Copulas with prescribed correlation matrix

Luc Devroye (1) , Gérard Letac (2)

1 McGill University = Université McGill [Montréal, Canada]
2 LSP - Laboratoire de Statistique et Probabilités

1 Foreword Marc Yor was also an explorer in the jungle of probability distributions, either in discovering a new species, or in landing on an unexpected simple law after a difficult trip on stochastic calculus: we remember his enthousiam after proving that ∞ 0 exp(2B(t) − 2st)dt −1 is gamma distributed with shape parameter s ('The first natural occurrence of a gamma distribution which is not a chi square!'). Although the authors have been rather inclined to deal with discrete time, common discussions with Marc were about laws in any dimension. Here are some remarks-actually initially coming from financial mathematics-where the beta-gamma algebra (a term coined by Marc) has a role. 2 Introduction The set of symmetric positive semi-definite matrices (r ij) 1≤i,j≤n of order n such that the diagonal elements r ii are equal to 1 for all i = 1,. .. , n is denoted by R n. Given a random variable (X 1 ,. .. , X n) on R n with distribution µ such that the second moments of the X i s exist, its correlation matrix R(µ) = (r ij) 1≤i,j≤n ∈ R n is defined by r ij as the correlation of X i and X j if i < j, and r ii = 1. A copula is a probability µ on [0, 1] n such that X i is uniform on [0, 1] for i = 1,. .. , n when (X 1 ,. .. , X n) ∼ µ. We consider the following problem: given R ∈ R n , does there exist a copula µ such that R(µ) = R? The aim of this note is to show that the answer is yes if n ≤ 9. The present authors believe that this limit n = 9 is a real obstruction and that for n ≥ 10 there exists R ∈ R n such that there is no copula µ such that R(µ) = R. Section 3 gives some general facts about the convex set R n. Section 4 proves that if k ≥ 1/2, if 2 ≤ n ≤ 5 and if R ∈ R n there exists a distribution µ on

Domaines: Mathématiques [math]

Luc Devroye, Gérard Letac. Copulas with prescribed correlation matrix. In Memoriam Marc Yor - Séminaire de Probabilités XLVII, pp.585-601, 2017, Lecture Notes in Mathematics. ⟨hal-01977665⟩