2023 [1]Martial Longla [1]Department of mathematics, University of
Mississippi, University Ave, University, 38677, Mississippi, US

# New copula families and mixing properties

<EMAIL_ADDRESS>*

###### Abstract

We characterize absolutely continuous symmetric copulas with square integrable
densities in this paper. This characterization is used to create new copula
families, that are perturbations of the independence copula. The full study of
mixing properties of Markov chains generated by these copula families is
conducted. An extension that includes the Farlie-Gumbel-Morgenstern family of
copulas is proposed. We propose some examples of copulas that generate non-
mixing Markov chains, but whose convex combinations generate $\psi$-mixing
Markov chains. Some general results on $\psi$-mixing are given. The Spearman’s
correlation $\rho_{S}$ and Kendall’s $\tau$ are provided for the created
copula families. Some general remarks are provided for $\rho_{S}$ and $\tau$.
A central limit theorem is provided for parameter estimators in one example. A
simulation study is conducted to support derived asymptotic distributions for
some examples.

###### keywords:

Perturbations of copulas, characterization of symmetric copulas, square
integrable copula density, reversible Markov chains, dependence modeling with
copulas, mixing for copula-based Markov chains, New copula families

###### pacs:

[

MSC Classification]62G08; 62M02; 60J35

##  1 Introduction

A bivariate copula, which is defined as a restriction to the unit square
$[0\mathchar 24635\relax\;1]^{2}$ of a bivariate joint cumulative distribution
function with uniform marginals, is a tool that has been gaining popularity in
dependence modeling. An equivalent definition can be found in Nelsen (2006) N
, where one can find several related notions. The popularity of copulas is due
to Sklar’s theorem that relates them to the joint distribution and marginals
of a bivariate random vector (see Sklar (1959) SK ).

Many authors have worked on building copulas with various properties. Some
constructions can be found in Nelsen (2006) N , where several construction
methods are presented. Arakelian and Karlis (2014)AK  studied mixtures of
copulas, that were investigated for mixing by Longla (2015)Lo2015 . Longla
(2014) L2  constructed copulas based on prescribed $\rho$-mixing properties
obtained from extending results of Beare (2010)Beare  on mixing for copula-
based Markov chains. Longla et al. (2022) L0  and Longla et al. (2022b)L00
constructed copulas in general based on perturbations. In these two works, two
perturbation methods were considered. For the first method, copulas are
perturbed at the level of the variables by adding some noise components to
each of the variables. For the second method, a copula $C(u,v)$ is perturbed
by creating $C_{D}(u,v)=C(u,v)+D(u,v)$ and requiring that $C_{D}(u,v)$
satisfies the definition of a copula. This last method of perturbation was
also presented in Komornik et al (2017) KK  and the references therein.
Several other authors have considered extensions of existing copula families
via other methods, that are not the focus of this work, see for instance
Morillas (2005)Morillas , Aas et al (2009) AAS , Klement et al (2017) KL ,
Chesneau (2021) Chesneau  among others.

Chesneau (2021)Chesneau  considered multivariate trigonometric copulas, which
seem close to one of the examples of copula families of this paper, but are in
fact very far from the work we provide here. We are concerned with the
important question of characterizing absolutely continuous symmetric copulas
with square integrable densities. After characterizing such copulas, we
extract some general copula families with specific functions, one of which is
made of trigonometric functions. These copula families come with pre-difined
mixing structure of the copula-based Markov chains they generate. This, in
general, can not be said about the copula families of Chesneau (2021)Chesneau
, which where published when this paper was already written. We also provide
the mixing structure of Markov chains generated by any of the copulas
constructed in this paper. Therefore, one of the main points of the paper is
the presentation of several copula families and the mixing structure of the
Markov chains they generate. Moreover, for each of these copulas, we provide
joint distributions of any two variables along the Markov chain they generate;
and show that the copulas of these variables also belong to the initial copula
family. We show that the set of all copulas for each of the selected basis of
functions is closed under the operation $*$, defined by $C^{1}(u,v)=C(u,v),$
for $n>1$

$C^{n}(u,v)=C^{n-1}*C(u,v)=\int_{0}^{1}C^{n-1}_{,2}(u,t)C_{,1}(t,v)dt\mathchar
24635\relax\;$

where $C_{,i}(u,v)$ is the derivative with respect to the $i^{th}$ component
of $(u,v)$. This product plays a central role in our study and its properties
can be found in Darsow et al. (1992) DAR . Recall that for $n=2$ this product
is the joint distribution of $(X_{1},X_{3})$ when $(X_{1},X_{2},X_{3})$ is a
stationary Markov chain with copula $C(u,v)$ and the uniform marginal
distribution. In general, the copula $C^{n}(u,v)$ is the joint distribution of
$(X_{0},X_{n})$ when $X_{0},\cdots,X_{n}$ is a stationary Markov chain
generated by $C(u,v)$ and the Uniform(0,1) distribution. This is a simple
consequence of applying recursively the product formula (see Longla (2022) L0
).

The product $*$ and related notions are used in this paper to derive
properties of Markov chains, including mixing and association. In Longla et
al. (2022) L0 , measures of association such as Spearman’s $\rho$ and
Kendall’s $\tau$ were studied for perturbations of copulas. Recall that in
terms of copulas, these coefficients are defined as

$\rho_{S}(C)=12\int_{0}^{1}\int_{0}^{1}C(u,v)dudv-3,$ (1)
$\text{and}\quad\tau(C)=1-4\int_{0}^{1}\int_{0}^{1}C_{,1}(u,v)C_{,2}(u,v)dudv.$
(2)

Formulas (1) and (2) are respectively formula (5.1.15c) and formula (5.1.12)
of Nelsen (2006)N . For definitions and practical use of these measures, we
refer the reader to Nelsen (2006) N  and references therein. In this paper,
our interest for these measures is in finding the relationship between them
and the parameters of the copulas, then propose an estimation procedure. Among
copula families, for which we derive measures of association, is the family of
copulas based on Legendre polynomials. This family is an extension of the
Farlie-Gumbel-Morgenstern (FGM) copula family. The FGM has been extensively
studied in the literature with several extensions (see Hurlimann (2017)H  or
Ebaid et al. (2022) Eb ). Our extension is unrelated to extensions that exist
in the literature. We use a new approach to extend this family in a way that
has not been done in the literature. Moreover, we show that copulas from this
family generate $\psi$-mixing stationary Markov chains for all values of the
family parameter $\theta\in[-1,1]$. This improves a previous result of Longla
et al (2022) L0 , who failed to provide the proof for the boundary points
$\theta=\pm 1$. The new characterization of copulas obtained in this paper
resulted in a new elegant proof of $\psi$-mixing that includes the boundary
points. For more on mixing coefficients, see Bradley (2005)Bradley , Beare
(2010) Beare  and the references therein. To avoid confusions in this paper,
we say that a function $f(x)$ is a version of the function $g(x)$ if and only
if the Lebesgue measure (in the appropriate dimension) of the set on which
they differ is 0.

This paper is structured as follows. In section 2, we present a
characterization of copulas with square integrable densities. The obtained
characterization is used to construct new copula families. Among these general
functional families of copulas, are finite and infinite sums, including new
sine and cosine copula families. A second group of copulas that is shown as
example here is based on Legendre polynomials and contains the FGM copula. We
also provide the Spearman’s $\rho_{S}$ and Kendall’s $\tau$ coefficients for
each of the examples of copula families along with some useful remarks. All
these results are new to the best of our knowledge. In section 3, we develop
mixing properties of the extended FGM copula family based on Legendre
polynomials. Several examples are provided here along with some copulas that
don’t generate mixing Markov chains, but have convex combinations that
generate Mixing Markov chains. These examples answer an open question on
convex combinations of copulas and mixing. A new general result on mixing for
Markov chains generated by copulas with square integrable densities is given.
We provide an extension of a previous result of Longla et al (2022c) LoMous
on $\psi$-mixing and $\psi^{\prime}$-mixing. Namely, we show that
$\psi$-mixing follows from the two conditions $c^{s_{1}}(u,v)>0$ on a set of
Lebesgue measure 1 and $c^{s_{2}}(u,v)<2$ for all $(u,v)\in[0,1]^{2}$. In
section 4 we provide some parameter estimators and their asymptotic
distributions via centra limit theorems. We provide here a simulation study.
Section 5 covers comments and conclusions. The Appendix of proofs ends the
paper.

##  2 Square integrable symmetric densities

In this section, we provide a characterization of copulas with square
integrable densities. The aim is to show that any such copula can be
represented as a sum of possibly non-zero infinitely many terms. Such a
representation indicates a new way to construct new copula families by
modifying the independence copula $\Pi(u,v)=uv$. We use this method in this
section to provide new sets of copulas and compute their coefficients
$\rho_{S}$ and $\tau$.

Recall that a copula in general is defined as a multivariate cumulative
distribution function, for which the restriction on $[0,1]^{2}$ has uniform
marginals on (0,1). For any bivariate random variable $(X,Y)$ with continuous
marginal distributions $(F_{X},F_{Y})$, the copula of $(X,Y)$ is the joint
cumulative distribution function of $(F_{X}(X),F_{Y}(Y))$. Note that both
$F_{X}(X)$ and $F_{Y}(Y)$ are uniform on $[0,1]$. This is why most works on
copula theory ignore the marginal distributions and are limited to uniform
marginals. This is done without loss of generality, when variables are assumed
continuous (see Nelsen (2006) N , Durante and Sempi (2016) DS  or Sklar (1959)
SK  for more on the topic). This work is concerned mainly with absolutely
continuous symmetric bivariate copulas $C(u,v)$ for which the density function
$c(u,v)$ exist and satisfies

$C(u,v)=\int_{0}^{u}\int_{0}^{v}c(s,t)dtds.$ (3)

Formula (3) follows from the fact that the singular part of the copula
vanishes for absolutely continuous copulas (see Darsow et al. (1992) DAR ). It
is known that the density of a copula is a positive bivariate function on
$[0,1]^{2}$. Many authors have worked on copulas and their properties. Copulas
are used to model dependence in various applied fields for problems that
include but are not limited to estimation, classification and statistical
tests of hypotheses. When used to model dependence, they help establish mixing
properties of Markov chains, that are useful in establishing central limit
theorems for sample averages of functions of observations (see Doukhan et al.
(2009)Doukhan  Chen and Fan (2006)Chen , Peligrad and Utev (1997)PelUt , and
Jones (2004) Jones ). For more on properties of copulas, see Darsow et al.
(1992) DAR  and Beare (2010) Beare .

In this paper, we say that a copula $C(u,v)$ is symmetric if and only if
$C(u,v)=C(v,u)$ for all $(u,v)\in[0,1]^{2}$. Such copulas are useful in
modeling reversible Markov chains. For a symmetric absolutely continuous
copula, the density $c(u,v)$ is square integrable if and only if

$\int_{0}^{1}\int_{0}^{1}[c(s,t)]^{2}dtds<\infty.$ (4)

Square integrable copula densities are kernels of Hilbert-Shmidt operators on
the Hilbert space $L^{2}(0,1)$. This means, that the linear operator defined
by

$Kf(x)=\int_{0}^{1}c(x,y)f(y)dy$ (5)

is a Hilbert-Shmidt operator in $L^{2}(0,1)$ (see Ferreira and Manegatto
(2009) Fer ). Let $\\{\varphi_{k}(x),k\in\mathbb{N}\\}$ be an orthonormal
basis made of eigenfunctions of the operator $K$.
$K\varphi_{k}(x)=\lambda_{k}\varphi_{k}(x),$ where $\lambda_{k}$ is called
eigenvalue of $K$ associated to the eigenfunction $\varphi_{k}(x)$ and the
equality is given in the sense of $L^{2}(0,1)$. Based on this, for a symmetric
copula with square integrable density, we have (see also Beare (2010) Beare ,
Longla (2014) L2  ):

  * •

The sequence of $\lvert{}\lambda_{k}\lvert{}$ has a finite number of values
including their multiplicities, or converges to $0$.

  * •

$\\{\varphi_{k}(x),k\in\mathbb{N}\\}$ can be selected to form an orthonormal
system in $L^{2}(0,1)$.

Due to the fact that $L^{2}(0,1)$ is a separable Hilbert space and the
orthonormal system made of eigenfunctions of this operator is complete, it
follows that formula (6) holds in the sense of $L^{2}(0,1)$. Therefore, by
Mercer’s theorem (see formula 27 page 445 of Mercer (1909) Mercer  or Theorem
2.4 of Ferreira and Manegatto (2009) Fer )

$c(u,v)=\sum_{k=1}^{\infty}\lambda_{k}\varphi_{k}(u)\varphi_{k}(v),$ (6)

where convergence of the series is point-wise and uniform on $(0,1)^{2}$ when
the opertor is non-negative definite (when all eigenvalues are non-negative).
The same conclusion can be achieved when all eigenvalues are non-positive (or
the operator is non-positive). In fact, Mercer (1909) Mercer  showed that for
any continuous non-negative definite kernel $k(x,y)$, the series converges
point-wise and uniformly. A non-negative definite kernel is a function
$k(x,y)$ such that

$\sum_{i,j=1}^{n}k(x_{i},x_{j})s_{i}s_{j}\geq 0\quad\text{for all}\quad
x_{i},x_{j},s_{i},s_{j}.$

Mercer (1909) Mercer  pointed that the above definition is equivalent to

$\int\int k(x(t),x(s))x(t)x(s)dsdt\geq 0.$

Moreover, it is obvious that for a continuous symmetric kernel $k(x,y)$, the
trace of the operator is

$\sum_{j=1}^{\infty}\lambda_{j}=\int_{0}^{1}k(x,x)dx.$

Therefore, the series of partial sums of eigenvalues is convergent when the
density of the copula is symmetric, square integrable, positive definite and
continuous on $[0,1]^{2}$. It is important to mention that these conditions
impose that eigenvalues be all positive (or all negative), therefore ruling
out some of the copulas. But the decomposition can be extended by continuity
to copulas with eigenvalues of both signs.

######  Proposition 2.1.

For any absolutely continuous copula $C(u,v)$ the following holds.

  1. 1.

The kernel operator (5) has and eigenvalue $\lambda=1$ associated to the
eigenfunction $\varphi(x)=1$;

  2. 2.

There exists a basis of eigenfunctions of (5) for any square integrable
symmetric copula density. Moreover, there exists a decomposition (6)
containing $\varphi_{1}(x)=1$ associated to the eigenvalue $\lambda_{1}=1$.

  3. 3.

A selected basis might not contain $\varphi(x)=1$ in the case when the
eigenvalue $\lambda=1$ has multiplicity at least equal to $2$, generating a
subspace of dimension higher than $1$ with orthonormal basis not containing
$\varphi(x)=1$.

Note that when the eigenvalue $\lambda=1$ has multiplicity $s>1$ in the
decomposition, $s$ has to be finite because of square integrability. It is
also worth mentioning that it is impossible for some functions to be
eigenfunctions associated to the eigen value $1$. For instance,
$\varphi_{k}(x)$ has to be bounded.

We devote the following subsections of this work to the case when
$\varphi(x)=1$ is used. This case by itself is interesting because it turns
into the question of perturbations of the independence copula (see Longla et
al. (2022) L0 ). As we will see below, the mixing structure of Markov chains
generated by our constructed copula families is easily established in general.
This eases the study of estimators of coefficients and measures of
association.

###  2.1 Perturbations of the copula $\Pi(u,v)$

The copula $\Pi(u,v)=uv$ is called independence copula. It is used to model
data under the assumption of independence. This means that the copula of
random variables that are independent is $\Pi(u,v)$. The notion of
perturbation has been recently used in many papers to address modifications
that are done to a copula to improve estimation results. This is done by
introducing some level of dependence in the case of the independence copula
(see Durante et al (2013) F.S ) through a function $D(u,v)$. When $\Pi(u,v)$
is perturbed, the obtained copula exhibits some dependence that can be
estimated to fit better the data at hands.

Various types of perturbations have been considered in Durante et al (2013)
F.S , Komornik et al (2017) KK , Longla et al (2022) L0 , Longla et al
(2022c)LoMous  and the references therein. In this section, we investigate
perturbations of the kind $C_{D}(u,v)=\Pi(u,v)+D(u,v)$, where $D(u,v)$ is the
perturbation factor to be determined using (6). This perturbation method was
studied in Komornik et al (2017) KK  and the references therein. Longla et al
(2022) L0  and Longla et al (2022c) LoMous  considered mixing for Markov
chains generated by perturbations with $D(u,v)=\theta(\Pi(u,v)-C(u,v))$ and
$D(u,v)=\theta(M(u,v)-C(u,v))$, where $M(u,v)=\min(u,v)$ is the Fréchet-
Hoeffding upper bound.

######  Remark 1.

The use of $M(u,v)$ to perturb a copula $C(u,v)$ for data with uniform
marginals assumes that values of the Markov chain have a positive probability
of repeating themselves, before providing new values from the transition
copula $C(u,v)$.

Considering the decomposition (6), when the first term uses $\varphi(x)=1$,
reindexing the remaining functions $\varphi_{k}(x)$ and integrating leads to

$C(u,v)=uv+\sum_{k=1}^{\infty}\lambda_{k}\int_{0}^{u}\varphi_{k}(x)dx\int_{0}^{v}\varphi_{k}(y)dy.$
(7)

This implies $C(u,v)=uv+D(u,v)$, where

$D(u,v)=\sum_{k=1}^{\infty}\lambda_{k}\int_{0}^{u}\varphi_{k}(x)dx\int_{0}^{v}\varphi_{k}(y)dy.$

Note that $D(u,0)=D(u,1)=0$ and $D(u,v)$ is symmetric. Therefore, the
perturbation is a copula when its second order mixed derivative is non-
negative (refer to Longla et al. (2022) L0 ). So, any representation (7) with
$\lambda_{k}\in\mathbb{R}$ defines a copula with square integrable density if
and only if

$1+\sum_{k=1}^{\infty}\lambda_{k}\varphi_{k}(x)\varphi_{k}(y)\geq
0\quad\mbox{for all $(x,y)\in[0,1]^{2}$ and}$ (8)
$\sum_{k=1}^{\infty}\lambda_{k}^{2}<\infty.$ (9)

Condition (9) is required for square integrability of the density and takes
into account the fact that $\varphi_{k}(x)$ form an orthonormal system of
functions. Condition (8) is satisfied when the following holds:

$1+\sum_{k=1}^{\infty}\lambda_{k}\alpha_{k}\geq
0,\quad\alpha_{k}=\begin{cases}\max\varphi^{2}_{k}&\text{if
}\quad\lambda_{k}<0\\\
\min\varphi_{k}\max\varphi_{k}&\text{if}\quad\lambda_{k}>0.\end{cases}$ (10)

Note that condition (10) is not necessary. For different values of $k$, the
functions $\varphi_{k}(x)$ might have their minima or maxima at different
points. It is a strong condition, but easy to handle in practice. Note that
when condition (10) holds, if $\varphi_{k}(x)$ are uniformly bounded, then (9)
holds. This can be shown by first showing that the sum of
$\lvert\lambda_{k}\lvert$ is convergent and larger than the sum in (9). For
all systems of functions that are considered in this section, all these
conditions are satisfied. Thus, the following characterization theorems hold.

######  Theorem 2.2.

Any symmetric absolutely continuous copula with square integrable density can
be represented by (7), where $\\{1,\varphi_{k}(x),k>1,k\in\mathbb{N}\\}$ is an
orthonormal basis associated to the eigenvalues
$\\{1,\lambda_{k},k>1,k\in\mathbb{N}\\}$ of the operator (5).

In Theorem 2.2, equality is understood as equality of operators in
$L^{2}(0,1)$. This equality turns into point-wise or uniform convergence when
the density of the copula is continuous and defines a positive semi-definite
operator.

######  Theorem 2.3.

Assume that $(\lambda_{k}\in\mathbb{R},k>1)$ is such that conditions (10) and
(9) hold for an orthonormal system of functions of $L^{2}(0,1)$ that contains
$\varphi_{1}(x)=1$. The function (7) is an absolutely continuous symmetric
copula with square integrable density that we call Type-I Longla copula with
notation $L(u,v,\Lambda,\Phi)$.

For any system of functions $\varphi_{k}(x)$ that is uniformly bounded,
condition (10) implies condition (9). Theorem 2.3 opens the ground for various
copula families with functions as parameters. These copulas can have finite or
infinite sums of terms. They depend both on the system of functions
$\varphi_{k}(x)$ and the coefficients $\lambda_{k}$. The characterization
formulated in the following general result summarizes the above analysis.

###  2.2 Examples of new copula families

Theorem 2.3 and Theorem 2.2 provide a relationship between extreme values of
the functions $\varphi_{k}(x)$ and the possible eigenvalues $\lambda_{k}$ of
(5). For example, a function $\varphi(x)$ with maximum $2$ and minimum $-3$
cannot be associated to an eigenvalue less than $-1/9$ or larger than $1/6$,
when all other coefficients are equal to 0. The larger the values of
$\lvert{}\varphi_{k}(x)\lvert{}$, the smaller the maximal range of
$\lambda_{k}$, when the sum (6) has only $\varphi_{k}(x)$ and $1$. Setting
$\Phi(x)=\int_{0}^{x}\varphi(s)ds$ for any mean-zero normal function
$\varphi(x)$ on $[0,1]$, we have the copula

$C(u,v)=uv+\lambda\Phi(u)\Phi(v),\quad\frac{-1}{\max\varphi^{2}(x)}\leq\lambda\leq\frac{-1}{\max\varphi(x)\min\varphi(x)}.$
(11)

An example of copula (11) is defined for
$-\min(\frac{1}{\alpha},\alpha)\leq\lambda\leq 1,$ $\alpha\in(0,\infty)$ by

$C_{\alpha,\lambda}(u,v)=\begin{cases}(1+\lambda\alpha)uv\quad\text{if $0\leq
u,v\leq\frac{1}{\alpha+1}$}\\\ uv+\lambda u(1-v)\quad\text{if $0\leq
u\leq\frac{1}{\alpha+1}<v\leq 1$}\\\ uv+\lambda v(1-u)\quad\text{if $0\leq
v\leq\frac{1}{\alpha+1}<u\leq 1$}\\\
uv+\frac{\lambda}{\alpha}(1-u)(1-v),\quad\text{if $\frac{1}{\alpha+1}<u,v\leq
1$}.\end{cases}$ (12)

Copula (12) is also a representative of the only class of copulas with square
integrable symmetric densities based on a single function $\varphi(x)$ that
takes $2$ values on $[0,1]$. This class of copulas is defined using

$\varphi(x)=\sqrt{\alpha}\mathbb{I}(x\in
A)-\frac{1}{\sqrt{\alpha}}\mathbb{I}(x\in[0,1]\setminus A),$

where $A$ is a set of measure $\frac{1}{\alpha+1}$ for some strictly postive
real number $\alpha$. Copula (12) is obtained via integration to get $\Phi(x)$
of formula (11) for

$\varphi(x)=\sqrt{\alpha}\mathbb{I}(0\leq
x<\frac{1}{\alpha+1})-\frac{1}{\sqrt{\alpha}}\mathbb{I}(1\geq
x\geq\frac{1}{\alpha+1}).$

For copula (12), while the parameter $\lambda$ answers for dependence (we will
show later that its absolute value is the maximal coefficient of correlation),
the parameter $\alpha$ answers for the portions of $[0,1]^{2}$ that have
constant likelihood values. In fact, the density of this copula is constant on
each of the four portions of its support. Via simple computations we establish
that for copula (12),

$\rho_{S}(C_{\alpha,\lambda})=\frac{3\alpha\lambda}{(1+\alpha)^{2}}\quad\text{and}\quad\tau(C_{\alpha,\lambda})=\frac{2\alpha\lambda}{(1+\alpha)^{2}}.$

######  Remark 2.

The coefficients $\rho_{S}(C_{\alpha,\lambda})$ and $\tau(C_{\alpha,\lambda})$
show that for the purpose of correlation or association, the range of $\alpha$
can be limited to $(0,1]$, because $1/\alpha$ and $\alpha$ produce the same
value for $\rho_{S}(C_{\alpha,\lambda})$ and $\tau(C_{\alpha,\lambda})$. The
maximum of these coefficients is reached at $\alpha=1$.

Interest in the copula family $C_{1,\lambda}(u,v)$ is in the fact that the
range of the Pearson correlation coefficient of functions of variables $(U,V)$
generated by this copula family is $[-1,1]$. For $\lambda\neq\pm 1$, the
procedure to generate a stationary Markov chain from $C_{1,\lambda}(u,v)$ with
the uniform marginal distribution is as follows. We generate an observation
$U_{0}$ from $Uniform(0,1)$; then for every integer $i$, generate $Q_{i}$ from
$Uniform(0,1)$ and define $U_{i}$ using the quantile function of the
conditional distribution of $U_{i}\lvert{}U_{i-1}$. The quantile function is
the inverse with respect to $v$ of the derivative with respect to $u$ of
$C(u,v)$ at the point $(U_{i-1},U_{i})$ (see Longla and Peligrad (2012) LoMag
). We obtain the formula

$U_{i}=\begin{cases}\frac{Q_{i}}{1+\lambda},\quad\text{if}\quad
0<U_{i-1}<.5,\quad Q_{i}<.5+.5\lambda\\\
\frac{Q_{i}-\lambda}{1-\lambda},\quad\text{if}\quad 0<U_{i-1}<.5,\quad
Q_{i}>.5+.5\lambda\\\ \frac{Q_{i}}{1-\lambda},\quad\text{if}\quad
1>U_{i-1}>.5,\quad Q_{i}<.5-.5\lambda\\\
\frac{Q_{i}+\lambda}{1+\lambda},\quad\text{if}\quad 1>U_{i-1}>.5,\quad
Q_{i}>.5-.5\lambda.\end{cases}$ (13)

Recall that the joint cumulative distribution function of $(U_{0},U_{n})$ is
the copula $C_{1,\lambda^{n}}(u,v)$. Therefore, the following holds.

######  Lemma 1.

For any stationary Markov chain generated by $C_{1,\lambda}(u,v)$ and the
uniform marginal distribution for $\lambda\neq\pm 1$ , we have $U_{n}-Q_{n}\to
0$ in probability when $n\to\infty$.

The limiting value of $U_{n}$ as $n\to\infty$ doesn’t depend on the initial
value $U_{0}$. This is true because $\lambda$ is replaced by $\lambda^{n}\to
0$ in formula (13) and $Q_{n}$ is a random sample obtained independently. This
fact is one of the selling points of the study of mixing properties. The
intial point of the Markov chain doesn’t affect the long run behavior under
mixing assumptions.

An extension of the copula family (12) is obtained considering for all
$s\in\mathbb{N^{*}}$, $\delta_{i}=\mathbb{I}(a_{i}\leq u,v<a_{i+1})$ and
$\lvert\theta_{i}\lvert\leq 1$. This extension has density

$\displaystyle
c(u,v)=1+\sum_{i=1}^{s}\theta_{i}sign(2u-a_{i}-a_{i+1})sign(2v-a_{i}-a_{i+1})\delta_{i}.$
(14)

Formula (14) defines the density a copula for any set
$\\{a_{i},i=1,\cdots,s:0\leq a_{1}<a_{2}<\cdots<a_{s}<a_{s+1}=1\\}$. Via
simple computations, we show that functions

$f_{i}(x)=\frac{sign(2x-a_{i}-a_{i+1})}{\sqrt{a_{i+1}-a_{i}}}$

satisfy the conditions of Theorem 2.3 . Note that
$\theta_{i}(a_{i+1}-a_{i})=\lambda_{i}$ implies $\lvert\lambda_{i}\lvert\leq
1/(a_{i+1}-a_{i})$. Moreover, this copula coincides with copula (12) when
$s=1$ and $a_{1}=0$. This copula departs from independence by adding local
perturbations to rectangles $[a_{i},a_{i+1}]\times[a_{i},a_{i+1}]$ of size
$\pm\theta$.

An interesting case is when all functions have the same maximum, same minimum
and the absolute values of these numbers are same. The range of the
$\lambda_{k}$ is symmetric. This is the case when $\varphi_{k}$ is $a\sin
w_{k}x$, or $a\cos w_{k}x$. We consider these cases in the subsections below.

####  2.2.1 Copulas based trigonometric on functions

Here, we look into copula families based on trigonometric bases of
$L^{2}(0,1)$. These copulas are different from those introduced in Chesneau
(2021)Chesneau , because they don’t involve sums of variables. Moreover, the
mixing structure of copulas of Chesneau (2021)Chesneau  is not as evident as
that of the copula families that we construct here. It is also worth
mentioning that the copula families that are introduced here can be finite or
infinite sums and that their fold products remain in the considered classes.
When modeling with these copulas, the existence of waves in the scatter plot
can indicate the number of functions to consider in the sum. An observation on
$\rho_{S}(C)$ and $\tau(C)$ shows that some terms can be added to increase
linear correlation without affecting association.

  * •

The sine-cosine copulas as perturbations of $\Pi(u,v)$

We call sine-cosine copulas perturbations of the independence copula that
consider the trigonometric orthonormal basis of $L^{2}(0,1)$ that consists of
functions: $\\{1,\sqrt{2}\cos 2\pi kx,\sqrt{2}\sin 2\pi
kx,k\in\mathbb{N^{*}}\\}$. Theorem 2.3 guarantees that

$\displaystyle
C(u,v)=uv+\frac{1}{2\pi^{2}}\sum_{k=1}^{\infty}\frac{\lambda_{k}}{k^{2}}\sin
2\pi ku\times\sin 2\pi kv+$
$\displaystyle+\frac{1}{2\pi^{2}}\sum_{k=1}^{\infty}\frac{\mu_{k}}{k^{2}}(1-\cos
2\pi ku)(1-\cos 2\pi kv)$ (15)

is a copula, when
$\displaystyle\sum_{k=1}^{\infty}(\lvert{}\lambda_{k}\lvert{}+\lvert{}\mu_{k}\lvert{})\leq
1/2$. For this example, condition (9) holds automatically. The series (2.2.1)
also converges uniformly and absolutely on $[0,1]^{2}$ thanks to Weierstrass’
$M$-test and the fact that the functions in the series are uniformly bounded
on $[0,1]^{2}$.

Figure 1: Examples of sine-cosine copula densities

Figure 1(a) represents the copula density

$c(u,v)=1+.3\cos(4\pi u)\cos(4\pi v)-.4\cos(2\pi u)\cos(2\pi v),$

Figure 1(b) is the graph of

$c(u,v)=1+.3\cos(4\pi u)\cos(4\pi v)-.4\sin(2\pi u)\sin(2\pi v),$

Figure 1(c) represents

$c(u,v)=1+.3\sin(4\pi u)\sin(4\pi v)-.4\sin(2\pi u)\sin(2\pi v)$

and Figure 1(d) represents the copula density

$c(u,v)=1+.2\cos(4\pi u)\cos(4\pi v)-.4\cos(2\pi u)\cos(2\pi v)$ $+.4\cos 6\pi
u)\cos(6\pi v).$

The Spearman’s $\rho$ and Kendall’s $\tau$ for this copula family have been
computed and are given below.

######  Lemma 2.

For any bivariate population with copula $C(u,v)$ defined by (2.2.1), the
Spearman’s $\rho_{S}(C)$ and Kendall’s $\tau(C)$ are

$\displaystyle\rho_{S}(C)=\frac{6}{\pi^{2}}\sum_{k=1}^{\infty}\frac{\mu_{k}}{k^{2}},\quad-\frac{3}{\pi^{2}}\leq\rho_{S}(C)\leq\frac{3}{\pi^{2}}$
$\displaystyle\tau(C)=\frac{1}{\pi^{2}}\sum_{k=1}^{\infty}\frac{4\mu_{k}+2\lambda_{k}\mu_{k}}{k^{2}},\quad-\frac{2}{\pi^{2}}\leq\tau(C)\leq\frac{2}{\pi^{2}}.$

We can see on Figure 1 that the range of the copula decreases as we increase
the number of terms in the sum. For sine-cosine copulas, Lemma 2 shows that
terms related to cosine in the density of the copula don’t affect
$\rho_{S}(C)$, while terms with sine linearly affect $\rho_{S}(C)$.

######  Remark 3.

Any copula from this family with $\mu_{k}=0$ for all integer $k\geq 1$
satisfies $\rho_{S}(C)=\tau(C)=0$. This is a subclass of copulas of the form

$C(u,v)=uv+\frac{1}{2\pi^{2}}\sum_{k=1}^{\infty}\frac{\lambda_{k}}{k^{2}}\sin
2\pi ku\times\sin 2\pi kv.$ (16)

Therefore, the Spearman’s $\rho$ and Kendall’s $\tau$ are not good parameters
for tests of independence because they don’t uniquely identify a member of
this family.

  * •

The sine copulas as perturbations of $\Pi(u,v)$

Consider the basis of $L^{2}(0,1)$ given by $\\{1,\sqrt{2}\cos k\pi
x,k>0,k\in\mathbb{N}\\}$. By Theorem 2.3, for any sequence
$\\{\lambda_{k},k\in\mathbb{N}\\}$ satisfying (18), the function (17) is a
copula.

$C(u,v)=uv+\frac{2}{\pi^{2}}\sum_{k=1}^{\infty}\frac{\lambda_{k}}{k^{2}}\sin
k\pi u\sin k\pi v\quad\text{with}$ (17)
$\sum_{k=1}^{\infty}\lvert{}\lambda_{k}\lvert{}\leq 1/2.$ (18)

Figure 2: Examples of sine copula densities

######  Lemma 3.

For copulas defined by formula (17), we have

$\displaystyle\tau(C)=\frac{64}{\pi^{4}}\sum_{k=0}^{\infty}\frac{\lambda_{2k+1}}{(2k+1)^{4}}+\frac{16}{\pi^{4}}\sum_{k=1}^{\infty}\sum_{j=k+1}^{\infty}\frac{2\lambda_{k}\lambda_{j}}{(k^{2}-j^{2})^{2}}(1-cos\pi(j+k)\pi)^{2},$
$\displaystyle\rho_{S}(C)=\frac{96}{\pi^{4}}\sum_{k=0}^{\infty}\frac{\lambda_{2k+1}}{(2k+1)^{4}},\quad-\frac{48}{\pi^{4}}\leq\rho_{S}(C)\leq\frac{48}{\pi^{4}},\quad-\frac{32}{\pi^{4}}\leq\tau(C)\leq\frac{32}{\pi^{4}}.$

This family of copulas seems close to the sine-cosine family, but the formula
of $\rho_{S}(C)$ or $\tau(C)$ in Lemma 3 shows that they are not comparable.
We see that for these copulas $\rho_{S}(C)=0$ when all odd coefficients are
equal to zero. Formula (16) defines a subclass of copulas from (17), for which
all odd coefficients are equal to zero. Any finite sum of terms from (17) that
includes the first term $uv$ and with coefficients that satisfy condition (18)
is a copula. An example is

$C(u,v)=uv+\frac{2\lambda}{\pi^{2}}\sin\pi u\sin\pi
v,\quad\lvert{}\lambda\lvert{}\leq 1/2.$ (19)

######  Remark 4.

The maximal coefficient of correlation of copula-based Markov chains generated
by copulas (19) is $\lvert{}\lambda\lvert{}$. We find
$\rho_{S}(C)=\frac{96}{\pi^{4}}\lambda$ and
$\tau(C)=\frac{64}{\pi^{4}}\lambda$. So,
$\lvert{}\tau(C)\lvert{}\leq\lvert{}\rho_{S}(C)\lvert{}\leq\lvert{}\lambda\lvert{}$.

####  2.2.2 The extended Farlie - Gumbel - Morgenstern copulas

In this subsection, we construct an example of copula family based on formula
(7) that extends the Farlie-Gumbel-Morgenstern copulas. The FGM copula is a
pertubation of the independence copula (see Longla et al (2022c) ). The form
of the FGM copula allows an extension based on shifted Legendre polynomials,
orthonormal basis of $L^{2}(0,1)$. It is important to mention that formula (7)
allows various families of extensions of the FGM copula that depend on the
used orthonormal systems of functions. Legendre polynomials are defined by
Rodrigues’ formula

$P_{k}(x)=\frac{1}{2^{k}k!}\frac{d^{k}}{dx^{k}}(x^{2}-1)^{k}.$ (20)

This formula was discovered by Rodrigues in (1816) Ro  and leads to

$P_{k}(x)=\frac{1}{2^{k}}\sum_{i=1}^{[\frac{k}{2}]}(-1)^{i}{k\choose
i}{2k-2i\choose k}x^{k-2i}.$ (21)

Legendre polynomials form a basis of $L^{2}(-1,1)$. When they are shifted to
$[0,1]$, a renormalization is used to obtain shifted orthonormal polynomials.
They appear in many fields of mathematics as solutions to a special
differential equation and are used for approximation of functions. For these
polynomials, $\int_{-1}^{1}P^{2}_{k}(x)dx=\frac{2}{2k+1}$. The transformation
$y=2x-1$ gives to the orthonormal basis of shifted Legendre polynomials as

$\varphi_{k}(x)=\sqrt{2k+1}P_{k}(2x-1).$ (22)

For $k>0,$ $\varphi_{k}(1)=\sqrt{2k+1}=\max\varphi_{k}$, for odd $k$,
$\min\varphi_{k}=-\sqrt{2k+1}=\varphi_{k}(0)$ and for even $k,$
$0>\min\varphi_{k}>-\sqrt{2k+1}$. This minimum is achieved at least at one
point. Therefore, when (7) contains $\varphi_{k}(x)$ alone, the range of
$\lambda_{k}$ is

$\frac{-1}{2k+1}\leq\lambda_{k}\leq\min(1,\frac{1}{(2k+1)\lvert{}\min
P_{k}(x)\lvert{}}).$

Based on the special structure of these functions, using the following
equality known as Bonnet’s recursion formula

$(2n+1)P_{n}(x)=\frac{d}{dx}(P_{n+1}(x)-P_{n-1}(x)),$

we establish the equality

$\displaystyle(2n+1)\int_{-1}^{x}P_{n}(t)dt=P_{n+1}(x)-P_{n-1}(x).$

This formula uses the fact that $P_{n}(-1)=(-1)^{n}$. Substituting
$u=\frac{1}{2}(x+1)$ and $t=2s-1,2ds=dt$, we obtain

$\displaystyle(2n+1)\int_{0}^{u}P_{n}(2s-1)2ds=P_{n+1}(2u-1)-P_{n-1}(2u-1).$

Thus, using the definition of shifted Legendre polynomials $\varphi_{k}(x)$,
we have

$\displaystyle
2\sqrt{2n+1}\int_{0}^{u}\varphi_{n}(s)ds=\frac{1}{\sqrt{2n+3}}\varphi_{n+1}(u)-\frac{1}{\sqrt{2n-1}}\varphi_{n-1}(u).$
(23)

For more on Legendre polynomials, see Belousov (1962) Belousov , Szëgo (1975)
Szego  and the references therein. Formula (23) is used to establish the
following.

######  Lemma 4.

For any copula of the extended FGM copula family,

$\rho_{S}(C)=\lambda_{1},\quad\text{and}\quad\tau(C)=\frac{2\lambda_{1}}{3}-\sum_{k=1}^{\infty}\frac{2\lambda_{k}\lambda_{k+1}}{(2k+1)\sqrt{2k+3}\sqrt{2k-1}}.$

Simple computations show that $-1/3\leq\rho_{S}(C)\leq 1/3$ and $\tau(C)$ can
be increased or decreased by adding more non-zero terms to the series. In
general, copulas with $\varphi_{k}(x)$ given in formula (22) satisfy condition
(10). We call this set of copulas extended Legendre-Farlie-Gumbel-Morgenstern
family of copulas. Some copulas of this form are as follows. For
$\lvert{}\lambda\lvert{}\leq 1/3,$
$3\lvert{}\lambda_{1}\lvert{}+5\lvert{}\lambda_{2}\lvert{}\leq 1$,

$\displaystyle C_{\lambda}(u,v)=uv+3\lambda(u-u^{2})(v-v^{2}),$ (24)
$\displaystyle
C_{\lambda_{1},\lambda_{2}}(u,v)=uv+3\lambda_{1}(u-u^{2})(v-v^{2})+$
$\displaystyle+5\lambda_{2}(2u^{3}-3u^{2}+u)(2v^{3}-3v^{2}+v).$ (25)

The first example above is the FGM copula. For more on this copula family, see
Farlie (1960) FARLIE , Gumbel (1960) GUMBEL , Jonhson and Kotz (1975) Johnson
or Morgenstern (1956) Morgenstern . In general this copula has parameter
$\theta$ in place of $3\lambda_{1}$. The second example is one of the possible
extensions of this copula family. For this extension, we have

$\rho_{S}(C_{\lambda_{1},\lambda_{2}})=\lambda_{1},\quad\tau(C_{\lambda_{1},\lambda_{2}})=\frac{2}{3}\lambda_{1}(1-\frac{\lambda_{2}}{\sqrt{5}}).$
(26)

Any subclass of the copula family defined with Legendre polynomials and
containing the copula given by formula (24) is an extension of the FGM family.
Depending on what the investigator is looking for, it might be better to set
some coefficients equal to zero, deal with the signs of the coefficients to
increase or reduce the strength of dependence. This is justified by formula
(26). The FGM copula family has parameter $\theta=3\lambda$, with
$\lvert\theta\lvert\leq 1$; so $\lvert\lambda\lvert\leq 1/3$. No copula from
our extension can have $\rho_{S}(C)$ out of this range because increasing the
number of non-zero parameters reduces the range of $\lambda_{1}$. Example (25)
has $-1/5\leq\lambda_{2}\leq 1/5$ when $\lambda_{1}=0$, but can be extended to
$-1/5\leq\lambda_{2}\leq 2/5$.

Figure 3: Densities of Legendre-Farlie-Gumbel-Morgenstern copulas

For any copula from this family, it is worth formulating the following.

######  Remark 5.

Lemma 4 shows that any copulas from this family with the same $\lambda_{1}$
have the same $\rho_{S}(C)$, while it is not the case for $\tau(C)$. Other
formulations of these copulas use the basis
$\varphi_{k}(x)=\sqrt{2j_{k}+1}P_{j_{k}}(2x-1)$, where $j_{k}$ is a sequence
of positive integers.

####  2.2.3 Counter example - the sine basis

This case is to show that not all orthonormal bases of $L^{2}(0,1)$ can be
used to construct copulas with square integrable densities. We take the case
of functions constructed using the sine basis. They are not copulas under any
assumptions. In fact, if

$C(u,v)=\frac{2}{\pi^{2}}\sum_{k=1}^{\infty}\frac{\lambda_{k}}{k^{2}}(1-\cos
k\pi u)(1-\cos k\pi v),\quad\text{then}$ (27)
$c(u,v)=2\sum_{k=1}^{\infty}\lambda_{k}\sin k\pi u\sin k\pi v.$ (28)

If this function is a copula, then using $C(u,1)=u$ and taking the derivative
of the series, we obtain

$1=2\sum_{k=1}^{\infty}\frac{\lambda_{k}}{k\pi}\sin k\pi u(1-\cos k\pi).$ (29)

Multiplying by $\sin k\pi u$ and integrating leads to $\frac{1-\cos
k\pi}{k\pi}=\frac{\lambda_{k}}{\pi k}(1-\cos k\pi).$ Therefore,
$\lambda_{2k+1}=1$. As said earlier, the function $\varphi(x)=1$ in this case
is embedded in a subspace of dimension $\infty$ of eigenfunctions associated
to the eigenvalue $1$. This is in contradiction with condition (9). So, no
such decomposition represents a square integrable copula density.

##  3 Copula-based Markov chains and $\psi$-mixing

We study here the long run behavior of several copula families constructed
with various sets of bases of $L^{2}(0,1)$. We provide the study of the mixing
structure of copula-based Markov chains generated by our constructed copulas.
In some cases, we provide a modification of the copulas to present them as
convex combinations of some members of the constructed families. This
transformation provides a link between this work and results of Longla et al.
(2022) L0  and Longla (2015) Lo2015 . Mixing coefficients are used in
probability theory to establish central limit theorems for sums of dependent
random variables (see Peligrad and Utev (1997)PelUt ), which are used to
construct confidence intervals for means of functions of Markov chains.
Copulas characterize the dependence structure of Markov chains or multivariate
random variables in general via Sklar’s theorem (See Sklar (1959) SK ) when
marginal distributions are continuous. Therefore, it is important to look into
some conditions on copulas that would guarantee a given dependence structure
(see Beare (2010) Beare , Longla (2015)Lo2015  among others). Dependence
structures considered in the literature include $\alpha$-mixing,
$\beta$-mixing, $\rho$-mixing, $\phi$-mixing, $\psi$-mixing,
$\psi^{\prime}$-mixing, $\psi^{*}$-mixing and others. We define only mixing
coefficients of interest in this paper. For more on mixing coefficients, see
Bradley (2007) BR . For a stationary Markov chain $X_{0},\cdots,X_{n}$,
$\mathbb{A}=\sigma(X_{0})$, mixing coefficients are defined as

$\phi_{n}=\sup_{P(A)P(B)>0,A,B\in\mathbb{A}}\lvert{}P^{n}(B\lvert{}A)-P(B)\lvert{},$
$\rho_{n}=\sup_{f}\\{\mathbb{E}(f(X_{0})f(X_{n})):\mathbb{E}f(X_{0})=0,\mathbb{E}f^{2}(X_{0})=1\\},$
$\psi_{n}=\sup_{P(A)P(B)>0,A,B\in\mathbb{A}}\frac{\lvert{}P^{n}(A\cap
B)-P(A)P(B)\lvert{}}{P(A)P(B)},$
$\psi^{\prime}_{n}=\inf_{P(A)P(B)>0,A,B\in\mathbb{A}}\frac{P^{n}(A\cap
B)}{P(A)P(B)}\quad\text{and}$
$\psi_{n}^{*}=\sup_{P(A)P(B)>0,A,B\in\mathbb{A}}\frac{P^{n}(A\cap
B)}{P(A)P(B)},$

where $P^{n}(B\lvert{}A)$ stands for $P(X_{n}\in B\lvert X_{0}\in A)$, and
$P^{n}(A\cap B)$ stands for $P(X_{0}\in A~{}and~{}X_{n}\in B)$. Using these
coefficients, Bradley (2005) Bradley  in his survey presented the following
result. Suppose $X_{0},\cdots,X_{n}$ is a strictly stationary sequence of
random variables. If there exists $n$ such that $\psi^{\prime}_{n}>0$ and
$\psi_{n}^{*}<\infty$, then $\psi_{n}\to 0$ (the sequence is $\psi$-mixing).
Based on this statement, results of Longla (2015) Lo2015  and Longla et al.
(2022c) LoMous , we can conclude the following.

######  Theorem 3.1.

An absolutely continuous copula C(u,v) generates $\psi$-mixing stationary
Markov chains when there exist integers $s_{1}$ and $s_{2}$ such that some
versions of the densities of $C^{s_{1}}(u,v)$ and $C^{s_{2}}(u,v)$ satisfy one
of the following conditions.

  1. 1.

There exists a constant real number $M$ such that $c^{s_{1}}(u,v)>0$ on a set
of Lebesgue measure $1$ and $c^{s_{2}}(u,v)<M$ for all $(u,v)\in[0,1]^{2}$.

  2. 2.

For all $(u,v)\in[0,1]^{2}$, $c^{s_{2}}(u,v)<M\leq 2$.

It is important in Theorem 3.1 that the version of the density is bounded
above on $[0,1]$. Failure of this condition on a set of Lebesgue measure $0$
can imply that there is no $\psi$-mixing.

######  Example 1.

The copula $C(u,v)=aM(u,v)+(1-a)W(u,v),$ with $0\leq a\leq 1,$ from the Mardia
family, when used with continuous marginals, doesn’t generate $\rho$-mixing
Markov chains as shown in Longla (2014) L2 . These copulas are not absolutely
continuous. Another copula that fails these conditions is (12) with
$\alpha=1=\lambda$.

This example is a convex combination of $W(u,v)=\max(u+v-1,0)$ and
$M(u,v)=\min(u,v)$. A convex combination of copulas $C_{1}(u,v)$, $\cdots,$
$C_{k}(u,v)$ is the copula $C(u,v)=a_{1}C_{1}(u,v)+\cdots a_{k}C_{k}(u,v),$
with $0<a_{i},$ for all $i$ and $a_{1}+\cdots+a_{k}=1.$ Using Theorem 3.1, we
get.

######  Corollary 1.

Let $C_{1}(u,v),\cdots,C_{k}(u,v)$ be copulas such that the density of
$C_{s_{1}}*C_{s_{1}+1}*\cdots*C_{s_{2}}(u,v)$ is bounded away from zero on a
set of Lebesgue measure $1$ for some positive integers $s_{1},s_{2}$. Assume
that a version of the density of $(a_{1}C_{1}+\cdots+a_{k}C_{k})^{s_{3}}$ is
bounded on $[0,1]^{2}$ for some positive integer $s_{3}$ or a version of the
density of $(\sum a_{i}C_{i})^{s_{4}}(u,v)$ is strictly less than $2$ for some
$s_{4}\in\mathbb{N}$. Then, any convex combination of copulas
$C_{1}(u,v),\cdots,C_{k}(u,v)$ generates exponential $\psi$-mixing stationary
Markov chains with continuous marginals.

###  3.1 Applications to our new copula families

In this subsection, we apply the result on mixing to Markov chains generated
by our newly created copula families. For any copula of the form (12), the
maximal coefficient of correlation is $\lvert\lambda\lvert$. Therefore, it
generates $\rho$-mixing Markov chains when $\lvert{}\lambda\lvert{}<1$.
Moreover,

  1. 1.

For $\alpha\neq 1$, $C_{\alpha,\lambda}(u,v)$ generates $\psi$-mixing when
$-\min(\frac{1}{\alpha},\alpha)\leq\lambda<1$;

  2. 2.

$C_{1,\lambda}(u,v)$ generates $\psi$-mixing when $-1<\lambda<1$;

  3. 3.

There is no $\psi^{\prime}$-mixing, $\psi^{*}$ or $\psi$-mixing for other
values of $(\alpha,\lambda)$.

Note that when $\alpha=1$, the range of $\lambda$ includes $\pm 1$, for which
there is no $\rho$-mixing, $\psi^{*}$-mixing or $\psi^{\prime}$-mixing. This
case is equivalent to

$c_{3}(u,v)=1+\lambda\cdot sign(2u-1)\cdot
sign(2v-1),\quad\quad\lvert{}\lambda\lvert{}\leq 1.$ (30)

When $\lvert{}\lambda\lvert{}<1$, the density (30) splits $[0,1]^{2}$ into 4
subsets. It is constant on each of the subsets, and the union of the 4 subsets
is of full Lebesgue measure. Clearly, for $\lvert{}\lambda\lvert{}=1$, this
density is not bounded away from zero.

For a stationary copula-based Markov chain $U_{0},\cdots,U_{n}$ based on (7)
with uniform marginals, the copula of $(U_{0},U_{n})$ is $C^{n}(u,v)$ (see
Longla et al. (2022)L0 ).

######  Theorem 3.2.

For any copula defined by (7) and satisfying condition (8),

$C^{n}(u,v)=uv+\sum_{k=1}^{\infty}\lambda_{k}^{n}\int_{0}^{u}\varphi(x)dx\int_{0}^{v}\varphi(y)dy.$
(31)

Moreover, $\rho_{n}=\sup_{k}\lvert{}\lambda_{k}\lvert{}^{n}$ and Markov chains
generated with uniform marginals are both geometrically ergodic and
exponential $\rho$-mixing if and only if
$\sup_{k}\lvert{}\lambda_{k}\lvert{}<1$. If the sum has a finite number of
non-zero terms with $\lvert\lambda_{k}\lvert<1$ for all $k$, then the Markov
chains generated with uniform marginals are $\psi$-mixing.

######  Corollary 2.

For any set $\\{a_{i},i=1,\cdots,s:0\leq
a_{1}<a_{2}<\cdots<a_{s}<a_{s+1}=1\\}$, and $\lvert{}\theta_{i}\lvert{}\leq
1$, formula (14) defines the density of a copula that generates $\psi-mixing$
stationary Markov chains with continuous marginal distributions.

###  3.2 Mixing properties of the extended FGM family

We investigate the long run behavior of Markov chains generated by extended
FGM copulas. Define $\alpha_{k}=1$ if $k$ is odd and $\alpha_{k}=\min
P_{k}(x)$, if $k$ is even.

######  Theorem 3.3.

For large enough values of $n$, a version of the copula density $c^{n}(u,v)$
is bounded above by $M<2$ when $\varphi_{k}(x)$ is defined using Legendre
polynomials with $\sum{\lambda_{k}^{2}}<\infty$,
$\sup_{k}\lvert{}\lambda_{k}\lvert{}<1$ and
$\sum\lvert{}\lambda_{k}\lvert{}(2k+1)\alpha_{k}\leq 1$. Therefore the copula
$C(u,v)$ generates $\psi^{*}$-mixing stationary Markov chains (which is
equivalent to $\psi$-mixing for stationary Markov chains) for all values of
its parameters.

######  Remark 6.

It follows from the proof of Theorem 3.3, that $c^{n}(x,y)\to 1$ as
$n\to\infty$, when the conditions of Theorem 3.3 are satisfied. For any system
of eigenfunctions $\varphi_{k}(x)$ and for all values of the parameters
described in Theorem 3.3, Theorem 3.2 holds because $\psi$-mixing implies
$\rho$-mixing. Theorem 3.2 holds when extended FGM copulas are considered. A
result similar to Theorem 3.3 is valid for our trigonometric extensions
(2.2.1) and (17) in the form: for
$\sum(\lvert{}\lambda_{k}\lvert{}+\lvert{}\mu_{k}\lvert{})\leq 1/2,$ generated
Markov chains are $\psi$-mixing.

Based on the analysis above, we can derive several facts about mixing
properties of Markov chains generated by absolutely continuous copulas with
square integrable densities. In general, if we assume that a Markov chain is
generated by a copula (7) satisfying conditions (9) and (10) with an
absolutely continuous marginal distribution, then the following holds.

######  Theorem 3.4.

Under the assumptions of Theorem (3.2),

  1. 1.

If the sequence $\varphi_{k}(x)$ is uniformly bounded, and
$\sup_{k}\lvert{}\lambda_{k}\lvert{}<1$ for all $k$, then the Markov chain is
$\psi$-mixing.

  2. 2.

If $\sup_{k}\lvert{}\lambda_{k}\lvert{}<1$ for all $k$, then the Markov chain
is $\rho$-mixing.

  3. 3.

If Condition (10) allows strict inequality, then the Markov chain is
$\psi^{\prime}$-mixing.

Each of the examples of Figure (1) is bounded away from zero on $[0,1]^{2}$
and is strictly less than $2$. Therefore, by Theorem (3.4), these examples
generate $\psi$-mixing Markov chains. This means that the sine copulas and
sine-cosine copulas generate $\psi$-mixing Markov chains. This is justified by
the fact that each of the basis functions is bounded by $\sqrt{2}$ and
condition (10) implies $\lvert\lambda_{k}\lvert\leq 1/2$. It is important to
note that the uniform bound on the sequence $\varphi_{k}(x)$ is crucial in
statement 1 of Theorem 3.4.

Figure 4: Density of copula $C_{1,\lambda}$

###  3.3 Answer to an open question on mixing

Longla (2015) was solely devoted to mixing properties of copula-based Markov
chains. Results were provided for several mixing coefficients, in one
direction. There was a proof that mixing for Markov chains generated by
copulas implies mixing for Markov chains generated by their convex
combinations. In this subsection, we show that mixing for Markov chains
generated by convex combinations doesn’t require mixing for Markov chains
generated by any of the copulas of the convex combination. To do this, we take
extreme cases of formula (30), defined by $\lambda=\pm 1$. None of them
exhibits mixing in the sense of the coefficients defined in this paper, but
they have square integrable densities.

######  Remark 7.

We provide here a connection between various results of Longla et al. (2022)
L0 , Longla (2015) Lo2015  and Longla et al (2022c) LoMous  on mixing.

  1. 1.

The definition of $C_{3}(u,v)$ through its density leads to

$C_{3}(u,v)=\frac{1+\lambda}{2}C_{1}(u,v)+\frac{1-\lambda}{2}C_{2}(u,v),$

where $C_{1}(u,v)$ and $C_{2}(u,v)$ are the copulas of this family defined by
$\lambda=\pm 1$. Moreover, $C_{3}(u,v)$ generates $\psi$-mixing Markov chains
for $\lvert{}\lambda\lvert{}<1$.

  2. 2.

None of the two copulas $C_{1}(u,v)$ and $C_{2}(u,v)$ has a density bounded
away from zero on a set of Lebesgue measure 1. Moreover, $C_{1}(u,v)$ and
$C_{2}(u,v)$ don’t generate $\psi^{\prime}$-mixing Markov chains with
continuous marginal distributions. This is because they don’t generate
$\rho$-mixing Markov chains with continuous marginals. Their convex
combinations with $\lvert{}\lambda\lvert{}<1$ generate $\psi^{\prime}$-mixing
(which implies that they generate $\rho$-mixing).

  3. 3.

None of the copulas $C_{1}(u,v)$ and $C_{2}(u,v)$ generates $\psi$-mixing,
because they would otherwise generate $\rho$-mixing; but their convex
combinations with $\lvert{}\lambda\lvert{}<1$ generate $\psi$-mixing Markov
chains.

  4. 4.

None of the copulas $C_{1}(u,v)$ and $C_{2}(u,v)$ generates $\phi$-mixing,
because they would otherwise generate $\rho$-mixing; but their convex
combinations with $\lvert{}\lambda\lvert{}<1$ generate $\phi$-mixing Markov
chains.

  5. 5.

None of the copulas $C_{1}(u,v)$ and $C_{2}(u,v)$ generates $\beta$-mixing,
because they would otherwise generate $\rho$-mixing; but their convex
combinations with $\lvert{}\lambda\lvert{}<1$ generate $\beta$-mixing Markov
chains. This is because they are symmetric and generate $\rho$-mixing (see
Longla and Peligrad (2012)) LoMag .

It is crucial in the above results that the copulas are absolutely continuous.
To see this, take the copula

$C_{\alpha,\theta}(u,v)=\alpha(uv+\theta(u-u^{2})(v-v^{2}))+(1-\alpha)\min(u,v)$
$=\alpha uv+\alpha\theta(u-u^{2})(v-v^{2})+(1-\alpha)\min(u,v).$

$C_{\alpha,\theta}(u,v)$ is a convex combination of copulas containing
$\Pi(u,v)$. Therefore, by Longla (2015) Lo2015 , it generates
$\psi{{}^{\prime}}$-mixing stationary Markov chains. $C_{\alpha,\theta}(u,v)$
is a convex combination that contains $M(u,v)$. Therefore, by Longla et al.
(2022) L0 , it doesn’t generate $\psi$-mixing stationary Markov chains with
continuous marginals. Its density
$c_{\alpha,\theta}(u,v)=\alpha(1+\theta(1-2u)(1-2v))<2$ for all
$(u,v)\in[0,1]^{2}$ et all $\alpha$ and $\theta$, such that
$0<\alpha<1,0<\lvert{}\theta\lvert{}<1$. Therefore, the conditions cannot be
weakened.

Formula (12) defines an absolutely continuous copula that generates
$\psi$-mixing stationary Markov chains. Note that its density is equal to 0 on
a set of non-zero Lebesgue measure for $\lambda=-\alpha>-1$ ($\lambda<1$)or
$\lambda=-\frac{1}{\alpha}>-1$ ($\lambda>1$). This by itself answers an open
question on $\psi$-mixing from Longla (2015)Lo2015 . Longla (2015) showed that
when the density is bounded away from $0$ on a set of Lebesgue measure $1$,
the copula generates $\psi^{\prime}$-mixing. This example shows that the
condition is very strong. We can obtain $\psi$-mixing even for copulas with
density equal to zero on a set of strictly positive Lebesgue measure using
copulas from these new families.

##  4 Central limit theorem and simulation study

In this section we consider some functions of the Markov chain and derive
central limit theorems for estimators of parameters of the copula and the
population mean. We use an example of copula from the derived copula families,
with special association properties. In fact, we select a copula that has
$\rho_{S}(C)=0$, and a modification that also includes $\tau(C)=0$. For these
examples, regular estimation procedures based on these measures of association
fail.

###  4.1 Central limit theorem for parameter estimators

We consider in this section central limit theorems for averages of functions
of the Markov chain generated by the copula

$C(u,v)=uv+\mu_{1}\Phi_{1}(u)\Phi_{1}(v)+\mu_{2}\Phi_{2}(u)\Phi_{2}(v),$ (32)

with $\varphi_{1}(x)=\sqrt{2}\sin 2\pi x$, $\varphi_{2}(x)=\sqrt{2}\sin 4\pi
x$ and $2\lvert\mu_{1}\lvert+2\lvert\mu_{2}\lvert\leq 1$. Let $f(x)$ be such
that $\mu=\int_{0}^{1}f(x)dx$ and $\int_{0}^{1}f^{2}(x)dx<\infty$. Define
$S_{n}(f)=\sum_{i=1}^{n}f(U_{i})$,

$\hat{\mu}_{k}=\frac{1}{n-1}\sum_{i=1}^{n-1}\varphi_{k}(U_{i})\varphi_{k}(U_{i+1}),\quad\text{for}\quad
k=1,2.$

######  Theorem 4.1.

Assume that $U_{1},\cdots U_{n}$ is a realization of the stationary Markov
chain generated by (32) and the uniform marginal distribution. The following
holds.

  1. 1.

$\sqrt{n-1}\large\left({\hat{\mu}_{1}\choose\hat{\mu}_{2}}-{\mu_{1}\choose\mu_{2}}\right)\to
N\left({0\choose 0},\begin{pmatrix}1&-\mu_{1}\mu_{2}\\\
-\mu_{1}\mu_{2}&1\end{pmatrix}\large\right).$ (33)

  2. 2.

The central limit theorem holds in the form

$\sqrt{n}(\frac{S_{n}(f)}{n}-\mu)\to
N(0,\sigma^{2}_{f}),\quad\text{where},\quad\sigma^{2}_{f}=\sigma^{2}+2(\frac{\mu_{1}A_{1}^{2}}{1-\mu_{1}}+\frac{\mu_{2}A_{2}^{2}}{1-\mu_{2}}),$
(34)

where $A_{i}^{2}=2(\int_{0}^{1}\sin 2i\pi uf(u)du)^{2}$.

Based on properties of the copulas that we have constructed here, the variance
$\sigma^{2}_{f}$ exists, is finite and is strictly positive. To construct a
confidence set or test a specific value of $(\mu_{1},\mu_{2})$, we can use the
following consequence of Theorem 4.1.

######  Corollary 3.

Assume that $U_{1},\cdots U_{n}$ is a realization of the stationary Markov
chain generated by (32) and the uniform marginal distribution. The following
holds.

$X_{n}=\frac{(n-1)}{1-{\mu}_{1}{\mu}_{2}}\large\left((\hat{\mu}_{1}-\mu_{1})^{2}+(\hat{\mu}_{2}-\mu_{2})^{2}+2{\mu}_{1}{\mu}_{2}(\hat{\mu}_{1}-\mu_{1})(\hat{\mu}_{2}-\mu_{2})\large\right)$

has an approximately $\chi^{2}(2)$ distribution. For confidence interval
purposes, replace $\mu_{1}\mu_{2}$ by $\hat{\mu}_{1}\hat{\mu}_{2}$.

Consider now $\lambda_{k}=0$ for all $k$ and $\mu_{k}=0$ for $k>2$. Moreover,
request that $\tau(C)=0$ for the sine-cosine copula. This implies
$\mu_{2}=-4\mu_{1}$, reducing the copula to the one parameter family with
$\lvert\mu_{1}\lvert\leq 0.11$

$C(u,v)=uv+\frac{\mu_{1}}{2\pi^{2}}[(1-\cos 2\pi u)(1-\cos 2\pi v)-(1-\cos
4\pi u)(1-\cos 4\pi v)].$ (35)

######  Remark 8.

Consider a Markov chain generated by (35). Theorem 4.1 implies the following.

  1. 1.

For $f(x)=\mathbb{I}(x\leq a)$, $a\in(0,1)$, we get a dependent Bernoulli
sequence and a CLT for the sample relative frequency of success holds with

$\sigma^{2}_{f}=a(1-a)+4\frac{\mu_{1}}{\pi^{2}}\large\left(\frac{\sin^{4}\pi
a}{1-\mu_{1}}-\frac{\sin^{4}2\pi a}{1+4\mu_{1}}\large\right).$ (36)

Note that for $\mu_{1}\neq 0$, the observations are dependent, but there
exists a value of
$a=\frac{1}{\pi}\arccos((\frac{1-4\mu_{1}}{4(1-\mu_{1})})^{1/4})$ for which
$\sigma^{2}_{f}=a(1-a)$. This means that the sample behaves as if it was a
simple random sample. A researcher who starts with this simple random sample
assumption would have wrong conclusions.

  2. 2.

For $f(x)=-\lambda\ln(1-x)$, we get a Markov chain with exponential marginal
distribution of parameter $\lambda$. The estimator of $\lambda$ is
$S_{n}(f)/n$ and the CLT holds with $(\int_{0}^{1}\sin 2\pi
t\ln(1-t)dt)^{2}=0.1505165$, $4(\int_{0}^{1}\sin 4\pi
t\ln(1-t)dt)^{2}=0.245684$ (these values where obtained using $R$ to
integrate).

$\sigma^{2}_{f}=\lambda^{2}+4\mu_{1}\lambda^{2}\large\left(\frac{0.1505165}{1-\mu_{1}}-\frac{0.245684}{1+4\mu_{1}}\large\right).$
(37)

This example shows that we can still estimate $\lambda$ using the sample mean,
but the variance of the estimator has to be treated more carefully. The
asymptotic variance is not $\lambda^{2}/n$. It is multiplied by a coefficient
that can be less than or grater than 1 depending on the choice of $\mu_{1}$.

  3. 3.

For $f(x)=x$, the asymptotic variance is always larger than it would be, if
the data was assumed from a simple random sample.

$\sigma^{2}_{f}=\frac{1}{12}+\frac{\mu_{1}/\pi^{2}}{1-\mu_{1}}-\frac{\mu_{1}/\pi^{2}}{1+4\mu_{1}}=\frac{1}{12}+\frac{5\mu_{1}^{2}}{\pi^{2}(1-\mu_{1})(1+4\mu_{1})}.$
(38)

  4. 4.

For $f(x,y)=2w\sin(2\pi x)\sin(2\pi y)-\frac{1}{2}(1-w)\sin(4\pi x)\sin(4\pi
y)$, $w\in[0,1]$, we have the estimator

$\mu_{w}=\sum_{i=2}^{n}\large\left(2w\sin(2\pi U_{i-1})\sin(2\pi
U_{i})-\frac{1}{2}(1-w)\sin(4\pi U_{i-1})\sin(4\pi U_{i})\large\right).$

The central limit theorem for this estimator holds with variance less than
that of $\hat{\mu}_{1}$,

$\sigma^{2}_{f}=1-2(1-4\mu_{1}^{2})(w-w^{2}).$ (39)

The smallest possible value of this variance is
$\sigma^{2}_{0}=0.5+2\mu_{1}^{2}$.

###  4.2 Simulation study

We generate a Markov chain ($U_{1},\cdots U_{1000}$) using the uniform
distribution on (0,1) and the copula (35) for $\mu_{1}=0.05$. We use the
following standard procedure to generate the Markov chain. Randomly generate
an observation from $Uniform(0,1)$ as $U_{1}$. and for every $i>1$, we
generate a new observation $W$ from $Uniform(0,1)$ and solve for $U_{i}$ the
equation $C_{,1}(U_{i-1},U_{i})=W$. This step is implemented using the R
function $uniroot$. After generating this sample, we performed a correlation
test. This test showed that one could conclude there is no correlation between
variables along the Markov chain. Moreover, the scatter plot would also
convince further the investigator that this data seems to be a random sample.
We use $\alpha=0.05$ for level of confidence of the intervals.

  1. 1.

Case 1: We use $a=01.,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9$ to obtain a set of
variables $Yi=\mathbb{I}(U\leq 0.i)$ for our sample. For this function of the
observations we get a sample of values $(Y1\dots Y9)$. We now consider the
problem of estimation of the parameters $a_{i}$ of the model using the
obtained dependent Bernoulli sequence. We get $\hat{a}_{i}=\bar{Y}i$ and the
variance is given by formula (36) for every $i$. A confidence interval is
constructed. We run $N$ such studies, and get the proportions of intervals
that contain the true value of the parameter. This is done under independence
assumption and under our model. We obtain the following comparative results,
that report the coverage probabilities under each of the models and for the
given values of $100$ and $1000$ for $N$.

N/a | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9
---|---|---|---|---|---|---|---|---|---
100 | 95 | 90 | 94 | 91 | 91 | 93 | 96 | 97 | 96
100 | 91 | 99 | 95 | 94 | 94 | 97 | 98 | 96 | 96
1000 | 95.7 | 95.0 | 94.8 | 93.3 | 92.9 | 92.7 | 94.5 | 95.0 | 95.5
Table 1: Coverage probabilities under assumption of independence N/a | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9
---|---|---|---|---|---|---|---|---|---
100 | 92 | 95 | 96 | 96 | 96 | 95 | 94 | 94 | 93
100 | 96 | 92 | 95 | 95 | 93 | 93 | 91 | 96 | 96
1000 | 94.6 | 95.0 | 95.0 | 96.4 | 95.3 | 93.4 | 92.9 | 94.3 | 94.7
Table 2: Coverage probabilities under our model C(u,v)

These tables show that the researcher would be off by a lot by assuming
independence for the considered data. Despite the small departure from
normality with $\mu_{1}=.05$ and $\mu_{2}=-.2$, there is a serious impact on
the coverage probability of the confidence interval. If dependence is
neglected, the interval would not be close to a $95\%$ confidence interval.

  2. 2.

Case 2: We create a sample of $Y=-\lambda\ln(1-U)$ with
$\lambda=0.5,1,5,10,20,30$. For each of the values of $\lambda$, we create $N$
samples. Each of the samples produces a confidence interval, and we report
proportions of samples that cover the true parameter $\lambda$ as coverage
probabilities. For the given value of $\mu_{1},$ we have
$\sigma^{2}_{f}=0.9702667\lambda^{2}$ and $\sigma_{f}=0.9953594\lambda$.

N/$\lambda$ | 0.5 | 1 | 5 | 10 | 20 | 30
---|---|---|---|---|---|---
100 | 96 | 95 | 98 | 94 | 95 | 92
100 | 96 | 94 | 95 | 96 | 90 | 96
100 | 94 | 97 | 94 | 93 | 99 | 94
Table 3: Coverage probabilities under the Exponential marginal

For this example, the low departure from independence doesn’t influence much
the distribution of the estimator or the confidence interval. Assuming
independence in this case would not have serious consequences. When the same
sample is used for all values of $\lambda$, we obtain equal coverage
probabilities.

  3. 3.

Case 3. In this case, we check that the Data that shows possible correlation
0, $\rho_{S}=0$ and $\tau=0$. We estimate the mean of the marginal
distribution and build a confidence interval for samples of 1000 observations.
This is repeated 100 times to obtain coverage probabilities. The procedure is
repeated for sample sizes 500 and 100.

N=1000 | 97 | 98 | 94 | 95 | 92 | 92
---|---|---|---|---|---|---
N=500 | 98 | 95 | 96 | 92 | 91 | 96
N=100 | 95 | 95 | 91 | 96 | 94 | 91
Table 4: Coverage probabilities under uniform marginal N=1000 | 99 | 91 | 95 | 97 | 99 | 95
---|---|---|---|---|---|---
N=500 | 95 | 94 | 97 | 97 | 91 | 97
N=100 | 97 | 97 | 97 | 94 | 95 | 98
Table 5: Coverage probabilities under uniform marginal and i.i.d

These tables also show that the confidence intervals under independence tend
to capture more than $94\%$ of the time the true value of the mean.

  4. 4.

Case 4: Estimation of $\mu_{1}$ for a sample with the uniform marginal
distribution. We construct 100 confidence intervals for each of the true
values of $\mu_{1}=.05,.1,.11$. The coverage probabilities are reported for
sample sizes $N=100,500,1000$. These intervals use $w=0.25,0.5,0.75$.

$w$/$\mu_{1}$ | .05 | .05 | .1 | .1 | .1 | .11 | .11 | .11
---|---|---|---|---|---|---|---|---
.25 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100
0.5 | 98 | 100 | 100 | 100 | 100 | 100 | 99 | 99
0.9 | 95 | 93 | 95 | 95 | 94 | 96 | 95 | 93
1 | 94 | 93 | 95 | 95 | 94 | 96 | 95 | 93
Table 6: Coverage probabilities for $\mu_{1}$ as function of $w$ and
$\mu_{1}$.

This table shows that the best estimator would use values of $w$ close to 1,
giving more weigh tho the impact of $\varphi_{1}(x)$ on the estimator of
$\mu_{1}$. This is true because when $w\leq 0.6$, even for a sample of size
$5000$, confidence intervals for $\mu_{1}$ are too wide and cover the true
value of $\mu_{1}$ almost always.

##  5 Conclusion and comments

We have provided a characterization of copulas with square integrable
symmetric densities. We have used this representation to construct several
families of copulas. One of the constructed copula families is an extension of
the FGM copula family. Other families include trigonometric functions and are
all new in the literature. Mixing properties of the constructed copula
families have been established. Theorems have been provided for $\rho$-mixing
and $\psi$-mixing of Markov chains generated by these copulas and the uniform
distribution. This result is equivalent to the said mixing for Markov chains
generated by means of a continuous marginal distribution.

Copulas based on $\\{1,\sin 2\pi kx,\cos 2\pi kx,k\in\mathbb{N}\\}$ have been
proposed. Their Spearman $\rho$ and Kendall $\tau$ indicate a way to modify
the copula without modifying $\tau(C)=\rho_{S}(C)=0$. The equalities
$\tau(C)=\rho_{S}(C)=0$ hold for the independence copula. We have investigated
central limit theorems for these Markov chains. An example with
$\lvert\mu_{1}\lvert\leq.11,\mu_{2}=-4\mu_{1}$, $\mu_{k}=0,k\geq 3$ was used
in simulations to illustrate departure from independence, while conserving
$\tau(C)=\rho_{S}(C)=0$. This simulation study has shown that in some cases,
the dependence is not seen graphically, and even on the correlation test.
However, assuming independence produces poor confidence intervals. Various
constructions of this form can be used to model the dependence structure of
the data while keeping the key factors of association that the researcher
doesn’t want to modify. We have provided in this case a central limit theorem
for the estimator of $(\mu_{1},\mu_{2})$; and indicated how it can be used for
testing and confidence intervals.

We have shown that examples of copulas based on finite sums of terms can
increase $\rho_{S}(C)$ up to $0.49$ and $\tau(C)$ to $0.32$. These values are
larger than those of the popular FGM copula family. A conclusion on the
obtained extension of the FGM copula family is that it can help modify the
Kendall coefficient while keeping the Spearman’s $\rho$ at the level
$\lambda_{1}$ by introducing non-zero coefficients. This fact is based on
$\rho_{S}(C)=\lambda_{1}$ for all copulas of the family.

We also mention that functions $\varphi_{k}(x)$ can act as parameters of the
constructed copula families, and be subject to estimation issues. This would
be the case when one needs to find the appropriate perturbation that fits the
best the relevant situation based on the information at hands. This question
will rely on mixing properties of the constructed copula families and is one
of the topics for further research on estimation problems based on these
copulas. A drawback for the constructed copula families is that, being
absolutely continuous, they do not exhibit any tail dependence. Consideration
of their tail dependent extensions is part of ongoing work.

##  6 Appendix of proofs

Proof of Proposition 2.1

  1. 1.

Note that for any copula, we have $C(x,1)=x$ and $C(x,0)=0$. Therefore, for
any absolutely continuous copula, $\varphi(x)=1$ satisfies

$K\varphi(x)=\int_{0}^{1}c(x,y)dy=C_{,1}(x,1)-C_{,1}(x,0)=1=\varphi(x).$

It follows that the function $\varphi(x)=1$ is an eigenfunction of any kernel
operator defined by an absolutely continuous copula associated to the
eigenvalue $1$.

  2. 2.

The proof of the second point relies on formula (6) and the first point of
Proposition 2.1.

  3. 3.

The proof of this third point is a consequence of the fact that when the
eigenvalue $1$ has multipilicity higher than $1$, it generates a subspace of
dimension greater than $1$. In this subspace, an orthonormal basis not
containing $\varphi(x)=1$ can be constructed. $\square$

Proof of Theorem 2.3

The statement of Theorem 2.3 implicitly claims that the sum has point-wise
convergence. To prove this, we recall Weierstrass’ M-test. By Weierstrass’
M-test, for a sequence $f_{n}(x)$ defined on the same support $E$, if
$\lvert{}f_{n}(x)\lvert{}\leq M_{n}$ for all $x$ and $\sum M_{n}<\infty$, then
$\sum f_{n}(x)$ converges uniformly and absolutely on $E$. For the sum (7),
Condition (10) is equivalent to

$\sum_{k=1}^{\infty}\lvert{}\lambda_{k}\alpha_{k}\lvert{}\leq 1.$

Note that
$\lvert{}\lambda_{k}\varphi_{k}(x)\varphi_{k}(y)\lvert{}\leq\lvert{}\lambda_{k}\alpha_{k}\lvert{}$
for all $x,y$ and $k$. Moreover,
$\sum_{k=1}^{\infty}\lvert{}\lambda_{k}\alpha_{k}\lvert{}$ converges as a
consequence of Condition (10). Thus, by Weierstrass M-test, the series that
defines the density of the copula converges absolutely and uniformly.
Moreover, (7) also holds with absolute convergence and uniform convergence.
The rest of the proof relies on the fact that functions form an orthonormal
system. Orthogonality implies $C(1,x)=C(x,1)=x$, because
$\int_{0}^{1}\varphi_{k}(s)ds=0$ and $C(x,0)=C(0,x)=0$ by definition.
$\square$

Proof of Theorem 3.1

The first condition of this theorem implies $\psi^{\prime}$-mixing as a
consequence of Longla (2015) Lo2015 . Longla (2015)Lo2015  showed that if the
density of the absolutely continuous part of $C^{s_{1}}(u,v)$ is bounded away
from $0$ on a set of Lebesgue measure 1, then $\psi^{\prime}_{s_{1}}>0$.
Moreover, $c^{s_{2}}(u,v)<M$ implies $\psi_{s_{2}}<\infty$. Therefore, by
Bradley (2005) Bradley , the first condition of Theorem 3.1 implies
$\psi$-mixing. The second condition implies $\psi$ as a result of Longla et
al. (2022c) LoMous . $\square$

Proof of Example 1.

If this copula generates $\psi$-mixing stationary Markov chains with
continuous marginal distributions, then it would have to generate
$\psi^{\prime}$-mixing and would therefore generate $\rho$-mixing
(contradiction) with Longla (2014)L2 . Therefore, it doesn’t generate
$\psi$-mixing. Though its density is $c(u,v)=0$ for all $(u,v)\in(0,1)^{2}$
such that $v\neq u\neq 1-v$, this density doesn’t exist on a set of Lebesgue
measure $0$ and is bounded on a set of Lebesgue measure $1$. The second
example is based on the fact that the density of the given copula is equal to
$2$ on a set of Lebesgue measure $1/2$ and equal to $0$ on its complement.
$\square$

Proof of Lemma 2

The upper bound on these values is based on the assumptions on the
coefficients of the series. These conditions imply that
$\lvert{}\mu_{k}\lvert{}\leq 1/2$ and the largest value in absolute value is
achieved when $\lvert{}\mu_{1}\lvert{}=1/2$, $\mu_{k}=0$ for $k>1$ and
$\lambda_{k}=0$ for $k\geq 1$. Moreover,

$\lvert{}\sum_{k=1}^{\infty}\frac{\mu_{k}}{k^{2}}\lvert{}\leq\sum_{k=1}^{\infty}\frac{\lvert{}\mu_{k}\lvert{}}{k^{2}}\leq\sum_{k=1}^{\infty}\lvert{}\mu_{k}\lvert{}\leq
1/2\quad\text{and}$
$\lvert{}\sum_{k=1}^{\infty}\frac{4\mu_{k}+2\lambda_{k}\mu_{k}}{k^{2}}\lvert{}\leq
2(2+\lambda_{1})\lvert{}\mu_{1}\lvert{}+\sum_{k=2}^{\infty}2\frac{(2+\lambda_{k})\lvert{}\mu_{k}\lvert{}}{k^{2}}$
$\leq
2(2+\lambda_{1})\lvert{}\mu_{1}\lvert{}+\sum_{k=2}^{\infty}2\lvert{}\mu_{k}\lvert{}=2(1+\lambda_{1})\lvert{}\mu_{1}\lvert{}+2\sum_{k=1}^{\infty}\lvert{}\mu_{k}\lvert{}.$

$\text{Therefore,}\quad\lvert{}\sum_{k=1}^{\infty}\frac{4\mu_{k}+2\lambda_{k}\mu_{k}}{k^{2}}\lvert{}\leq
2(1+\lambda_{1})\lvert{}\mu_{1}\lvert{}+1\leq 2.$

The last inequality uses
$\lvert{}\lambda_{1}\lvert{}+\lvert{}\mu_{1}\lvert{}\leq 1/2$. Equality
happens for $\lvert{}\mu_{1}\lvert{}=1/2$ and $\lambda_{1}=0$. These
inequalities justify the boundary on $\rho_{S}(C)$ and $\tau(C)$. $\square$

Proof of Theorem 3.2

The proof of Theorem 3.2 relies on the fact that $C^{n}(u,v)$ is the
$n^{th}$-fold product of $C(u,v)$ , $\varphi_{k}(x)$ form an orthonormal basis
of $L^{2}(0,1)$, $\lambda^{n}_{k}$ are all eigenvalues of the Hilbert-Schmidt
operator associated to $c^{n}(u,v)$. Note that though we have square
integrability, the supremum in Theorem 3.2 can be equal to $1$. The rest is an
application of Longla and Peligrad (2012) LoMag . $\square$.

Proof of Theorem 3.3

The proof of Theorem 3.3 is an application of Longla et al (2022c) LoMous . In
Longla et al (2022c) LoMous  it was shown that if the density $c^{n}(u,v)<2$
on $[0,1]$ for some $n$, then the copula $C(u,v)$ generates $\psi^{*}$-mixing.
Let $\underline{\lambda}=\sup_{k}\lvert{}\lambda_{k}\lvert{}<1$. From the fact
that the density is square integrable, we have

$c^{n}(x,y)\geq 1-\sum\lvert{}\lambda_{k}\lvert{}^{n}(2k+1)\alpha_{k}>$
$1-\underline{\lambda}^{n-3}\sum\lvert{}\lambda_{k}\lvert{}^{3}(2k+1)\alpha_{k}>1-M\underline{\lambda}^{n-3}\geq
0.$

Therefore, $c^{n}(u,v)$ is bounded away from 0 on a set of Lebesgue measure
$1$. This implies $\psi^{\prime}$-mixing. Moreover,
$\lvert{}\lambda_{k}\lvert{}\leq(2k+1)^{-1}$ when $k$ is even. Thus,

$c^{n}(x,y)\leq 1+\sum\lvert{}\lambda_{k}\lvert{}^{n}(2k+1)\leq
1+\sum_{odd}\lvert{}\lambda\lvert{}_{k}^{n-1}+\sum_{even}\lvert{}\lambda_{k}\lvert{}^{n}(2k+1)\leq$
$1+\underline{\lambda}^{n-3}\sum_{odd}\lambda_{k}^{2}+\sum_{even}\lvert{}\lambda_{k}\lvert{}^{n}(2k+1).$

The last inequality uses the fact that the sequence $\lambda_{k}$ converges to
$0$ or has finitely many values. The second portion is also bounded. It is
written separately to emphasize the fact that for even values of $k$, it is
possible to have larger values when $\lvert{}\lambda_{k}\lvert{}>(2k+1)^{-1}$.
For odd values of $k$, this is not possible because
$\lvert{}\lambda_{k}\lvert{}\leq(2k+1)^{-1}$. If we denote
$M=\sum_{k=1}^{\infty}\lambda_{k}^{2}$, then $M<\infty$ because $c(x,y)$ is
square integrable. So, we obtain

$1-M\underline{\lambda}^{n-3}\leq c^{n}(x,y)\leq
1+M\underline{\lambda}^{n-3}.$

$M$ is a constant free of $n$. So, we conclude that as $n\to\infty$,
$M\underline{\lambda}^{n-3}\to 0$. Therefore, $c^{n}(x,y)<2$ for sufficiently
large values of $n$. By Longla et al (2022c) LoMous  it follows that the
Markov chains are $\psi$-mixing.

Note that there are still some values of $\lambda_{k}>(2k+1)^{-1}$ that are
not considered by Theorem 3.3 which can increase up to the minimum of the
considered even Legendre polynomials. Using the same arguments, we extend the
theorem to the cases when the supremum of $\lambda_{k}$ is less than $1$. Due
to square integrability, we have $\sum\lambda_{k}^{2}<\infty$. The comparison
test for series leads to $\lambda_{k}^{2}(2k+1)\to 0$ as $k\to\infty$ because
$\sum(2k+1)^{-1}$ diverges. Therefore, there exists an integer K, such that
for $k>K$, $\lambda^{2}_{k}\leq\varepsilon/(2k+1)$. Now, for $k\leq K$, it is
easy to find $N$ such that for all $n\geq N$,
$\sum\lambda_{k}^{n}(2k+1)<\varepsilon_{1}$, where the sum is taken over even
integers less than or equal to $K$. This leads to

$c^{n}(u,v)\leq 1+\underline{\lambda}^{n-3}\sum_{odd}\lambda^{2}_{k}+$
$\underline{\lambda}^{n-N}\sum_{evenk\leq
K}\lvert{}\lambda_{k}\lvert{}^{N}(2k+1)+\underline{\lambda}^{n-4}\sum_{evenk>K}\lvert{}\lambda_{k}\lvert{}^{4}(2k+1).$
$\text{So,}\quad c^{n}(u,v)\leq
1+\underline{\lambda}^{n-3}M+\underline{\lambda}^{n-N}\varepsilon_{1}+\underline{\lambda}^{n-3}M\varepsilon.$

The last inequality has parameters $\varepsilon,\varepsilon_{1},N$ and $M$
that are free of $n$. Therefore, as a sequence that converges to 1, the right
hand side can be made strictly less than 2 for any values of these parameters
as $n$ gets sufficiently large. So, $c^{n}(u,v)<2$ for some integer $n$.
$\square$

Proofs of Theorem 4.1 and Corollary 3

Assume $(U_{1},\cdots,U_{n})$ is a Markov chain with uniform marginals
generated by copula (32).

Proof of Theorem 4.1

Let $f_{i}(u,v)=\varphi_{i}(u)\varphi_{i}(v)$, for $i=1,2$ and
$f=af_{1}+bf_{2}$. It is obvious that
$\mathbb{E}f(U_{i},U_{i+1})=a\mu_{1}+b\mu_{2}$ and $var(f(U_{1},U_{2}))$ is
finite. Moreover, $Y_{i-1}=(U_{i-1},U_{i}),$ $i=2\cdots n$ is a Markov chain.
This Markov chain is ergodic because the original Markov chain is
$\psi$-mixing. Therefore, by Kipnis and Varadhan (1986)KV , the CLT holds with

$\sigma^{2}_{f}=var(f(U_{1},U_{2}))+2\sum_{i=1}^{\infty}cov(f(U_{1},U_{2}),f(U_{1+i},U_{2+i})).$

Note that for a Markov chain generated by (32), the density of the cumulative
distribution function of $(U_{1},U_{2},U_{1+i},U_{2+i})$ is obtained using the
Markov property as follows. For $i\geq 2$, using formula (31),
$(U_{2},U_{1+i})$ has density

$h_{1}(u,v)=1+\mu^{i-1}_{1}\varphi_{1}(u)\varphi_{1}(v)+\mu^{i-1}_{2}\varphi_{2}(u)\varphi_{2}(v).$

Therefore, the joint density of $(U_{1},U_{2},U_{1+i},U_{2+i})$ is

$h(s,u,v,w)=c(s,u)h_{1}(u,v)c(v,w).$ (40)

Formula (40) implies

$\mathbb{E}(f_{1}^{2}(U_{1},U_{2}))=1+\mu_{1}\large\left(\int_{0}^{1}\varphi_{1}^{3}(t)dt\large\right)^{2}+\mu_{2}\large\left(\int_{0}^{1}\varphi_{2}(t)\varphi_{1}^{2}(t)dt\large\right)^{2},$
$\mathbb{E}(f_{1}(U_{1},U_{2})f_{1}(U_{2},U_{3}))=\mu_{1}^{2}\int_{0}^{1}\varphi_{1}^{4}(t)dt,$
$\mathbb{E}(f_{1}(U_{1},U_{2})f_{1}(U_{1+i},U_{2+i}))=\mu_{1}^{2}+\mu^{1+i}_{1}\large\left(\int_{0}^{1}\varphi_{1}^{3}(t)dt\large\right)^{2},\quad
i>1\mathchar 24635\relax\;$
$\mathbb{E}(f_{1}(U_{1},U_{2})f_{2}(U_{1},U_{2}))=\mu_{1}\large\left(\int_{0}^{1}\varphi_{1}^{2}(t)\varphi_{2}(t)dt\large\right)^{2}+\mu_{2}\large\left(\int_{0}^{1}\varphi_{1}(t)\varphi_{2}^{2}(t)dt\large\right)^{2},$
$\mathbb{E}(f_{1}(U_{1},U_{2})f_{2}(U_{2},U_{3}))=\mu_{1}\mu_{2}\int_{0}^{1}\varphi_{1}^{2}(t)\varphi_{2}^{2}(t)dt,$
$\mathbb{E}(f_{1}(U_{1},U_{2})f_{2}(U_{1+i},U_{2+i}))=\mu_{1}\mu_{2}^{i}\int_{0}^{1}\varphi_{1}^{2}(t)\varphi_{2}(t)dt\int_{0}^{1}\varphi_{2}^{3}(t)dt+$
$+\mu_{2}\mu_{1}^{i}\int_{0}^{1}\varphi_{2}^{2}(t)\varphi_{1}(t)dt\int_{0}^{1}\varphi_{1}^{3}(t)dt+\mu_{1}\mu_{2}.$

Corresponding formulas for $f_{2}$ are not provided here. They are obtained by
symmetry. Using $\varphi_{1}(x)=\sqrt{2}\sin(2\pi x)$ and
$\varphi_{2}(x)=\sqrt{2}\sin(4\pi x)$, we obtain

$\int_{0}^{1}\varphi^{3}_{i}(x)dx=0,\quad\int_{0}^{1}\varphi^{2}_{1}(x)\varphi^{2}_{2}(x)dx=1$
$\text{and}\quad\int_{0}^{1}\varphi_{1}^{2}(x)\varphi_{2}(x)dx=\int_{0}^{1}\varphi_{1}(x)\varphi^{2}_{2}(x)dx=0,\quad\int_{0}^{1}\varphi_{i}^{4}(x)dx=\frac{3}{2}.$

Therefore, substituting and simplifying leads to
$\sigma^{2}_{f}=a^{2}+b^{2}-2ab\mu_{1}\mu_{2}.$ It follows that for
$\bar{f_{n}}=\frac{1}{n-1}\sum_{i=1}^{n-1}(a\varphi_{1}(U_{i})\varphi_{1}(U_{i+1})+b\varphi_{2}(U_{i})\varphi_{2}(U_{i+1}))$,

$\sqrt{n-1}(\bar{f_{n}}-a\mu_{1}-b\mu_{2})\to
N(0,a^{2}+b^{2}-2ab\mu_{1}\mu_{2}).$ (41)

Therefore, the Cramer-Wold device concludes the proof of Theorem 4.1.
$\square$

Proof of Corollary 3

According to our work, this Markov chain is $\psi$-mixing. Let $f(x)$ be a
function such that $\mu=\int_{0}^{1}f(x)dx$ and
$\int_{0}^{1}f^{2}(x)dx<\infty$. Define $S_{n}(f)=\sum_{i=1}^{n}f(U_{i})$. By
Kipnis and Varadhan (1986)KV , the central limit theorem holds in the form

$\sqrt{n}(\frac{S_{n}(f)}{n}-\mu)\to
N(0,\sigma^{2}_{f}),\quad\text{where},\quad\sigma^{2}_{f}=\lim_{n\to\infty}\frac{var(S_{n}(f))}{n}.$
(42)

Based on the provided formula of joint cumulative distribution of
$(U_{0},U_{n})$ and stationarity, $\varphi_{1}(x)=\sqrt{2}\sin 2\pi x$ and
$\varphi_{2}(x)=\sqrt{2}\sin 4\pi x$ imply

$cov(f(U_{1}),f(U_{k}))=\mu_{1}^{k-1}\int_{0}^{1}\int_{0}^{1}\varphi_{1}(u)\varphi_{1}(v)f(u)f(v)dudv+$
$\mu_{2}^{k-1}\int_{0}^{1}\int_{0}^{1}\varphi_{2}(u)\varphi_{2}(v)f(u)f(v)dudv.$

Moreover, in the context of reversible Markov chains, we have

$\sigma^{2}_{f}=\sigma^{2}+2\sum_{k=2}^{\infty}cov(f(U_{1}),f(U_{k}))=\sigma^{2}+2(\frac{\mu_{1}A_{1}^{2}}{1-\mu_{1}}+\frac{\mu_{2}A_{2}^{2}}{1-\mu_{2}}),$

where $A_{i}=\int_{0}^{1}\varphi_{i}(u)f(u)du$ and $\sigma^{2}$ is the
variance of $f(X_{i})$. In this case, $\mu_{2}=-4\mu_{1}$. Simple computations
lead to the formula of $\sigma^{2}_{f}$.

  * •

Case 1: Dependent Bernoulli observations. Select $f(x)=\mathbb{I}(x\leq a)$
for $a\in(0,1)$. Via simple computations, we obtain
$\mu=\mathbb{E}f(U_{1})=a$, $\sigma^{2}=a(1-a)$ and $A_{i}^{2}=\frac{(1-\cos
2\pi ia)^{2}}{2\pi^{2}i^{2}}=\frac{2\sin^{4}\pi ia}{\pi^{2}i^{2}}$. Thus,
equation (36) holds.

  * •

Case 2: Trigonometically dependent exponential sequences. Consider
$f(x)=-\lambda\ln(1-x)$. This defines a sequence of variables
$X_{1},\cdots,X_{n}$ that has exponential marginal distributions
($X_{i}=-\lambda\ln(1-U_{i})$). $\mu=\mathbb{E}(X_{i})=\lambda$,
$\sigma^{2}=\lambda^{2}$. Simple computations give formula (37)

  * •

Case 3. Dependent uniform Data. Consider now $f(x)=x$. The generated Markov
chain itself. $\mathbb{E}(X)=1/2$, $\sigma^{2}=1/12$.
$A_{i}^{2}=\frac{1}{2i^{2}\pi^{2}}$. So, formula (38) holds. For
$\lvert\mu_{1}\lvert\leq.11$, the variance (38) is strictly positive.

  * •

Estimating $\mu_{1}$. Consider estimating the parameter $\mu_{1}$ based on the
generated Markov chain. Take any $w\in\mathbb{R}$ and

$f(x,y)=2w\sin(2\pi x)\sin(2\pi y)-\frac{1}{2}(1-w)\sin(4\pi x)\sin(4\pi y).$

Theorem 4.1 and Formula (41) conclude the proof. $\square$

## Data availability and conflict of interest

This manuscript has no associated data and there is no conflict of interest.

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