©Electronic Publishing House ON APPROXIMATION OF COPULAS

New sufficient conditions for strong approximation of copulas, 
generated by sequences of partitions of unity, are given. Results are applied to the checkerboard and Bernstein approximations.


Introduction.
A copula is a distribution function of a doubly stochastic measure µ on the unit square [0, 1] 2 , i.e., C(x, y) = µ([0,x] × [0,y]) for x, y ∈ [0, 1].Copulas are of interest because they link joint distributions to marginal distributions.Sklar showed in [8,9] that, for any real-valued random variables X 1 and X 2 with joint distribution F 12 , there is a copula C such that where F 1 and F 2 denote the cumulative distribution functions of X 1 and X 2 , respectively.
Copulas are Lipschitz functions and the set of copulas is a convex compact subset of the space of continuous functions with uniform norm.Therefore, a natural way of approximating copulas is approximation in the topology of uniform convergence.The copula captures information about the dependence structure of X 1 and X 2 .It is surprising that any copula, even the copula which relates a pair of independent random variables, can be approximated arbitrarily closely in the uniform sense by copulas which correspond to the deterministic dependence between a pair of random variables.Li, Mikusiński, Sherwood, and Taylor, in their work [5,6], proposed another type of convergence of copulas.Namely, since the set of copulas is isomorphic to the set of Markov operators on L ∞ [0, 1], a strong convergence of copulas is defined by the strong convergence of the corresponding Markov operators.This convergence does not lead to the paradox mentioned above.
In [5,6], Li, Mikusiński, Sherwood, and Taylor discussed sequences of approximation copulas given by partitions of unity.The aim of this paper is to give sufficient conditions for these sequences to be convergent in the strong sense.The convergence in the strong operator topology of L p (p ≥ 1) is also discussed.
The paper is organized as follows: Section 2 contains preliminary definitions and results.In Section 3, we formulate and prove theorems concerning the convergence of Markov operators.In Section 4, Corollary 4.5 gives sufficient conditions for the strong convergence of Markov operators related to partitions of unity.In Section 5, we give three examples of approximation: the checkerboard, Bernstein, and tent approximations.Similar results for the checkerboard and Bernstein approximations were proved in [6] by different methods.

Preliminaries. A copula is a function
and the monotonicity condition We say that T : (2.2) A Markov operator is bounded and the norm of T in L ∞ is 1.We remark that T can be extended to L p (p ≥ 1) and it is easy to verify (by (2.2) and (2.4)) that the norm of T in L 1 is 1.Therefore, by the Riesz-Thorin interpolation theorem, the norm of T in L p is also 1.The set of copulas is isomorphic to the set of Markov operators T on L ∞ [0, 1] via the correspondence where C ,2 = ∂C/∂y.We say that C n converges to C in the strong sense if T Cn converges to T C in the strong operator topology of L 1 .This strong convergence has a probabilistic interpretation.Let T be a Markov operator.T corresponds to some doubly stochastic measure on [0, 1] 2 so that we may regard [0, 1] 2 as a probability space.Then that is, T f (x) is the mean value of f (Y ) given that X = x, where f is a real-valued function on [0, 1] and X, Y are random variables on [0, 1] 2 defined by X(u, v) = u and Y (u, v) = v.Therefore, strong convergence for copulas amounts to convergence of conditional expectations.

Convergence of Markov operators.
We study the following situation.Let k n : [0, 1] 2 → R be a sequence of nonnegative measurable functions satisfying the following two conditions It is easy to verify that the operators P n , are Markov operators.Now, we formulate two theorems.

be a sequence of nonnegative measurable functions satisfying (3.1), (3.2), and, for every
This shows that Thus, P n → I in the strong operator topology of L p .The proof that P * n → I is analogous.

Partition of unity operators.
In some applications, we are interested in approximations of copulas by simple ones.One type of approximation is related to the partitions of unity.We recall the definition This approximation of copulas using the sequence of partitions of unity is given in [5,Thm. 6].Proposition 4.2.Let φ 1 ,...,φ n ∈ L 1 ([0, 1]) be nonnegative functions.The following statements are equivalent (i) φ 1 ,...,φ n is a partition of unity.(ii) For every copula C, the operator T n (C) : L 1 → L 1 defined by is a Markov operator, where This shows that T n (C) = P n • T C • P * n , which completes the proof.
We are interested in the question of when T n (C) → T C in the strong operator topology of L 1 and also we ask when T n (C) → T C in the strong operator topology of L p for all p ∈ [1, ∞).Since P n ≤ 1 and P * n ≤ 1, the next result follows immediately from (4.7).

Applications
1.A checkerboard approximation.Let C be a copula and let n ∈ N. Define We call Čn (C) a checkerboard approximation to C. The associated Markov operator can be written as It is easy to see that χ 1,n ,...,χ n,n is a partition of unity for all n ∈ N. If T n (C) is the Markov operator corresponding to this partition of unity, then T n (C) = T Čn(C) holds.An associated sequence of kernels k n is given by We show that the kernels k n satisfy the assumptions of Corollary 4. 2. Bernstein approximation.We recall that Bernstein polynomials are defined with the help of the following expressions The polynomials b 1,n ,...,b n,n form a partition of unity for n = 1, 2,... .We approximate any copula C by The associated kernels k n are given by k n (x, y) = n n i=1 b i,n (x)χ i,n (y), (5.8) and the corresponding Markov operator by (5.12)Let ξ n be a sequence of independent random variables such that Prob(ξ n = 1) = x and Prob(ξ ( (5.18) From (4.1), we get for sufficiently large n.Combining this with (5.16) and (5.18), we get (4.11).This shows that T Bn(C) → T C in strong operator topology of L p for every p ∈ [1, ∞).

3.
A tent approximation.We define a sequence of tent functions by

Corollary 4 . 4 . 4 . 4 . 5 .
Let p ∈ [1, ∞) be given.Suppose that the operators P n ,P * n are generated by kernels k n given by(4.6).Assume that P n → I and P * n → I in the strong operator topology of L p .Then T n (C) → T C in the strong operator topology of L p .Now, we can formulate sufficient conditions for approximation by copulas corresponding to Markov operators generated by partitions of unity.Using definition (4.6), we can write the assumption (3.5) of Theorem 3n (x)χ i,n (y)dx dy = 1]+1,n (x)dx = 1 for a.e.y ∈ [0, n (x)A(ε,x) χ i,n (y)dy = 1 for a.e.x ∈ [0, 1],(4.11)where[z] denotes the largest integer not larger than z.The following corollary is a consequence of Theorem 3.1 and 3.2, Proposition 4.3 and Corollary 4.Corollary Let φ 1,n ,...,φ n,n be a partition of unity for n = 1, 2,... and let T n (C) be a sequence of Markov operators given by(4.4).(i) If (4.9) holds, then for every copula C, T n (C) → T C in the strong operator topology of L 1 .(ii) If (4.10) and (4.11) hold, then for every copula C, T n (C) → T C in the strong operator topology of L p for every p ∈ [1, ∞).
Since kernels k n are symmetrical, (4.11) also holds.Consequently, T Čn(C) → T C in the strong operator topology of L p for every p ∈ [1, ∞).