(cid:1)(cid:9), {a }2h are the end points of
0 0 1 1 0 hn,k 0 hn,k+1 hn,k k=0
(thxe,n··o·de,sxo)f∈TsIhnn−, 1f,orb i =(x2,··)·=,n,mjin=(cid:8)0c,·(·x·2;hhi −,k1)(cid:9),,Twhii,tjhis given by (3.2) and for a vector
2 n hn,k 2:n i=2,···,n i i n
(cid:40)
a , if B (x ;h ,k)=∅;
c (x ;h ,k)= hn,k i i n
i i n
B (x ;h ,k), otherwise.
i i n
and
(cid:8) (cid:9)
B (x ;h ,k)= min T (x ):T (x )>a and a <a .
i i n j=0,···,2hi−1 hi,j i hi,j i hn,k hi,j hn,k+1
S.R.C. Lopes and G. Pumi 9
Proof: Let{Xn}n∈N beanMPprocesswithparameters ∈ (0,1)andϕ ∈ L1(µs)beanalmost
everywhereincreasingfunction. Withoutlossofgenerality,wecanassumethat0 < t < ··· < t .
1 n
bInevtiheewdoisftTrihbeuotrioemn f2u.n2c,tiitosnuoffifc(cid:0)eTsstt1o(Uw0o)r,k··w·i,tThsttnh(eU0v)e(cid:1)c.toLre(cid:0)tThst1i(=U0t)i,−···t1,,Tfstonr(Uea0c)h(cid:1).iL=et1H,·t1·,····,,ntn,
and notice that h > 0 since t < t , for all i = 2,··· ,n. Let (x ,··· ,x ) ∈ (0,1)n and for the
i 1 i 1 n
sake of simplicity, let Yt1 := Tst1(U0), so that we have
H (x ,··· ,x )=P(cid:0)Tt1(U )≤x ,··· ,Ttn(U )≤x (cid:1)
t1,···,tn 1 n s 0 1 s 0 n
=P(cid:0)Y ≤x ,Th2(Y )≤x ,··· ,Thn(Y )≤x (cid:1)
t1 1 s t1 2 s t1 n
(cid:16) (cid:17)
=P Y ∈[0,x ],Y ∈(cid:83)2h2−1A (x ),··· ,Y ∈(cid:83)2hn−1A (x )
t1 1 t1 k=0 h2,k 2 t1 k=0 hn,k n
=P(cid:16)Y ∈[0,x ](cid:84)n (cid:2)(cid:83)2hi−1A (x )(cid:3)(cid:17)
t1 1 i=2 k=0 hi,k i
=P(cid:16)U ∈(cid:84)n (cid:83)2hi−1(cid:2)[0,x ](cid:84)A (x )(cid:3)(cid:17), (4.2)
0 i=2 k=0 1 hi,k i
where A ’s are given by (3.3) and the last equality is a consequence of the T -invariance of
hi,k s
µs. For k = 0,··· ,2hn−1, let
A(cid:101)hn,k(x2:n)=Ahn,k(xn)(cid:84)ni=−21(cid:2)(cid:83)2j=hi0−1Ahi,j(xi)(cid:3).
In order to simplify the notation, for i = 2,··· ,n and k = 0,··· ,2hn −1, let
(cid:8) (cid:9)
B (x ;h ,k)= min T (x ):T (x )>a and a <a .
i i n j=0,···,2hi−1 hi,j i hi,j i hn,k hi,j hn,k+1
For each k and i, B (x ;h ,k) is either the smallest T (x ) which is greater than a and
i i n hi,j i hn,k
such that the correspondent A (x ) has non-empty intersection with A (x ), or empty. Let
hi,j i hn,k n
(cid:40)
a , if B (x ;h ,k)=∅;
c (x ;h ,k)= hn,k i i n
i i n
B (x ;h ,k), otherwise.
i i n
Then, for each k = 1,··· ,2hn −1, setting bhn,k(x2:n) = i=m2,i··n·,n(cid:8)ci(xi;hn,k)(cid:9), it follows that
(cid:2) (cid:3)
A(cid:101)hn,k(x2:n)= ahn,k,bhn,k(x2:n) ,
which is a closed subset of [a ,a ]. Also notice that, from the definition ofb (x ), we
hn,k hn,k+1 hn,k 2:n
couldhaveA(cid:101)hn,k(x2:n) = {ahn,k},inwhichcasewesetA(cid:101)hn,k(x2:n) = ∅(althoughfromameasure-
theoretical point of view, this correction makes no difference). Again we are omitting the
dependence in s from the notation on both, bhn,k and A(cid:101)hn,k. Each bhn,k(x2:n) above determines
the smallest Thi,j(xi) that lies on the k-th node of Tshn (which has the smallest nodes among all
Tshi’s), so that A(cid:101)hn,k’s are just the intersection of all Ahi,k(xi)’s with end point in the k-th node
of Tshn. Also notice that the A(cid:101)hn,k’s are pairwise disjoints. One can rewrite (4.2) as
Ht1,···,tn(x1,··· ,xn)=P(cid:16)U0 ∈(cid:83)k2h=n0−1(cid:2)A(cid:101)hn,k(x2:n)(cid:84)[0,x1](cid:3)(cid:17). (4.3)
Now, let n1 := n1(x1;n) = (cid:8)k : x1 ∈ [ahn,k,ahn,k+1)(cid:9) ∈ {0,··· ,2hn −1}, and assume for the
moment that n ≥ 1. Then (4.3) becomes
1
Ht1,···,tn(x1,··· ,xn)=n(cid:88)1−1P(cid:0)U0 ∈A(cid:101)hn,k(x2:n)(cid:1)+P(cid:0)U0 ∈A(cid:101)hn,n1(x2:n)(cid:84)[ahn,n1,x1](cid:1)
k=0
n(cid:88)1−1 (cid:0) (cid:1) (cid:0) (cid:1)
= µ [a ,b (x )] +µ [a ,min{x ,b (x )}]
s hn,k hn,k 2:n s hn,n1 1 hn,n1 2:n
k=0
n(cid:88)1−1(cid:2) (cid:0) (cid:1) (cid:3) (cid:8) (cid:9)
= F b (x ) −F (a ) +min F (x ),F (b (x )) −F (a ).
0 hn,k 2:n 0 hn,k 0 1 0 hn,n1 2:n 0 hn,n1
k=0
10 Copulas Related to Manneville-Pomeau Processes
If n = 0, then
1
(cid:8) (cid:9)
H (x ,··· ,x )=min F (x ),F (b (x )) −F (a ).
t1,···,tn 1 n 0 1 0 hn,0 2:n 0 hn,0
In any case, we can write
(cid:32)n(cid:88)1−1 (cid:0) (cid:1) (cid:33)
Ht1,···,tn(x1,··· ,xn) = F0 bhn,k(x2:n) −F0(ahn,k) δN∗(n1)+
k=0
(cid:8) (cid:9)
+ min F (x ),F (b (x )) −F (a ).
0 1 0 hn,n1 2:n 0 hn,n1
Recall that the distribution function of Tt(U ) is also F by the T -invariance of µ . Now
s 0 0 s s
applying Sklar’s Theorem, it follows that,
C (u ,··· ,u )=H (cid:0)F−1(u ),··· ,F−1(u )(cid:1)
Xt1,···,Xtn 1 n t1,···,tn 0 1 0 n
=(cid:32)n(cid:88)0−1F0(cid:16)bhn,k(cid:0)F0−1(u2:n)(cid:1)(cid:17)−F0(ahn,k)(cid:33)δN∗(n1)+
k=0
+min(cid:8)u ,F (cid:0)b (cid:0)F−1(u )(cid:1)(cid:1)(cid:9)−F (a ).
1 0 hn,n0 0 2:n 0 hn,n0
where n := n (cid:0)F−1(u ),n(cid:1) = (cid:8)k : u ∈ (cid:2)F (a ),F (a )(cid:1)(cid:9), which is the desired formula.
0 1 0 1 1 0 hn,k 0 hn,k+1
(cid:4)
Remark 4.1. Notice that the proof of Theorem 4.1 from equation (4.3) on is exactly the same
as the one in Proposition 3.2 with the obvious notational adaptations.
Now we turn our attention to the case where ϕ is an almost everywhere decreasing function.
In view of Theorem 2.2, one cannot expect a simple expression for the copula. What happens
is that the copula in this case will be the sum of the lower dimensions copulas related to the
iterations Tk(U ), as the next proposition shows.
s 0
Proposition 4.1. Let {Xn}n∈N be an MP process with parameter s ∈ (0,1), and ϕ ∈ L1(µs)
be an almost everywhere decreasing function. Let t,h ,··· ,h ∈ N, 0 < h < ··· < h
1 n 1 n
and set Y0 := U0 and Yk := Tshk(U0). Denote the copula associated to the random vector
(X ,X ,··· ,X ) by C . Then the following relation holds
t t+h1 t+hn t
n n n
(cid:88) (cid:88) (cid:88)
C (u ,··· ,u )=1−n+ u + C (1−u ,1−u )+···+
t 0 n i Yi,Yj i j
i=0 i=0j=i+1
n n n
(cid:88) (cid:88) (cid:88)
+(−1)n−1 ··· C (1−u ,··· ,1−u )+
Yk1,···,Ykn−1 k1 kn−1
k1=0k2=k1+1 kn−1=kn−2+1
+(−1)nC (1−u ,··· ,1−u ), (4.4)
U0,Y1,···,Yn 0 n
everywhere in In+1.
Proof: Let t,h1,··· ,hn ∈ N, 0 < h1 < ··· < hn, t (cid:54)= 0. Set Y0 := U0, Yk := Tshk(U0) and
y := ϕ(x ). We have
k k
H (x ,x ,··· ,x )=P(cid:0)U ≥y ,Y ≥y ,··· ,Y ≥y (cid:1)
X0,Xh1,···,Xhn 0 1 n 0 0 1 1 n n
= P(cid:16)U0 ≥y0(cid:12)(cid:12)Y1 ≥y1,Y2 ≥y2,··· ,Yn ≥yn(cid:17)P(cid:16)Y1 ≥y1,··· ,Yn ≥yn(cid:17)
= P(cid:16)Y ≥y ,··· ,Y ≥y (cid:17)−P(cid:0)U ≤y ,Y ≥y ,··· ,Y ≥y (cid:1). (4.5)
1 1 n n 0 0 1 1 n n
Upon applying a long chain of a conditioning argument on both terms in (4.5), we arrive at
n n n
(cid:88) (cid:88) (cid:88)
H (x ,x ,··· ,x )=1− F (y )+ H (y ,y )+
X0,Xh1,···,Xhn 0 1 n 0 i Yi,Yj i j
i=0 i=0j=i+1