Innovations in Quantitative Risk Management pp 393-409 | Cite as

# Some Consequences of the Markov Kernel Perspective of Copulas

## Abstract

The objective of this paper is twofold: After recalling the one-to-one correspondence between two-dimensional copulas and Markov kernels having the Lebesgue measure \(\lambda \) on \([0,1]\) as fixed point, we first give a quick survey over some consequences of this interrelation. In particular, we sketch how Markov kernels can be used for the construction of strong metrics that strictly distinguish extreme kinds of statistical dependence, and show how the translation of various well-known copula-related concepts to the Markov kernel setting opens the door to some surprising mathematical aspects of copulas. Secondly, we concentrate on the fact that iterates of the star product of a copula \(A\) with itself are Cesáro convergent to an idempotent copula \(\hat{A}\) with respect to any of the strong metrics mentioned before and prove that \(\hat{A}\) must have a very simple form if the Markov operator \(T_A\) associated with \(A\) is quasi-constrictive in the sense of Lasota.

## Keywords

Copula Doubly stochastic measure Markov kernel Markov operator## 1 Introduction

## 2 Notation and Preliminaries

*copulas*, \(d_\infty \) will denote the uniform metric on \(\fancyscript{C}\). For properties of copulas, we refer to [8, 22, 26]. For every \(A\in \fancyscript{C}\), \(\mu _A\) will denote the corresponding

*doubly stochastic measure*, \(\fancyscript{P}_\fancyscript{C}\), the class of all these doubly stochastic measures. Since copulas are the restriction of two-dimensional distribution functions with \(\fancyscript{U}(0,1)\)-marginals to \([0,1]^2\), the Lebesgue decomposition of every element in \(\fancyscript{P}_\fancyscript{C}\) has no discrete component. The Lebesgue measure on \([0,1]\) and \([0,1]^2\) will be denoted by \(\lambda \) and \(\lambda _2\), respectively. For every metric space \((\varOmega ,d)\), the Borel \(\sigma \)-field on \(\varOmega \) will be denoted by \(\fancyscript{B}(\varOmega )\). A

*Markov kernel*from \(\mathbb {R}\) to \(\fancyscript{B}(\mathbb {R})\) is a mapping \(K: \mathbb {R} \times \fancyscript{B}(\mathbb {R})\rightarrow [0,1]\) such that \(x \mapsto K(x,B)\) is measurable for every fixed \(B \in \fancyscript{B}(\mathbb {R})\) and \(B \mapsto K(x,B)\) is a probability measure for every fixed \(x \in \mathbb {R}\). Suppose that \(X,Y\) are real-valued random variables on a probability space \((\varOmega , \fancyscript{A}, \fancyscript{P})\), then a Markov kernel \(K:\mathbb {R}\times \fancyscript{B}(\mathbb {R}) \rightarrow [0,1]\) is called

*regular conditional distribution of*\(Y\)

*given*\(X\) if for every \(B \in \fancyscript{B}(\mathbb {R})\)

*regular conditional distribution of*\(A\) or as

*the Markov kernel of*\(A\). Note that for every \(A\in \fancyscript{C}\), its conditional regular distribution \(K_A(\cdot ,\cdot )\), and every Borel set \(G \in \fancyscript{B}([0,1]^2)\) we have

\(\fancyscript{T}\) will denote the family of all \(\lambda \)-preserving transformations \(h:[0,1]\rightarrow [0,1]\) (see [34]), \(\fancyscript{T}_p\) the subset of all bijective \(h \in \fancyscript{T}\). A copula \(A\in \fancyscript{C}\) will be called *completely dependent* if and only if there exists \(h \in \fancyscript{T}\) such that \(K(x,E):=\mathbf {1}_E(hx)\) is a regular conditional distribution of \(A\) (see [17, 29] for equivalent definitions and main properties). For every \(h \in \fancyscript{T}\), the corresponding completely dependent copula will be denoted by \(C_h\), the class of all completely dependent copulas by \(\fancyscript{C}_d\).

*Markov operator*([3, 23] if it fulfills the following three properties:

- 1.
\(T\) is positive, i.e., \(T(f) \ge 0\) whenever \(f\ge 0 \)

- 2.
\(T(\mathbf {1}_{[0,1]})=\mathbf {1}_{[0,1]}\)

- 3.
\(\int _{[0,1]} (Tf)(x) d\lambda (x) =\int _{[0,1]} f(x) d\lambda (x)\)

*there is a one-to-one correspondence between*\(\fancyscript{C}\)

*and*\(\fancyscript{M}\)—in fact, the mappings \(\varPhi : \fancyscript{C}\rightarrow \fancyscript{M}\) and \(\varPsi : \fancyscript{M}\rightarrow \fancyscript{C}\), defined by

## 3 Some Consequences of the Markov Kernel Approach

In this section, we give a quick survey showing the usefulness of the Markov kernel perspective of two-dimensional copulas.

### 3.1 Strong Metrics on \(\fancyscript{C}\)

### **Theorem 1**

- (a)
\(\lim _{n \rightarrow \infty } D_1(A_n,A)=0\)

- (b)
\(\lim _{n \rightarrow \infty } D_\infty (A_n,A)=0\)

- (c)
\(\lim _{n \rightarrow \infty } \Vert T_nf - Tf \Vert _1 =0\) for every \(f \in L^1([0,1])\)

- (d)
\(\lim _{n \rightarrow \infty } D_2(A_n,A)=0\)

### **Theorem 2**

([29]) The metric space \((\fancyscript{C},D_1)\) is complete and separable. The same holds for \((\fancyscript{C},D_2)\) and \((\fancyscript{C},D_\infty )\). The topology induced on \(\fancyscript{C}\) by \(D_1\) is strictly finer than the one induced by \(d_\infty \).

### *Remark 3*

### 3.2 Induced Dependence Measures

The main motivation for the consideration of conditioning-based metrics like \(D_1\) was the need for a metric that, contrary to \(d_\infty \), is capable of distinguishing extreme types of statistical dependence, i.e., independence and complete dependence. For the uniform metric \(d_\infty \), it is straightforward to construct sequences \((C_{h_n})_{n \in \mathbb {N}}\) of completely dependent copulas (in fact, even sequences of shuffles of \(M\), see [9, 22]) fulfilling \(\lim _{n \rightarrow \infty } d_\infty (C_{h_n},\varPi )=0\)—for \(D_1\), however, the following result holds:

### **Theorem 4**

([29]) For every \(A \in \fancyscript{C}\) we have \(D_1(A,\varPi ) \le 1/3\). Furthermore, equality \(D_1(A,\varPi ) = 1/3\) holds if and only if \(A \in \fancyscript{C}_d\).

### *Example 5*

### *Example 6*

### *Remark 7*

The dependence measure \(\tau _1\) is nonmutual, i.e., we do not necessarily have \(\tau _1(A)=\tau _1(A^t)\), whereby \(A^t\) denotes the transpose of \(A\) (i.e., \(A^t(x,y)=A(y,x)\)). This reflects the fact that the dependence structure of random variables might be strongly asymmetric, see [29] for examples as well as [27] for a measure of mutual dependence.

### *Remark 8*

Since most properties of \(D_1\) in dimension two also hold in the general \(m\)-dimensional setting it might seem natural to simply consider \(\tau _1(A) := aD_1(A, \varPi )\) as dependence measure on \( \fancyscript{C}^m\) (a being a normalizing constant). It is, however, straightforward to see that this yields no reasonable notion of a dependence quantification in so far that we would also have \(\tau _1(A) > 0\) for copulas \(A\) describing independence of \(X\) and \(\mathbf {Y} = (Y_1,\ldots ,Y_{m-1})\). For a possible way to overcome this problem and assign copulas describing the situation in which each component of a portfolio \((Y_1,\ldots ,Y_{m-1})\) is a deterministic function of another asset \(X\) maximum dependence we refer to [11].

### *Remark 9*

It is straightforward to verify that for samples \((X_1,Y_1),\ldots ,(X_n,Y_n)\) from \(A \in \fancyscript{C}\) the empirical copula \(\hat{E}_n\) (see [22, 28]) cannot converge to \(A\) w.r.t. \(D_1\) unless we have \(A \in \fancyscript{C}_d\). Using Bernstein or checkerboard aggregations (smoothing the empirical copula) might make it possible to construct \(D_1\)-consistent estimators of \(\tau _1(A)\). Convergence rates of these aggregations and other related questions are future work.

### 3.3 The IFS Construction of (Very) Singular Copulas

Using Iterated Function Systems, one can construct copulas exhibiting surprisingly irregular analytic behavior. The aim of this section is to sketch the construction and then state two main results. For general background on Iterated Function Systems with Probabilities (IFSP, for short), we refer to [16]. The IFSP construction of two-dimensional copulas with fractal support goes back to [12] (also see [2]), for the generalization to the multivariate setting we refer to [30].

### **Definition 10**

([12]) A \(n\times m\)-matrix \(\tau =(t_{ij})_{i=1,\ldots , n, \,j=1,\ldots , m }\) is called *transformation matrix* if it fulfills the following four conditions: (i) \(\max (n,m)\ge 2\), (ii) all entries are non-negative, (iii) \(\sum _{i,j} t_{ij}=1\), and (iv) no row or column has all entries \(0\). \(\mathfrak {T}\) will denote the family of all transformations matrices.

*invariant copula*, such that \(\fancyscript{V}_\tau (\mu _{A_\tau ^\star })=\mu _{A_\tau ^\star }\) holds. The IFSP construction also converges w.r.t. \(D_1\)—the following result holds:

### **Theorem 11**

([29]) Let \(\tau \in \mathfrak {T}\) be a transformation matrix. Then \(V_\tau \) is a contraction on the metric space \((\fancyscript{C},D_1)\) and there exists a unique copula \(A_\tau ^\star \) such that \(V_\tau A_\tau ^\star = A_\tau ^\star \) and for every \(B \in \fancyscript{C}\) we have \(\lim _{n \rightarrow \infty } D_1(V_\tau ^nB,A_\tau ^\star )=0\).

### *Example 12*

Moreover (again see [12]) the support \(Supp(\mu _{A_\tau ^\star })\) of \(\mu _{A_\tau ^\star }\) fulfills \(\lambda _2(Supp(\mu _{A_\tau ^\star }))=0\) if \(\tau \) contains at least one zero. Hence, in this case, \(\mu _{A_\tau ^\star }\) is singular w.r.t. the Lebesgue measure \(\lambda _2\), we write \(\mu _{A_\tau ^\star } \perp \lambda _2\). On the other hand, if \(\tau \) contains no zeros we may still have \(\mu _{A_\tau ^\star } \perp \lambda _2\) although in this case \(\mu _{A_\tau ^\star }\) has full support \([0,1]^2\). In fact, an even stronger and quite surprising singularity result holds—letting \(\hat{\mathfrak {T}}\) denote the family of all transformation matrices \(\tau \) (i) containing no zeros, (ii) fulfilling that the row sums and column sums through every \(t_{ij}\) are identical, and (iii) \(\mu _{A_\tau ^\star } \not = \lambda _2\) we have the following striking result:

### **Theorem 13**

([33]) Suppose that \(\tau \in \hat{\mathfrak {T}}\). Then the corresponding invariant copula \(A_\tau ^\star \) is singular w.r.t. \(\lambda _2\) and has full support \([0,1]^2\). Moreover, for \(\lambda \)-almost every \(x \in [0,1]\) the conditional distribution function \(y\mapsto F_x^{A_\tau ^\star }(y)=K_{A_\tau ^\star }(x,[0,y])\) is continuous, strictly increasing and has derivative zero \(\lambda \)-almost everywhere.

### 3.4 The Star Product of Copulas

*star product*\(A*B \in \fancyscript{C}\) is defined by (see [3, 23] )

*idempotent*if \(A*A=A\) holds, the family of all idempotent copulas will be denoted by \(\fancyscript{C}^{ip}\). For a complete characterization of idempotent copulas we refer to [4] (also see [26]). The star product can easily be translated to the Markov kernel setting—the following result holds:

### **Lemma 14**

### *Remark 15*

Let \(A \in \fancyscript{C}\) be arbitrary. If \((X_n)_{n \in \mathbb {N}}\) is a stationary Markov process on \([0,1]\) with (stationary) transition probability \(K_A(\cdot ,\cdot )\) and \(X_1 \sim \fancyscript{U}(0,1)\) then \((X_n,X_{n+1}) \sim A\) for every \(n \in \mathbb {N}\) and Lemma 14 implies that \((X_1,X_{n+1}) \sim A * A * \cdots * A=:A^{*n}\), i.e., the \(n\)-step transition probability of the process is given by the Markov kernel of \(A^{*n}\).

### *Remark 16*

Since the star product of copulas is a natural generalization of the multiplication of doubly stochastic matrices and doubly stochastic idempotent matrices are fully characterizable (see [10, 25]) the following result underlines how much more complex the family of idempotent copulas is (also see [12] for the original result without idempotence).

### **Theorem 17**

- 1.
The invariant copula \(A_{\tau _s}^\star \) is idempotent.

- 2.
The Hausdorff dimension of the support of \(A_{\tau _s}^\star \) is \(s\).

### *Example 18*

For the transformation matrix \(\tau \) from Example 12 the invariant copula \(A^\star _\tau \) is idempotent and its support has Hausdorff dimension \(\ln {5}/\ln {3}\). Hence, setting \(A:=A^\star _\tau \) and considering the Markov process outlined in Remark 15 we have \((X_i,X_{i+n}) \sim A\) for all \(i,n \in \mathbb {N}\). The same holds if we take \(A:=\fancyscript{V}^j_\tau (\varPi )\) for arbitrary \(j \in \mathbb {N}\) since this \(A\) is idempotent too.

### **Theorem 19**

As nice by-product, Theorem 19 also offers a very simple proof of the fact that idempotent copulas are necessarily symmetric (originally proved in [4]).

## 4 Copulas Whose Corresponding Markov Operator Is Quasi-constrictive

Studying asymptotic properties of Markov operators quasi-constrictiveness is a very important concept. To the best of the authors’ knowledge, there is no natural/simple characterization of copulas whose Markov operator is quasi-constrictive. The objective of this section, however, is to show that the \(D_1\)-limit \(\hat{A}\) of \(s_{*n}(A)\) has a very simple form if \(T_A\) is quasi-constrictive. We start with a definition of quasi-constrictiveness in the general setting. In general, \(T\) is a Markov operator on \(L^1(\varOmega ,\fancyscript{A},\mu )\) if the conditions (M1)-(M3) from Sect. 2 with \([0,1]\) replaced by \(\varOmega \), \(\fancyscript{B}([0,1])\) replaced by \(\fancyscript{A}\), and \(\lambda \) replaced by \(\mu \) hold.

### **Definition 20**

*quasi-constrictive*if there exist constants \(\delta >0\) and \(\kappa <1\) such that for every probability density \(f \in \fancyscript{D}(\varOmega ,\fancyscript{A},\mu )\) the following inequality is fulfilled:

*asymptotic periodicity*—in particular they proved the following

*spectral decomposition theorem*: For every quasi-constrictive Markov operator \(T\) there exist an integer \(r\ge 1\), densities \(g_1,\ldots ,g_r \in \fancyscript{D}(\varOmega ,\fancyscript{A},\mu )\) with pairwise disjoint support, essentially bounded non-negative functions \(h_1,\ldots ,h_r \in L^\infty (\varOmega ,\fancyscript{A},\mu )\) and a permutation \(\sigma \) of \(\{1,\ldots ,r\}\) such that for every \(f \in L^1(\varOmega ,\fancyscript{A},\mu )\)

### *Example 21*

### *Example 22*

Before returning to the copula setting we prove a first proposition to the spectral decomposition that holds for general Markov operators on \(L^1(\varOmega ,\fancyscript{A},\mu )\) with \((\varOmega ,\fancyscript{A},\mu )\) being a probability space.

### **Lemma 23**

### *Proof*

### **Lemma 24**

### *Proof*

Using the fact that \(\hat{A}\) is idempotent we get the following stronger result:

### **Lemma 25**

### *Proof*

- (a)
\(\sum _{i,j=1}^r m_{i,j} \lambda (E_i)\lambda (E_j)=1\)

- (b)
\(\sum _{j=1}^r m_{i,j} \lambda (E_j)=1\) for every \(i \in \{1,\ldots ,r\}\)

- (c)
\(\sum _{i=1}^r m_{i,j} \lambda (E_i)=1\) for every \(j \in \{1,\ldots ,r\}\)

- (d)
\(\sum _{i=1}^r \vert m_{i,j}-m_{i,l}\vert >0\) whenever \(j \not = l\). \(\square \)

### **Theorem 26**

### *Proof*

### *Remark 27*

Consider again the transformation matrix \(\tau \) from Example 12. Then \(\fancyscript{V}^1_\tau (\varPi ), \fancyscript{V}^2_\tau (\varPi ),\ldots \) are examples of the ordinal-sum-of-\(\varPi \)-like copulas mentioned in the last theorem.

## Notes

### Acknowledgments

The second author acknowledges the support of the Ministerio de Ciencia e Innovación (Spain) under research project MTM2011-22394.

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