Copulas with continuous, strictly increasing singular conditional distribution functions Wolfgang Trutschniga,∗, Juan Fern´andezS´anchezb aDepartment of Mathematics, University of Salzburg, Hellbrunner Strasse 34, 5020 Salzburg, Austria, Tel.: +43 662 8044-5312, Fax: +43 662 8044-137 bGrupo de Investigaci´onde An´alisisMatem´atico, Universidad de Almer´ıa,La Ca~nadade San Urbano, Almer´ıa,Spain Abstract Using Iterated Function Systems induced by so-called modifiable transfor- mation matrices T and tools from Symbolic Dynamical Systems we first con- ? struct mutually singular copulas AT with identical (possibly fractal or full) support that are at the same time singular with respect to the Lebesgue mea- 2 sure λ2 on [0; 1] . Afterwards the established results are utilized for a simple ? proof of the existence of singular copulas AT with full support for which ? AT all conditional distribution functions y 7! Fx (y) are continuous, strictly increasing and have derivative zero λ-almost everywhere. This result under- lines the fact that conditional distribution functions of copulas may exhibit surprisingly irregular analytic behavior. Finally, we extend the notion of em- pirical copula to the case of non i.i.d. data and prove uniform convergence 0 of the empirical copula En corresponding to almost all orbits of a Markov ? process usually referred to as chaos game to the singular copula AT . Several examples and graphics illustrate both the chosen approach and the main results. XXX Keywords: Copula, Doubly stochastic measure, Singular function, Markov kernel, Symbolic Dynamical System 2010 MSC: 62H20, 60E05, 28A80, 26A30 ∗Corresponding author Email addresses: [email protected] (Wolfgang Trutschnig), [email protected] (Juan Fern´andezS´anchez) Preprint submitted to Journal of Mathematical Analysis and ApplicationsSeptember 2, 2013 1 1. Introduction 2 The construction of copulas with fractal support via Iterated Function 3 Systems (IFSs) induced by so-called transformation matrices goes back to 4 Fredricks et al. in [17]. Among other things the authors proved the existence 5 of families (Ar)r2(0;1=2) of two-dimensional copulas fulfilling that for every 6 s 2 (1; 2) there exists rs 2 (0; 1=2) such that the Hausdorff dimension of the 7 support Zrs of Ars is s. Using the fact that the same IFS-construction also 8 converges with respect to the strong metric D1 (a metrization of the strong 9 operator topology of the corresponding Markov operators, see [30]) on the 10 space C of two-dimensional copulas Trutschnig and Fern´andez-S´anchez [31] 11 showed that the same result holds for the subclass of idempotent copulas. 12 Thereby idempotent means idempotent with respect to the star-product in- 13 troduced by Darsow et al. in [9], i.e. A 2 C is idempotent if A ∗ A = A. 14 Families (Ar)r2(0;1=2) of copulas with fractal support were also studied by 15 de Amo et al. in [1] and in [2]. In the latter paper, using techniques from 16 Probability and Ergodic Theory, the authors discussed properties of subsets 17 of the corresponding fractal supports and constructed mutually singular cop- 18 ulas having the same fractal set as support. Moments of these copulas were 19 calculated in [4]; some surprising properties of homeomorphisms between 20 fractal supports of copulas were studied in [5]. 21 In the current paper we first generalize some results concerning the construc- 22 tion of mutually singular copulas with identical (fractal or full) support by 23 a different method of proof than the one chosen in [2]. In particular we 24 show that for each so-called modifiable transformation matrix T with cor- ? ? 25 responding invariant copula AT and attractor ZT we can find (uncountable ? ? 26 many) copulas B having the same support ZT but being singular w.r.t. AT . 27 Afterwards in Section 4 we focus on transformation matrices T having non- 28 zero entries (hence being modifiable) and the corresponding singular copulas ? 2 29 AT with full support [0; 1] and study singularity properties of their con- ? AT ? 30 ditional distribution functions y 7! F (y) = K ? (x; [0; y]) of A . Using x AT T 31 the one-to-one correspondence between copulas and Markov kernels hav- 32 ing the Lebesgue measure λ on [0; 1] as fixed point and the fact that the 33 IFS construction can easily be expressed as operation on the corresponding 34 Markov kernels we prove that (λ-almost) all conditional distribution func- ? AT ? 35 tions y 7! F (y) = K ? (x; [0; y]) of A are continuous, strictly increasing, x AT T 36 and have derivative zero λ-almost everywhere. In other words, we prove the ? 37 existence of copulas AT for which all conditional distribution functions are 2 38 continuous, strictly increasing and singular in the sense of [16, 26] as well as 39 [19] (pp. 278-282). Note that this complements some results in [12] and [13] 40 since the singular copulas with full support considered therein have discrete 41 conditional distributions. For a general study of the interrelation between 42 2-increasingness and differential properties of copulas we refer to [18]. 43 Finally, in Section 5 we first extend the notion of empirical copulas to non ^ 44 i.i.d. data and then consider sequences of empirical copulas (En(k))n2N in- 45 duced by orbits (Yn(k))n2N of the so-called chaos game (a Markov process 46 induced by transformation matrices T , see [15, 24]). We prove that, with ^ ? 47 probability one, (En(k))n2N converges uniformly to the copula AT . Several 48 examples and graphics illustrate the main results. 49 2. Notation and preliminaries 50 For every metric space (Ω; ρ) the family of all non-empty compact sets is 51 denoted by K(Ω), the Borel σ-field by B(Ω) and the family of all probability 52 measures on B(Ω) by P(Ω). We will call two probability measures µ1; µ2 on 53 B(Ω) singular with respect to each other (and will write µ1 ? µ2) if there 54 exist disjoint Borel sets E; F 2 B(Ω) with µ1(E) = 1 = µ2(F ). λ and 2 55 λ2 will denote the Lebesgue measure on B([0; 1]) and B([0; 1] ) respectively. 56 For every set E the cardinality of E will be denoted by #E. C will denote 57 the family of all two-dimensional copulas, see [11, 25, 29], Π will denote 58 the product copula. d1 will denote the uniform distance on C; it is well 59 known that (C; d1) is a compact metric space. For every A 2 C µA will 60 denote the corresponding doubly stochastic measure defined by µA([0; x] × 61 [0; y]) := A(x; y) for all x; y 2 [0; 1], PC the class of all these doubly stochastic 62 measures. A Markov kernel from R to B(R) is a mapping K : R × B(R) ! 63 [0; 1] such that x 7! K(x; B) is measurable for every fixed B 2 B(R) and 64 B 7! K(x; B) is a probability measure for every fixed x 2 R. Suppose 65 that X; Y are real-valued random variables on a probability space (Ω; A; P), 66 then a Markov kernel K : R × B(R) ! [0; 1] is called a regular conditional 67 distribution of Y given X if for every B 2 B(R) K(X(!);B) = E(1B ◦ Y jX)(!) (1) 68 holds P-a.e. It is well known that for each pair (X; Y ) of real-valued random 69 variables a regular conditional distribution K(·; ·) of Y given X exists, that X X 70 K(·; ·) is unique P -a.s. (i.e. unique for P -almost all x 2 R) and that 3 X⊗Y 71 K(·; ·) only depends on P . Hence, given A 2 C we will denote (a version 72 of) the regular conditional distribution of Y given X by KA(·; ·) and refer to 73 KA(·; ·) simply as regular conditional distribution of A or as Markov kernel 74 of A. Note that for every A 2 C, its conditional regular distribution KA(·; ·), 2 75 and every Borel set G 2 B([0; 1] ) we have (Gx := fy 2 [0; 1] : (x; y) 2 Gg 76 denoting the x-section of G for every x 2 [0; 1]) Z KA(x; Gx) dλ(x) = µA(G); (2) [0;1] 77 so in particular Z KA(x; F ) dλ(x) = λ(F ) (3) [0;1] 78 for every F 2 B([0; 1]). On the other hand, every Markov kernel K : [0; 1] × 2 79 B([0; 1]) ! [0; 1] fulfilling (3) induces a unique element µ 2 PC([0; 1] ) via 2 C 2 7! A 80 (2). For every A and x [0; 1] the function y Fx (y) := KA(x; [0; y]) 81 will be called conditional distribution function of A at x. For more details 82 and properties of conditional expectation, regular conditional distributions, 83 and disintegration see [21, 22]. 84 Expressing copulas in terms of their corresponding regular conditional dis- 85 tributions a metric D1 on C can be defined as follows: Z Z D1(A; B) := KA(x; [0; y]) − KB(x; [0; y]) dλ(x) dλ(y) (4) [0;1] [0;1] 86 It can be shown that (C;D1) is a complete and separable metric space and 87 that the topology induced by D1 is strictly finer than the one induced by d1 88 (for an interpretation and various properties of D1 see [30]). 89 Before sketching the construction of copulas with fractal support via so- 90 called transformation matrices we recall the definition of an Iterated Function 91 System (IFS for short) and some main results about IFSs (for more details 92 see [6, 14, 24]).