# On some mixing properties of copula-based Markov chains

Martial Longla<EMAIL_ADDRESS>University of Mississippi, Department of
mathematics  Mous-Abou Hamadou<EMAIL_ADDRESS>University of Maroua,
Department of mathematics  Seraphin Isidore Ngongo<EMAIL_ADDRESS>University of Yaounde I, ENS, Department of Mathematics

###### Abstract

This paper brings some insights of $\psi^{\prime}$-mixing, $\psi^{*}$-mixing
and $\psi$-mixing for copula-based Markov chains and the perturbations of
their copulas. We provide new tools to check Markov chains for $\psi$-mixing
or $\psi^{\prime}$-mixing, and also show that perturbations of
$\psi^{\prime}$-mixing copula-based Markov chains are $\psi^{\prime}$-mixing
while perturbations of $\psi$-mixing Markov chains are not necessarily
$\psi$-mixing markov chains, even when the perturbed copula is $\psi-mixing$.
Some examples of copula families are considered. A statistical study is
provided to emphacize the impact of perturbations on copula-based Markov
chains. Moreover, we provide a correction to a statement made in Longla and
al. (2021) on $\psi$-mixing.

Key words: Perturbations of copulas, mixtures of copulas, convex combinations
of Copulas, Mixing rates, Lower-psi mixing.

Mathematical Subject Classification (2000): 62G08, 62M02, 60J35

##  1 introduction

Modelling dependence among variables or factors in economics, finance, risk
management and other applied fields has benefited over the last decades from
the study of copulas. Copulas, these multivariate cumulative distributions
with uniform marginals on $[0,1]^{n}$, have been widely used as strength of
the dependence between variables. Sklar (1959) first showed that by rescalling
the effect of marginal distributions, one obtains a copula from the joint
distribution of random variables. This rescalling implies that when variables
are transformed using increasing functions, the copula of their
transformations remains same as that of the original variables. For many
dependence coefficients, this copula is all that affects the computations
(random vectors with common copulas have common dependence coefficients). This
justifies why dealing with the uniform distribution as stationary distribution
of a Markov chain is same as studying a Markov chain with any absolutely
continuous stationary distribution. Following the ideas of Durante and al.
(2013), Longla and al. (2021) and Longla and al. (2022) have considered the
perturbation method that adds to a copula an extra term called perturbation.
They also considered other classes of modifications and their impact on the
dependence structure as studied by Komornik and al. (2017). The long run
impact of such perturbations on the dependence structure and the measures of
association was investigated. In fact,They have investigated the impact of
perturbations of copulas on the mixing structure of the Markov chains that
they generate. The case was presented for $\rho$-mixing, $\alpha$-mixing,
$\psi$-mixing and $\beta$-mixing in Longla and al. (2021) and Longla and al.
(2022).

###  1.1 Facts about Copulas

The definition of a 2-copula and related topics can be found in Nelsen (2006).
2-copulas are in general just referred to as copulas when there is no reason
for confusion. We will follow this assumption throughout this paper. A
function $C:[0,1]^{2}\rightarrow[0,1]$ is called a bivariate copula if it
satisfies the following conditions:

  1. i.

$C(0,x)=C(x,0)=0$ (meaning that $C$ is grounded);

  2. ii.

$C(x,1)=C(1,x)=x,\forall x\in[0,1]$ (meaning each coordinate is uniform on
[0,1]);

  3. iii.

$C(a,c)+C(b,d)-C(a,d)-C(b,c)\geq 0,\forall\ [a,b]\times[c,d]\subset[0,1]^{2}.$

The last condition basically states that the porbability of any rectangular
subset of $[0,1]\times[0,1]$ is non-negative. This is an obvious condition,
given that $C(x,y)$ is a cumulative probability distribution function on
$[0,1]\times[0,1]$. The first condition states that the probability of any
rectangle that doesn’t cross $[0,1]\times[0,1]$ is equal to 0 (this covers
that fact that such a rectangle doesn’t intersect the support of the
distribution function). The second condition basically asserts that the
marginal distribution is uniform on $[0,1]$ for each of the components of the
considered vector.

Darsaw and al. (1992) derived the transition probabilities for stationary
Markov chains with uniform marginals on $[0,1]$ as
$P(X_{n}\in(-\infty,x]|X_{n-1}=x)=C_{,1}(x,y),\forall n\in\mathbb{N}$, where
$C_{,i}(x,y)$ denotes the derivative of $C(x,y)$ with respect to the $i^{th}$
variable. This property has been used by many authors to establish mixing
properties of copula-based Markov chains. We can cite Longla (2015), Longla
(2014), Longla and Peligrad (2012) who provided some results for reversible
Markov chains, Beare (2010) who presented results for $\rho$-mixing among
others.

It’s been shown in the literature (see Darsow and al. (1992) and the
references therein) that if $(X_{1},\cdots,X_{n})$ is a Markov chain with
consecutive copulas $(C_{1},\cdots,C_{n-1})$, then the fold product given by

$C(x,y)=C_{1}*C_{2}(x,y)=\int^{1}_{0}C_{1,2}(x,t)C_{2,1}(t,y)dt$

is the copula of $(X_{1},X_{3})$ and the $\star$-product given by

$C(x,y,z)=C_{1}\star C_{2}(x,y,z)=\int_{0}^{y}C_{1,2}(x,t)C_{2,1}(t,z)dt$

is the copula of $(X_{1},X_{2},X_{3})$. The $n$-fold product of $C(x,y)$
denoted $C^{n}(x,y)$ is defined by the recurrence $C^{1}(x,y)=C(x,y)$,

$C^{n}(x,y)=C^{n-1}*C(x,y).$

The most popular copulas are $\Pi(u,v)=uv$ (the independent copula), the
Hoeffding lower and upper bounds $W(u,v)=\max(u+v-1,0)$ and $M(u,v)=\min(u,v)$
respectively. Convex combinations of copulas
$\\{C_{1}(x,y),\cdots,C_{k}(x,y)\\}$ defined by
$\displaystyle\\{C(x,y)=\sum_{j=1}^{k}a_{j}C_{j}(x,y),0\leq
a_{j},\sum_{j=1}^{k}a_{j}=1\\}$ are also copulas. For any copula $C(x,y)$,
there exists a unique representation $C(x,y)=AC(x,y)+SC(x,y)$, where $AC(x,y)$
is the absolute continuous part of $C(x,y)$ and $SC(x,y)$ is the singular part
of the copula $C(x,y)$. $AC(x,y)$ induces on $[0,1]^{2}$ a measure $P_{c}$
defined on borel sets by

$\displaystyle P_{c}(A\times B)=\int_{A}\int_{B}c(x,y)dxdy\quad\text{and}\quad
P(A\cap B)=P_{c}(A\times B)+SC(A\times B),\quad\text{(see Longla (2015)).}$

An absolutely continuous copula is one that has singular part $SC(x,y)=0$ and
a singular copula is one that has absolutely continuous part $AC(x,y)=0$. This
work is concerned mostly by absolutely continuous copulas and mixing
properties of the Markov chains they generate.

###  1.2 Mixing coefficients of interest

The mixing coefficients of interest in this paper are $\psi^{\prime}$ and
$\psi$. The $\psi$-mixing condition has its origin in the paper by Blum and
al. (1963). They studied a different condition (“$\psi$*-mixing”) similar to
this mixing coefficient. They showed that for Markov chains satisfying their
condition, the mixing rate is exponential. The coefficient took its present
form in the paper of Philipp (1969). For examples of mixing sequences, see
Kesten and O’Brien (1976), who showed that in general, the mixing rate could
be arbitrarily slow, a large class of mixing rates can occur for stationary
$\psi$-mixing. The general definitions of these mixing coefficients are as
follows. Given any $\sigma$-fields $\mathscr{A}$ and $\mathscr{B}$ and a
defined probability measure $P$,

$\psi(\mathscr{A},\mathscr{B})=\sup_{B\in\mathscr{B},A\in\mathscr{A},P(A)\cdot
P(B)>0}\frac{|P(A\cap B)-P(A)P(B)|}{P(A)P(B)},$
$\psi^{\prime}(\mathscr{A},\mathscr{B})=\inf_{B\in\mathscr{B},A\in\mathscr{A},P(A)>0}\frac{P(B\cap
A)}{P(A)P(B)},\quad\text{and}\quad\psi^{*}(\mathscr{A},\mathscr{B})=\sup_{B\in\mathscr{B},A\in\mathscr{A},P(A)\cdot
P(B)>0}\frac{P(A\cap B)}{P(A)P(B)}.$

In case of stationary copula-based Markov chains generated by an absolutely
continuous copula, the $\psi^{\prime}$-mixing dependence coefficient takes the
form

$\psi^{\prime}_{n}(C)=\underset{\underset{\lambda(A)\lambda(B)>0}{A,B\in\mathscr{B}}}{\inf}\dfrac{\int_{A}\int_{B}c_{n}(x,y)dxdy}{\lambda(A)\lambda(B)},$

where $c_{n}(x,y)$ is the density of the of $C^{n}(x,y)$ and $\lambda$ is the
Lebesgue measure on $I=[0,1]$. For every positive integer $n$, let $\mu_{n}$
be the measure induced by the distribution of $(X_{0},X_{n})$. Let $\mu$ be
the measure induced by the stationary distribution of the Markov chain and
$\mathscr{B}$ the $\sigma$-algebra generated by $X_{0}$. The
$\psi^{\prime}$-mixing dependence coefficient takes the form

$\psi_{n}^{\prime}(C)=\underset{A,B\in\mathscr{B},\mu(A).\mu(B)>0~{}}{\inf}\dfrac{\mu_{n}(A\times
B)}{\mu(A)\mu(B)}$, and
$\psi_{n}^{*}(C)=\underset{A,B\in\mathscr{B},\mu(A).\mu(B)>0~{}}{\sup}\dfrac{\mu_{n}(A\times
B)}{\mu(A)\mu(B)}$

###  1.3 About perturbations

In applications, knowing approximately a copula $C(u,v)$ appropriate to the
model of the observed data, minor perturbations of $C(u,v)$ are considered.
Komornik and al. (2017) have investigated some perturbations that were
introduced by Mesiar and al. (2015). These perturbations were also considered
by Longla and al. (2021) and (2022). Perturbations that we consider in this
work have been studied by many authors. Sheikhi et al. (2020) looked at the
perturbations of copulas via modification of the random variables that the
copulas are used to represent the dependence structure of. Namely, they
perturbed the copula of $(X,Y)$ by looking at the copula of $(X+Z,Y+Z)$ for
some $Z$ independent of $(X,Y)$ that can be considered as noise. Mesiar and
al. (2019) worked on the perturbations induced by modification of one of the
random variables of the pair. Namely, the copula of $(X,Y)$ was perturbed to
obtain the copula of $(X+Z,Y)$. In this work, we look at he impact of
perturbations on $\psi$-mixing and $\psi^{\prime}$-mixing. We provide
theoretical proofs and a simulation study that justifies the importance of the
study of perturbations and their impact on estimation problems. This is done
through the central limit theorem that varies from one kind of mixing
structure to another and is severely impacted by perturbations in the case of
$\psi$-mixing for instance.

###  1.4 Structure of the paper

This paper consists of six sections, each of which concern a specific topic of
interest and are structured as follows. Introduction in Section 1 is divided
into several parts. Facts about copulas are introduced in subsection 1.1,
mixing coefficient of interest ($\psi^{\prime}$-mixing and $\psi$-mixing) are
defined in subsection 1.2 and Subsection 1.3 is dedicated to facts about
perturbation of copulas. Section 2 is devoted to the impact of perturbations
on $\psi^{\prime}$-mixing and $\psi$-mixing copula-based Markov chains,
addressing $\psi^{\prime}$-mixing in Subsection 2.1 and $\psi$-mixing in
Subsection 2.2. We emphasize on the fact that perturbations of
$\psi^{\prime}$-mixing copula-based Markov chains are $\psi^{\prime}$-mixing
while perturbations of $\psi$-mixing Markov chains are not necessarily
$\psi$-mixing, even when the perturbed copula is $\psi$-mixing. We present
here the case of $\psi^{*}$-mixng. This section ends by an explicit example
showing this fact. In Section 3 we provide some graphs to show the effect of
perturbations. In Section 4, we showcase a simulation study to emphasize the
importance of this topic. Comments on the paper’s results and their
relationship with current state of art are presented in Section 5 and Section
6 provided the proofs of our main results. Throughout this work $\psi_{n}(C)$
is replaced by $\psi_{n}$ when there is no reason for confusion.

##  2 Facts about $\psi^{\prime}$-mixing and $\psi$-mixing

###  2.1 All about $\psi^{\prime}$-mixing

A result of Bradley (1983) states the following

###### Theorem 2.1.1

For any strictly stationary Markov chain, either $\psi^{\prime}_{n}\to 1$ as
$n\to\infty$ or $\psi^{\prime}_{n}=0$ $\forall n\in\mathbb{N}$.

Based on this result, we show the following.

###### Theorem 2.1.2

Let $\lambda$ be the Lebesgue measure on $[0,1]$. If the copula $C(u,v)$ of
the stationary Markov chain $(X_{k},k\in\mathbb{N})$ is such that the density
of its absolutely continuous part
$c(u,v)\geq\varepsilon_{1}(u)+\varepsilon_{2}(v)$ on a set of Lebesgue measure
$1$ and $\displaystyle\inf_{A\subset
I}\frac{\int_{A}\varepsilon_{1}d\lambda}{\lambda(A)}>0$ or
$\displaystyle\inf_{A\subset
I}\frac{\int_{A}\varepsilon_{2}d\lambda}{\lambda(A)}>0$, then the Markov chain
is $\psi^{\prime}$-mixing.

Theorem 2.1.2 is an extension of Theorem 2.5 of Longla (2014). It extends the
result from $\rho$-mixing to $\psi^{\prime}$-mixing. Longla and al. (2021)
state that for a copula $C$ perturbed by means of the independence copula
$\Pi$, the following result holds.

###### Theorem 2.1.3

The perturbed copula with parameter $\theta$ has the following properties:

$C_{\theta,\Pi}^{n}(u,v)=(1-\theta)^{n}C^{n}(u,v)+(1-(1-\theta)^{n})uv.$ (2.1)

As a result of Theorem 2.1.3, following Longla (2015), based on the fact that
the density of the copula $C_{\theta,\Pi}^{n}(u,v)$ is bounded away from zero
on a set of Lebesgue Measure $1$, we can conclude the following:

###### Corollary 2.1.4

$C_{\theta,\Pi}^{n}(u,v)$ generates lower $\psi$ mixing stationary Markov
chains.

In general, for any convex combination of copulas, the following result holds.

###### Theorem 2.1.5

For any set of copulas $C_{1}(u,v)\cdots C_{k}(u,v)$, if there exists a subset
of copulas $C_{k_{1}}\cdots C_{k_{s}},$ $s\leq k\in\mathbb{N}$ such that
$\psi^{\prime}(\hat{C})>0\quad\text{for}\quad\hat{C}=C_{k_{1}}*\cdots*C_{k_{s}},$
then $\psi^{\prime}_{s}(C)>0$ and any Markov chain generated by

$C=a_{1}C_{1}+\cdots+a_{k}C_{k}\quad\text{for }\quad
0<a_{1},\dots,a_{k}<1\quad\text{is
exponential}\quad\psi^{\prime}-\text{mixing}.$

###### Theorem 2.1.6

For any set of copulas $C_{1}(u,v)\cdots C_{k}(u,v)$, if there exists a subset
of copulas $C_{k_{1}}\cdots C_{k_{s}},$ $s\leq k\in\mathbb{N}$ such that the
density of the absolutely continuous part of $\hat{C}(u,v)$ is bounded away
from $0$ $\text{for}\quad\hat{C}=C_{k_{1}}*\cdots*C_{k_{s}},$ then
$\psi^{\prime}_{s}(C)>0$ and any Markov chain generated by

$C=a_{1}C_{1}+\cdots+a_{k}C_{k}\quad\text{for }\quad
0<a_{1},\dots,a_{k}<1\quad\text{is
exponential}\quad\psi^{\prime}-\text{mixing}.$

###  2.2 All about $\psi$-mixing and $\psi^{*}$-mixing

It’s been shown in the literature that $\psi$-mixing implies
$\psi^{\prime}$-mixing, $\psi^{*}$-mixing and other mixing conditions. See for
instance Bradley (2007). We emphasize here that the above theorems cannot be
extended to $\psi$-mixing in general by exhibiting cases when the conditions
of the theorems are satisfied, but the $\psi$-mixing condition is not. A
result of Bradley (1983) states the following.

###### Lemma 2.2.1

For a strictly stationary mixing sequence, either $\psi^{*}_{n}=\infty$ for
all $n$ or $\psi^{*}_{n}\rightarrow 1$ as $n\rightarrow\infty$.

Based on this finding, if we want to show that a stationary Markov chain is
$\psi^{*}-mixing$, it is enough to show that it is mixing and
$\psi^{*}_{1}\neq\infty$. It needs to be clear that this is not a necessary
condition. In fact, there is $\psi^{*}$-mixing whenever we can show that for
some positive integer $n$, $\psi^{*}_{n}\neq\infty$. A remark of Longla and
al. (2021) states the following.

###### Remark 2.2.2

In general, for any convex convolution of two copulas (here $0\leq a\leq 1)$,
the $\psi-mixing$ coefficient satisfies the following inequalities:

$\displaystyle\psi(aC_{1}+(1-a)C_{2})$ $\displaystyle\leq$ $\displaystyle
a\psi(C_{1})+(1-a)\psi(C_{2});$ (2.2) $\displaystyle\psi(aC_{1}+(1-a)C_{2})$
$\displaystyle\geq$ $\displaystyle a\psi(C_{1})-(1-a)\psi(C_{2}).$ (2.3)

A result of Longla and al. (2021) states the following.

###### Theorem 2.2.3

A convex combination of copulas generates stationary $\psi-mixing$ Markov
chains if each of the copulas of the combination generates $\psi-mixing$
stationary Markov chains.

This Theorem as stated was not fully proved. Based on the provided proof, the
correct statement should be the following.

###### Theorem 2.2.4

A convex combination of copulas generates stationary $\psi-mixing$ Markov
chains if each of the copulas of the combination generates $\psi-mixing$
stationary Markov chains with $\psi_{1}<1$.

We now state the following new result.

###### Theorem 2.2.5

If a copula $C(u,v)$ is absolutely continuous and for some positive integer
$n$, the density of $C^{n}(u,v)$ is bounded above on $[0,1]^{2}$, then it
generates $\psi^{*}$-mixing stationary Markov chains. Alternatively, if for
every $n$ the density of $C^{n}(u,v)$ is continuous and not bounded above on
some subset of $[0,1]^{2}$, then $C(u,v)$ doesn’t generate $\psi^{*}$-mixing
or $\psi$-mixing Markov chains.

####  2.2.1 Examples

  1. 1.

The bivariate Gaussian copula and the Markov chains it generates.

The Bivariate Gaussian Copula density is defined as

$c_{R}(u,v)=\frac{1}{\sqrt{|R|}}e^{-\frac{1}{2}(\Phi^{-1}(u)\quad\Phi^{-1}(v))(R^{-1}-\mathbb{I}){\Phi^{-1}(u)\choose\Phi^{-1}(v)}},$

where $R$ is a bivariate variance-covariance matrix and $\mathbb{I}$ is the
$2\times 2$ identity matrix and $\Phi^{-1}(x)$ is the quantile function of the
standard normal distribution. The example when $R={2\quad 1\choose 1\quad 1}$
is

$c_{R}(u,v)=e^{\Phi^{-1}(u)\Phi^{-1}(v)-.5(\Phi^{-1}(v))^{2}}.$

It is clear that this density is not bounded above because for $v=.51$ and
$u\to 1$, we have $c_{R}(u,.51)\to\infty$. By simple computations, we can show
that any bivariate Gaussian copula that is not the independence copula has a
density that is not bounded above. And a $*$-product of Gaussian copulas is
the independence copula only when one of the two copulas is the independence
copula. This is important for the following clain.

###### Lemma 2.2.6

Any Copula-based Markov chain generated by a Gaussian copula that is not the
product copula is not $\psi^{*}$-mixing.

The proof of Lemma 2.2.6 is an application of Theorem 2.2.5 and the fact that
the joint distribution of $(X_{0},X_{n})$ is the consecutive $*$-product of
Gaussian copulas.

  2. 2.

The Ali-Mikhail-Haq copula and the Markov chains they generate.

Copulas from the Ali-Mikhail-Haq family are defined for $\theta\in[-1,1]$ by

$C_{\theta}(u,v)=\frac{uv}{1-\theta(1-u)(1-v)}\quad\text{with density}\quad
c_{\theta}(u,v)=\frac{(1-\theta)(1-\theta(1-u)(1-v))+2\theta
uv}{(1-\theta(1-u)(1-v))^{3}}.$

It is easy to see that this density is continuous and satisfies
$c_{\theta}(u,v)\leq\frac{1+\theta^{2}}{(1-\theta)^{3}}$ when $1>\theta>0$ or
$c_{\theta}(u,v)\leq 1+\theta^{2}$ when $\theta\leq 0$. Therefore, the
following result follows from Theorem 2.2.5.

###### Lemma 2.2.7

Any copula from the Ali-Mikhail-Haq family of copulas with $\theta\neq 1$
generates $\psi^{*}$-mixing stationary Markov chains.

  3. 3.

Copulas with densities $m_{1},m_{2},m_{3}$ and $m_{4}$ of Longla (2014) and
the Markov chains they generate.

Because each of these copulas is bounded when the functions $g(x)$ and $h(x)$
used in their definitions are bounded, we have the following result.

###### Lemma 2.2.8

All copulas with densities $m_{1},m_{2},m_{3}$ and $m_{4}$ of Longla (2014)
with bounded functions $g(x)$ and $h(x)$ generate $\psi^{*}$-mixing Markov
chains.

####  2.2.2 The Farlie-Gumbel-Morgenstern copula Family

This family of copulas is defined by $C_{\theta}(u,v)=uv+\theta uv(1-u)(1-v)$,
for $\theta\in[0,1]$.

###### Theorem 2.2.9

For any member of the Farlie-Gumbel-Morgenstern family of copula with
parameter $\theta$, the joint distribution of $(X_{0},X_{n})$ for a stationary
copula-based Markov chain generated is

$C_{\theta}^{n}(u,v)=uv+3\large(\frac{\theta}{3}\large)^{n}uv(1-u)(1-v).$
(2.4)

The density of this copula is
$c^{n}_{\theta}(u,v)=1+3\large(\frac{\theta}{3}\large)^{n}(1-2u)(1-2v)$. Via
simple calculations, il follows that

$0\leq 1-3\large(\frac{|\theta|}{3}\large)^{n}\leq c^{n}_{\theta}(u,v)\leq
1+3\large(\frac{|\theta|}{3}\large)^{n}.$ (2.5)

###### Theorem 2.2.10

Any Copula-based Markov chain generated by a copula from the Farlie-Gumbel-
Morgenstern family of copulas is $\psi$-mixing (for any $\theta\in[-1,1]$).

It has been established, using the first inequality of (2.5) when $n=1$ and a
weaker form of Theorem 2.1.6, that any copula from this family with
$|\theta|\neq 1$ generates exponential $\psi^{\prime}$-mixing. We can now show
via integration that for any copula-based Markov chain $(X_{1},\cdots,X_{k})$
generated by $C_{\theta}(u,v)$, if $A\in\sigma(X_{1})$ and
$B\in\sigma(X_{n+1})$, then

$1-3\large(\frac{|\theta|}{3}\large)^{n}\leq\frac{P^{n}(A\cap
B)}{P(A)P(B)}\leq 1+3\large(\frac{|\theta|}{3}\large)^{n}.$ (2.6)

Formula (2.6) implies that $\displaystyle\sup_{A,B}\frac{P^{n}(A\cap
B)}{P(A)P(B)}\leq 1+3\large(\frac{|\theta|}{3}\large)^{n}<2$, for $n>1$ and
any $|\theta|\leq 1$. It follows from Theorem 3.3 of Bradley (2005) that this
Markov chain is exponential $\psi$-mixing for all values of $\theta$ in the
range.

####  2.2.3 The Mardia and Frechet Families of Copula

Any copula from the Mardia family is represented as $\displaystyle
C_{\alpha,\beta}(u,v)=\alpha M(u,v)+\beta W(u,v)+(1-\alpha-\beta)\Pi(u,v),$
with $0\leq\alpha,\beta,1-\alpha-\beta\leq 1$. The Frechet family of copulas
is a subclass of the Mardia family with $\alpha+\beta=\theta^{2}$. The two
families thus enjoy the same mixing properties and their analysis is
theoretically identical. The density of any copula of these families is
bounded away from zero on a set of Lebesgues measure 1. Therefore, the results
of this paper imply that these families generate $\psi^{\prime}$-mixing. Now,
Consider $(X_{1},X_{2})$ with joint distribution $C_{\alpha,\beta}(u,v)$ and
the sets $A=(0,\varepsilon)$ and $B=(1-\varepsilon,1)$. Via simple
calculations, we obtain

$P(A\cap B)=(1-\alpha-\beta)\varepsilon^{2}+\beta\varepsilon.$ (2.7)

Thus,

$\sup_{A,B}\frac{P(A\cap B)-P(A)P(B)}{P(A)P(B)}\geq
sup_{\varepsilon}(-\alpha-\beta+\frac{\beta}{\varepsilon})=\infty.$ (2.8)

To complete the proof, we use the fact that based on the result of Longla
(2014), the joint distribution of $(X_{1},X_{n+1})$ is $C^{n}(u,v)$ \- member
of the Mardia family of copulas. This fact and formula (2.8) imply that
$\psi_{n}=\infty$ for all $n$. Therefore, this copula doesn’t generate
$\psi$-mixing and therefore as a result of Lemma 2.2.1. Hence, the results of
this work cannot be extended to $\psi$-mixing. The idea of this proof leads to
the following.

###### Theorem 2.2.11

Let $C(u,v)$ be a copula that generates non $\psi^{*}$-mixing stationary
Markov chains. Any convex combination of copulas containing $C(u,v)$ generates
non $\psi^{*}$-mixing Markov chains.

Theorem 2.2.11 combined with Longla and al (2022) imply the following result.

###### Theorem 2.2.12

A convex combination of copulas generates $\psi^{*}$-mixing stationary Markov
chains if every copula it contains generates $\psi^{*}$-mixing stationary
Markov chains with $\psi^{*}_{1}<1$.

####  2.2.4 General case of lack of $\psi$-mixing in presence of
$\psi^{\prime}$-mixing

We want here to present a large class of copulas that generate
$\psi^{\prime}$-mixing Markov chains, but doesn’t generate $\psi^{*}$-mixing
or $\psi$-mixing Markov chains. Based on the results of this work, we can
state the following general corollary.

###### Corollary 2.2.13

Any convex combination of copulas containing the independence copula
$\Pi(u,v)$ and $M(u,v)$ or $W(u,v)$ generates exponential
$\psi^{\prime}$-mixing, but doesn’t generate $\psi$-mixing or
$\psi^{*}$-mixing stationary Markov chains.

##  3 Some graphs of copulas and their perturbations

In this section, we provide graphical representations of the impact of
perturbations on Markov chains generated by the copulas of interest. The case
is presented for some examples from the Frechet and Farlie-Gumbel-Morgenstern
families of copulas. Examples are chosen for the values of the parameters that
are close to independence and the extreme case of each of the families. Two
graphs of data on $(0,1)^{2}$ are provided as well as two graphs for the
standard mornal distribution as marginal distribution of the Markov chains. To
generate a Markov chain with a copula from this family, we proceed as follows.

(a) Generate $U_{1}$ from $Uniform(0,1)$;

(b) For $t=2,\cdots n,$ generate $W_{t}$ from $Uniform(0,1)$ and solve for
$U_{t}$ the equation $W_{t}=U_{t}+\theta(1-2U_{t-1})U_{t}(1-U_{t})$;

(c) $Y_{t}=G^{-1}(U_{t})$, where $G(t)$ is the common marginal distribution of
the variables of the stationary Markov chain.

Longla and al. (2021) worked on perturbation of copulas and their
perturbations. For a copula $C(u,v)$, some of the studied perturbations are as
follows. Assume $\alpha,\theta\in[o,1]$.

$\displaystyle\tilde{C}_{\alpha}(u,v)$ $\displaystyle=$ $\displaystyle
C(u,v)+\alpha\left(\Pi(u,v)-C(u,v)\right),$ (3.1)
$\displaystyle\hat{C}_{\alpha}(u,v)$ $\displaystyle=$ $\displaystyle
C(u,v)+\alpha\left(\text{M}(u,v)-C(u,v)\right).$ (3.2)

Formulas (3.1) and (3.2) lead to the following.

###### Proposition 3.0.1

Let $\theta\in[0,1]$, $\alpha\in[-1,1]$ and $C_{\theta}(u,v)$ be a Farlie-
Gumbel-Morgenstern copula.

$\displaystyle\tilde{C}_{\alpha,\theta}(u,v)$ $\displaystyle=$ $\displaystyle
C_{\theta}(u,v)+\alpha\left(\Pi(u,v)-C_{\theta}(u,v)\right);$ (3.3)
$\displaystyle\hat{C}_{\alpha,\theta}(u,v)$ $\displaystyle=$ $\displaystyle
C_{\theta}(u,v)+\alpha\left(\text{M}(u,v)-C_{\theta}(u,v)\right).$ (3.4)

  1. 1.

$\tilde{C}_{\alpha,\theta}(u,v)=C_{\theta(1-\alpha)}(u,v)$ \- is a member of
the Farlie-Gumbel-Morgenstern family of copulas and generates $\psi$-mixing
Markov chains.

  2. 2.

$\hat{C}_{\alpha,\theta}(u,v)$ is not a member of the Farlie-Gumbel-
Morgenstern family of copulas and does not generates $\psi-mixing$ Markov
chains.

On Fifure 1 we have a 3-dimensional graph of the Farlie-Gumbel-Morgenstern
copula with parameter $\theta=.6$ and its level curves on the left and the
corresponding graphs for the perturbation with parameter $\alpha=.4$ on the
right. Figure 2 represents a simulated Markov chain form the Farlie-Gumbel-
Morgenstern copula with $\theta=.4$ and the one generated by its perturbation
with parameter $\alpha=.7$. Here, the marginal distribution of the Markov
chain is standard normal. We can see on the graphs that the mixing structure
is not the same when the copula is perturbed by $M(u,v)$. This supports the
theoretical results.

Figure 1: Farlie-Gumbel-Morgenstern copula and level curves Figure 2: Data
from the Farlie-Gumbel-Morgenstern copula and its perturbations.

The Mardia family of copulas is defined by

$C_{a,b}(u,v)=aM(u,v)+bW(u,v)+(1-a-b)\Pi(u,v)$ (3.5)

and the Frechet copulas are a subfamily with
$a=\dfrac{\theta^{2}(1+\theta)}{2}$, $b=\dfrac{\theta^{2}(1-\theta)}{2}$ and
$|\theta|\leq 1$. Unlike Farlie-Gumbel-Morgenstern copulas, these copulas are
not absolutely continuous. To generate an observation $(U,V)$ from
$C_{\theta}(u,v)$, one needs to generate independent observations
$(U,V_{1},V_{2})$ from the uniform distribution on $(0,1)$. Then, do the
following:

$V=\left\\{\begin{array}[]{lcl}V_{2}&\text{if}&V_{1}<1-\theta^{2},\\\
U&\text{if}&1-\theta^{2}<V_{1}<1-\theta^{2}+\theta^{2}(1+\theta)/2,\\\
1-U&\text{if}&V_{1}>1-\theta^{2}+\theta^{2}(1+\theta)/2.\end{array}\right.$

Figure 3: Frechet copula represenation and level curves

Figure 3 gives a representation of the Frechet copula for $\theta=.6$ and its
perturbation with $\alpha=.4$, together with level curves. Figure 4 represents
a Markov chain with 500 observations simulated from the Frechet copula with
$\theta=.6$ and its perturbation with parameter $\alpha=.7$.

Figure 4: Markov chain generated by Frechet copulas and its perturbations.

Perturbations of the Frechet copula will have the form:

$\displaystyle\tilde{C}_{\theta,\alpha}(u,v)=C_{\theta}(u,v)+\alpha(\Pi(u,v)-C_{\theta}(u,v));$
(3.6)
$\displaystyle\hat{C}_{\theta,\alpha}(u,v)=C_{\theta}(u,v)+\alpha(M(u,v)-C_{\theta}(u,v)).$
(3.7)

It is good to notice that these perturbations are not Frechet copulas, but
remain in the class of Mardia copulas. Figure 4 represents a Markov chain
generated by a Frechet copula and the ones generated by its perturbations
using the standard normal distribution for marginal distributions.

##  4 Simulation study

This simulation study shows the importance of this topic. We simulate a
dependent data set that exhibits $\psi$-mixing or $\psi^{\prime}$-mixing and
show how the mixing structure influences the statistical study. Based on the
fact that the considered mixing coefficient converges exponentially to $0$, we
can bound the variance o partial sums and obtain the condition of the central
limit theorem and confidence interval of Longla and Peligrad (2020). Thanks to
this central limit theorem, we construct confidence intervals without having
to estimate the limiting variance of the central limit theorem of Kipnis and
Varadhan (1986) that holds here because the Markov chains are reversible and
$nvar(\bar{Y})\to\sigma<\infty$. The standard central limit theorem is useless
in this case because the limiting variance is not necessarily that of $Y$. Let
us recall here the formulations of Longla and Peligrad (2020). They have
proposed a new robust confidence interval for the mean based on a sample of
dependent observations with a mild condition on the variance of partial sums.
This confidence interval needs a random sample $(X_{i},1\leq i\leq n)$,
generated independently of $(Y_{i},1\leq i\leq n)$ and following the standard
normal distribution, the Gaussian Kernel and the optimal bandwidths

$h_{n}=\left[\dfrac{\bar{y^{2}_{n}}}{n\sqrt{2}\bar{y}^{2}_{n}}\right]^{1/5}.$

Let us check the conditions required for use of this proposed estimator of the
mean and the confidence interval. They are as follows:

  1. 1.

$(Y_{i})_{i\in\mathbb{Z}}$ is an ergodic sequence;

  2. 2.

$(Y_{i})_{i\in\mathbb{Z}}$ have finite second moments;

  3. 3.

$nh_{n}var(\bar{Y}_{n})\rightarrow 0$ as $n\rightarrow\infty$.

For the sake of clarity, we will use $C^{FGM}_{\theta}(u,v)$ to denote the
Farlie-Gumbel-Morgenstern copula with parameter $\theta$.

Verification of the conditions

  1. 1.

Ergodicity

    1. (a)

It has been shown by Theorem 2.3 and Example 2.4 of Longla (2014) that the
copula $C_{\theta}^{FGM}(u,v)$ generates geometrically ergodic Markov chains.

    2. (b)

From this current work, we can deduce that the perturbed
$\hat{C}^{FGM}_{\theta,\alpha}(u,v)$ generates $\psi^{\prime}-mixing$ Markov
chains. In fact, this copula is a linear combination of two copulas such that
one is $\psi^{\prime}-mixing$. In addition, (see Bradley (2005) and Longla and
Peligrad (2012)) $\psi^{\prime}-mixing$ implies $\phi-mixing$ and $\phi-
mixing$ implies geometric ergodicity for reversible Markov chains. So the
Markov chain generated by $\hat{C}^{FGM}_{\theta,\alpha}(u,v)$ is
geometrically ergodic.

    3. (c)

According to Theorem 2.16 and Remark 2.17 of Longla (2014), the Frechet copula
$C_{\theta}(u,v)$ generates geometrically ergodic Markov chains.

    4. (d)

The perturbed Frechet copula $\hat{C}_{(\theta_{1},\theta_{2},\alpha)}(u,v)$
is a linear combination of copulas $C_{\theta_{1}}(u,v)$ and
$C^{FGM}_{\theta_{2}}(u,v)$. These two copulas are symmetric and each one
generates geometrically ergodic sequences as said above. Then, according to
Theorem 5 of Longla and Peligrad (2012), this copula generates geometrically
ergodic Markov chains.

  2. 2.

The marginal distribution used in this work is normal with mean 30 and
variance 1. Therefore, it has second moments.

  3. 3.

The condition on the variance ($nh_{n}var(\bar{Y})\to 0$) is checked in the
appropriate section below.

For data simulation, we set $Y_{i}\sim N(30,1)$ for all copulas and the
perturbation parameter $\alpha=0.4$ in all cases. For Farlie-Gumbel-
Morgenstern and Frechet copulas we set $\theta=0.6$. For the Frechet perturbed
copula, $\theta_{1}=\theta_{2}=0.6$. For $1\leq i\leq n$, $X_{i}\sim N(0,1)$
is a sequence of independent random variables that is independent of the
Markov chain $(Y_{i},1\leq i\leq n)$.

According to above considerations, the estimator of $\mu_{Y}$ is
$\tilde{r}_{n}=\dfrac{1}{nh_{n}}\sum\limits_{i=1}^{n}Y_{i}\exp\left(-0.5(\dfrac{X_{i}}{h_{n}})^{2}\right)$
and the confidence interval is
$\left(\tilde{r}_{n}\sqrt{1+h_{n}^{2}}-z_{\alpha/2}\left(\dfrac{\bar{Y_{n}^{2}}}{nh_{n}\sqrt{2}}\right)^{1/2};\tilde{r}_{n}\sqrt{1+h_{n}^{2}}+z_{\alpha/2}\left(\dfrac{\bar{Y_{n}^{2}}}{nh_{n}\sqrt{2}}\right)^{1/2}\right)$.

The following table is the result of the simulation study for the Markov
chains generated by the two considered copulas and their perturbations.

Copula | size | n=100 | n=5000 | n=10000 | n=20000
---|---|---|---|---|---
$C^{FGM}_{\theta}$ |  Estimator of  $\mu_{Y}$ | 23.25 | 28.41 | 29.85 | 29.54
Confidence interval | (16.72, 32.88) | (27.12, 30.51) | (28.89, 31.46) | (28.81, 30.76)
$\hat{C}^{FGM}_{\theta,\alpha}$ |  Estimator of  $\mu_{Y}$ | 23.70 | 28.39 | 29.80 | 29.52
Confidence interval | (17.08, 33.50) | (27.10, 30.50) | (28.84, 31.42) | (28.79, 30.74)
$C_{\theta}$ |  Estimator of  $\mu_{Y}$ | 31 | 29.40 | 30.30 | 30.29
Confidence interval | (24.97, 41,16) | (28.13, 31.52) | (29.34, 31.91) | (29.56, 31.51)
$\hat{C}_{(\theta_{1},\theta_{2},\alpha)}$ |  Estimator of  $\mu_{Y}$ | 31.15 | 29.39 | 30.29 | 30.23
Confidence interval | (25.08, 41.37) | (28.11, 31.51) | (29.33, 31.90) | (29.51, 31.46)

##  5 Conclusion and remarks

The graphs and simulations presented in this paper have been obtained using
$R$. We have provided some insights on $\psi^{*}$-mixing,
$\psi^{\prime}$-mixing and $\psi$-mixing. Though we have presented extensive
examples and results for $\psi^{\prime}$-mixing and $\psi^{*}$-mxing, we have
not been able to answer the question on convex combinations of $\psi$-mixing.
The following question remains open: Does a convex combination of
$\psi$-mixing generating copulas generate $\psi$-mixing? A positive answer to
this question has been presented for the case when each of the copulas
satistfy $\psi_{1}<1$. It would also be interesting to find a general
condition on the copula for $\psi$-mixing like the one presented for
$\psi^{*}$-mixing.

##  6 Appendix of proofs

###  6.1 Proof of Theorem 2.1.2

Recall that the function $c(x,y)$ defined on $I^{2}$ is said to be bounded
away from zero on a set of Lebesgue measure 1 iff $\exists
m>0,m\in\mathbb{R},\exists Q\subset I^{2}:\lambda(Q)=1,\forall(x,y)\in Q$,
$c(x,y)\geq m.$

According to Theorem 2.1.1 a strictly stationary Markov chain
$(X_{k},~{}k\in\mathbb{N})$ is $\psi^{\prime}$-mixing if : $\text{ for some
}n\in\mathbb{N},~{}\psi_{n}^{\prime}(C)\neq 0.$ Let $A\subset I,~{}B\subset
I$, by easy calculation we obtain:

$\displaystyle\dfrac{\int_{A}\int_{B}c_{1}(x,y)dxdy}{\lambda(A)\lambda(B)}$
$\displaystyle=$
$\displaystyle\dfrac{\int_{A}\int_{B}c(x,y)dxdy}{\lambda(A)\lambda(B)}$ (6.1)
$\displaystyle\geq$
$\displaystyle\dfrac{\int_{A}\int_{B}(\varepsilon_{1}(x)+\varepsilon_{2}(y))dxdy}{\lambda(A)\lambda(B)}$
$\displaystyle=$
$\displaystyle\dfrac{\int_{A}\int_{B}\varepsilon_{1}(x)dxdy}{\lambda(A)\lambda(B)}+\dfrac{\int_{A}\int_{B}\varepsilon_{2}(y)dxdy}{\lambda(A)\lambda(B)}$
$\displaystyle=$
$\displaystyle\dfrac{\int_{A}\varepsilon_{1}(x)dx\int_{B}dy}{\lambda(A)\lambda(B)}+\dfrac{\int_{B}\varepsilon_{2}(y)dy\int_{A}dx}{\lambda(A)\lambda(B)}$
$\displaystyle=$
$\displaystyle\dfrac{\lambda(B)\int_{A}\varepsilon_{1}(x)dx}{\lambda(A)\lambda(B)}+\dfrac{\lambda(A)\int_{B}\varepsilon_{2}(y)dy}{\lambda(A)\lambda(B)}$

So for all $A\subset I,~{}B\subset I$, the following inequality holds:

$\dfrac{\int_{A}\int_{B}c_{1}(x,y)dxdy}{\lambda(A)\lambda(B)}\geq\dfrac{\int_{A}\varepsilon_{1}(x)dx}{\lambda(A)}+\dfrac{\int_{B}\varepsilon_{2}(y)dy}{\lambda(B)}$
(6.2)

We also have:

$\dfrac{\int_{A}\varepsilon_{1}(x)dx}{\lambda(A)}\geq\underset{\underset{\lambda(A)>0}{A\subset
I}}{\inf}\dfrac{\int_{A}\varepsilon_{1}(x)dx}{\lambda(A)}\text{ and
}\dfrac{\int_{B}\varepsilon_{2}(y)dy}{\lambda(B)}\geq\underset{\underset{\lambda(B)>0}{A\subset
I}}{\inf}\dfrac{\int_{B}\varepsilon_{2}(y)dy}{\lambda(B)}.$

Thus, for all $A\subset I,~{}B\subset I$ :

$\dfrac{\int_{A}\int_{B}c_{1}(x,y)dxdy}{\lambda(A)\lambda(B)}\geq\underset{\underset{\lambda(A)>0}{A\subset
I}}{\inf}\dfrac{\int_{A}\varepsilon_{1}d\lambda}{\lambda(A)}+\underset{\underset{\lambda(B)>0}{A\subset
I}}{\inf}\dfrac{\int_{B}\varepsilon_{2}d\lambda}{\lambda(B)}.$

Which means:

$\underset{\underset{\lambda(A)\lambda(B)>0}{A\subset I,~{}B\subset
I}}{\inf}\dfrac{\int_{A}\int_{B}c_{1}(x,y)dxdy}{\lambda(A)\lambda(B)}\geq
M+N.$

Where $M=\underset{\underset{\lambda(A)>0}{A\subset
I}}{\inf}\dfrac{\int_{A}\varepsilon_{1}d\lambda}{\lambda(A)}$ and
$N=\underset{\underset{\lambda(B)>0}{A\subset
I}}{\inf}\dfrac{\int_{B}\varepsilon_{2}d\lambda}{\lambda(B)}$. Hence,
$\psi^{\prime}_{1}(C)\geq M+N.$ According to the theorem assumptions, $M>0$ or
$N>0$. So, $\psi^{\prime}_{1}(C)\geq M+N>0.$ We can conclude
$(X_{k},k\in\mathbb{N})$ is $\psi^{\prime}$-mixing.

###  6.2 Theorem 2.1.5 and Theorem 2.1.6

To prove these theorems, we will use the following proposition from Longla and
al (2022)

###### Proposition 6.2.1

For a convex combination of copulas $\displaystyle
C(x,y)=\sum_{i=1}^{k}a_{i}C_{i}(x,y),$ where $0<a_{1},...,a_{k}<1$ and
$\displaystyle\sum_{i=1}^{k}a_{i}=1$, the following formula holds. For any
$s\in\mathbb{N}$

$C^{s}(x,y)=\sum_{j=1}^{k^{s}}b_{j}\times~{}_{1}C_{j}\ast...\ast~{}_{s}C_{j}(x,y),$
(6.3)

where $\sum_{j=1}^{k^{s}}b_{j}=1,~{}0<b_{1},...,b_{k_{s}}<1$, and each of the
copulas $~{}_{i}C_{j}(x,y)=C_{j_{i}}(x,y)$ for some $j_{i}\in\\{1,...,k\\}$
and the sum is over all possible products of $s$ copulas selected from the
original $k$ copulas with replacement.

The notation $~{}_{i}C_{j}$ indicates that the copula $C_{j_{i}}$ was selected
in the given $j^{th}$ element of $B=\\{C_{1},...,C_{k}\\}^{n}$.

(1)  Suppose that there exists a subset of copulas
$C_{k_{1}},...,C_{k_{s}}~{},s\leq k\in\mathbb{N}$ such that
$\psi^{\prime}(\hat{C})>0$ for $\hat{C}=C_{k_{1}}\ast...\ast C_{k_{s}}$.
Equation (6.3) can be written as follows:

$C^{s}(x,y)=b_{i}\hat{C}(x,y)+\sum_{\underset{j\neq
i}{j=1}}^{k^{s}}b_{j}\hat{C}_{j}(x,y),~{}~{}~{}\text{where}$ (6.4)

$\hat{C}(x,y)=C_{i_{1}}\ast...\ast C_{i_{s}}(x,y)$ and
$\hat{C}_{j}(x,y)=C_{j_{1}}\ast...\ast C_{j_{s}}(x,y)$

Let $(X_{k},k\in\mathbb{N})$ be a copula-based Markov chain generated by the
copula $C(x,y)$; $(\hat{X}^{j}_{k},k\in\mathbb{N})$ a Markov chain generated
by copula $\hat{C}_{j}$ for $1\leq j\leq k^{s}$, $\hat{C}_{i}=\hat{C}$. For
$A\in\sigma(X_{0})$ and $B\in\sigma(X_{s})$, equation (6.4) yields

$\displaystyle P^{s}(A\cap B)$ $\displaystyle=$ $\displaystyle
b_{i}\hat{P}(A\cap B)+\sum_{\underset{j\neq
i}{j=1}}^{k^{s}}b_{j}\hat{P}_{j}(A\cap B)\geq b_{i}\hat{P}(A\cap B),$ (6.5)

where $P^{s}(A\cap B)=P(X_{1}\in A,X_{s+1}\in B)$; $\hat{P}_{j}(A\cap
B)=P(X^{j}_{1}\in A,X^{j}_{s+1}\in B)$ and $\hat{P}(A\cap B)=P(X^{i}_{1}\in
A,X^{i}_{s+1}\in B)$. Thus,

$\psi^{\prime}_{s}(C)=\underset{A\subset I,B\subset
I,P(A)(B)>0~{}}{\inf}\dfrac{P^{s}(A\cap B)}{P(A)P(B)}\geq
b_{i}\psi^{\prime}(\hat{C})$.

By our assumptions, $\psi^{\prime}(\hat{C})>0$. The conclusion follows from
Theorem 2.1.1.

(2) Suppose there exists a subset of copulas $C_{k_{1}},...,C_{k_{s}}~{},s\leq
k\in\mathbb{N}$ such that the density of the absolutely continuous part of the
copula $\hat{C}=C_{k_{1}}\ast...\ast C_{k_{s}}$ is bounded away from zero.
From equation (6.4) we have:

$c^{s}(x,y)\geq b_{i}\hat{c}(x,y).$ (6.6)

Moreover, the density of the absolutely continuous part of $\hat{C}(u,v)$ is
bounded away from zero. Thus, there exists $c>0$: $\forall(x,y)\in[0,1]^{2}$,
$\hat{c}(x,y)\geq c$ almost surely. Hence, from (6.6), we have $c^{s}(x,y)\geq
b_{i}c$. Now, if $(X_{k},k\in\mathbb{N})$ is a copula-based Markov chain
generated by the copula $C(x,y)$ and an absolutely continuous distribution,
then for $A\in\sigma(X_{1})$ and $B\in\sigma(X_{s+1})$, we have

$\displaystyle P^{s}(A\cap B)\geq b_{i}cP(A)\times P(B)\quad\text{ and
}\quad\dfrac{P^{s}(A\cap B)}{P(A)\times P(B)}\geq b_{i}c,$ (6.7)

where $P^{s}(A\cap B)=P(X_{1}\in A,X_{s+1}\in B)$. It follows from equation
(6.7) that

$\displaystyle\psi^{\prime}_{s}(C)=\underset{P(A)(B)>0~{}}{\inf}\dfrac{P^{s}(A\cap
B)}{P(A)P(B)}\geq b_{i}c>0.$

This concludes the proof of Theorem 2.1.6.

###  6.3 Proof of Theorem 2.5

The following decomposition if true for Farle-Gumbel-Morgenstern copulas with
$\lambda=1-\theta$:

$C_{\theta}(u,v)=(1-\lambda)(uv+uv(1-u)(1-v))+\lambda uv.$

Given that $C(u,v)=(uv+uv(1-u)(1-v))$ is a copula, we can apply Theorem 2.1.3
to obtain

$C^{n}_{\theta}(u,v)=(1-\lambda)^{n}C^{n}(u,v)+(1-(1-\lambda)^{n})uv.$

It remains to show that
$C^{n}(u,v)=uv+3\large(\frac{1}{3}\large)^{n}uv(1-u)(1-v)$ by mathematical
induction. It is clear that the formula is correct for $n=1$. Assume that for
$n=k$, we have

$C^{k}(u,v)=uv+3\large(\frac{1}{3}\large)^{k}uv(1-u)(1-v).$

Using the fold product, we obtain

$C^{k+1}=C^{k}*C(u,v)=\int_{0}^{1}C^{k}_{,2}(u,t)C_{,1}(t,v)dt.$
$C^{k}_{,2}(u,t)=u+3\large(\frac{1}{3}\large)^{k}u(1-u)(1-2t)\quad\text{and}\quad
C_{,1}(t,v)=v+v(1-v)(1-2t).$

Plugging these functions into the integral and computing yields the needed
results. The proof ends by replacing $C^{n}(u,v)$ by its value and using
$\lambda=1-\theta$.

###  6.4 Proof of Theorem 2.2.5

Assume that the copula $C(u,v)$ is such that for all $(u,v)\in[0,1]^{2}$,
$c^{n}(u,v)\leq K$, where $K$ is a constant and $c^{n}$ is the density of the
copula $C^{n}(u,v)$. Let $A\in\sigma(X_{0})$ and $B\in\sigma(X_{n})$, where
$(X_{0},X_{n})$ has copula $C^{n}(u,v)$. Assume that the stationary
distribution of the Markov chain has distribution $F(x)$. Using Sklar’s
Theorem (see Sklar (1959)), we have

$P^{n}(A\cap B)=P(X_{0}\in A,X_{n}\in
B)=\int_{A}\int_{B}c^{n}(F(x),F(y))dF(y)dF(x).$

Therefore, $P^{n}(A\cap B)\leq KP(A)P(B)$. This implies $\psi_{1}(C)\leq
K\neq\infty$ and by Lemma 2.2.1, $C(u,v)$ generates stationary $\psi$-mixing
Markov chains. Now, if we assume that There exists a set of non-zero measure
$\Omega\subset[0,1]^{2}$ such that $A\times B\subset\Omega$,
$A\in\sigma(X_{0})$, $B\in\sigma(X_{1})$ and the density of $C^{n}$ is not
bounded above on $\Omega$, but bounded below by a any given non-zero real
number $M$. This construction is possible due to continuity of the density of
$C(u,v)$. It follows that for any constant $M$,

$P^{n}(\Omega)\geq P^{n}(A\times B)\geq M\int_{A\times B}d\Pi(x,y)=MP(A)P(B).$

It is obvious here that As $M$ grows, the size of $P(A)P(B)$ reduces as their
product has to be at most $1$. From here, we obtain

$\frac{P(A\cap B)}{P(A)P(B)}\geq M.\quad\text{This leads to
}\quad\psi^{*}_{n}(C)>M.$

Because this is true for every $M$ and every $n$, we can conclude that
$\psi^{*}_{n}(C)=\infty$ and $\psi_{n}(C)=\infty$ for all $n$. Thus, the
generated Markov chain is not $\psi^{*}$-mixing and not $\psi$-mixing.

###  6.5 Proof of Theorem 2.2.11 and Theorem 2.2.12

Without loss of generality the proof can be done for a convex combination of
two copulas, one of which is $C(u,v)$ and doesn’t generate $\psi^{*}$-mixing
Markov chains. This is true because any convex combination of copulas can be
written as a convex combination of two copulas. Now, assume that

$C_{2}(u,v)=\alpha C(u,v)+(1-\alpha)C_{1}(u,v).$

By Lemma 2.2.1 , $\psi^{*}_{n}(C)=\infty$ for all $n\in\mathbb{N}$. We need to
show that $\psi^{*}_{n}(C_{2})=\infty$ for all $n\in\mathbb{N}$. By Longla and
al (2021), there exist $b_{in},C_{1in}(u,v)$, such that $b_{in}>0$,
$\alpha^{n}+\sum_{i=2}^{2^{n}}b_{in}=1$ and

$C_{2}^{n}(u,v)=\sum_{i=2}^{2^{n}}b_{in}C_{1in}(u,v)+\alpha^{n}C^{n}(u,v).$

Therefore, The probability distribution $P^{n}_{2}$ of $(X_{1},X_{n+1})$ from
the Markov chain generated by $C_{2}(u,v)$ and the probability distributions
$P^{n}_{1i}$ of $(\tilde{X}_{i1},\tilde{X}_{in+1})$ for the Markov chains
generated by the copulas $C_{1in}(u,v)$ satisfy the following relationship for
every $A\in\sigma(X_{0})$ and $B\in\sigma(X_{n+1})$:

$P_{2}^{n}(A\cap B)=\sum_{i=2}^{2^{n}}b_{in}P^{n}_{1i}(A\cap
B)+\alpha^{n}P^{n}(A\cap B).$

Therefore, $P_{2}^{n}(A\cap B)\geq\alpha^{n}P^{n}(A\cap B)$. Given that
$\psi^{*}_{n}(C)=\infty$ for all $n$, it follows that
$\sup_{A,B}\frac{P^{n}(A\cap B)}{P(A)P(B)}=\infty$, leading to

$\sup_{A,B}\frac{P^{n}_{2}(A\cap
B)}{P(A)P(B)}=\infty\quad\text{and}\quad\psi^{*}_{n}(C_{2})=\infty\quad\text{for
all}\quad n\in\mathbb{N}.$

This concludes the proof of Theorem 2.2.11. Now, to prove Theorem 2.2.12, as
for the previous case, it is enough to consider a convex combination of two
copulas. Assume that $C_{1}(u,v)$ and $C_{2}(u,v)$ generate each
$\psi^{*}$-mixing (or $\psi$-mixing) stationary copula-based Markov chains
qith $\psi^{*}_{1}<1$ (or $\psi_{1}<1$) respectively. Define $C(u,v)=\alpha
C_{1}(u,v)+(1-\alpha)C_{2}(u,v)$. Once more, we will use Lemma 2.2.1.
$\psi_{1}(C)\leq\alpha\psi_{1}(C_{1})+(1-\alpha)\psi_{1}(C_{2})<(\alpha)+(1-\alpha)=1$.
The same argument works for $\psi_{1}^{*}(C)$.

###  6.6 Checking the condition $nh_{n}var(\bar{Y})\to 0$.

Given the the Markov chains that we consider here are reversible and ergodic
(see Haggstrom and Rosenthal (2007), Kipnis and Varadhan (1986)),

$nvar(\bar{Y})\quad\text{behaves as}\quad
var(Y_{0})+2\sum_{k=1}^{\infty}cov(Y_{0},Y_{k}).$

Moreover, if the series converges, then the central limit theorem holds with
variance equal to its sum. On the other side, Markov chains generated by
Farlie-Gumbel-Morgenstern copulas, Frechet copulas and their considered
perturbations are exponential $\psi^{\prime}$-mixing. This implies that they
are all exponential $\rho$-mixing. Exponential $\rho$-mixing implies
convergence of the considered series. Therefore $nvar(\bar{Y})\to C$, leading
to $nh_{n}var(\bar{Y})$.

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