[Paper Review] Random discrete copulas

Damjana Kokol Bukovšek, Blaž Mojškerc+1 more|arXiv (Cornell University)|Mar 14, 2026

Probability and Risk Models0 citations

TL;DR

The paper defines and analyzes random discrete copulas on an equidistant mesh, first via random permutations and then via random convex combinations of permutation-based copulas, deriving distributions, means, and variances; it also extends these to bilinear checkerboard copulas on the unit square.

ABSTRACT

We introduce the notion of a bivariate random discrete copula on an equidistant mesh and explore its stochastic properties. A random discrete copula is a discrete random field, hence, its value at a given point on the mesh is a random variable. We determine the distribution of this random variable and calculate its expected value and variance. We also consider bilinear extension of a random discrete copula to a random field over the whole unit square.

Motivation & Objective

• Motivate the use of random copulas to model uncertain dependence structures when data are scarce or the dependence may change over time.
• Introduce a rigorous construction of a bivariate random discrete copula on an equidistant mesh.
• Characterize the distribution, expected value, and variance of the copula values at mesh points.
• Extend to general random discrete copulas as convex combinations of base copulas and analyze their moments.
• Explore bilinear (checkerboard) extensions to extend discrete random copulas to the full unit square.

Proposed method

• Define a bivariate discrete copula on an equidistant mesh and relate it to bistochastic matrices (Birkhoff polytope).
• Construct a random discrete copula induced by a permutation by placing mass 1/k in each square corresponding to the permutation and compute the distribution of the value at a mesh point.
• Prove that E[X_k(u,v)]=uv and Var[X_k(u,v)]=(uv(1−u)(1−v))/(k−1).
• Generalize to random discrete copulas as convex combinations of permutation-based copulas with coefficients uniformly drawn from the simplex (Dirichlet(1,...,1)); show E[Y_k(u,v)]=uv and Var[Y_k(u,v)]=uv(1−u)(1−v)/((k!+1)(k−1)).
• Provide an explicit framework for the CDF of Y_k(u,v) via Dirichlet aggregation and discuss a bilinear extension to the full unit square.
• Discuss the bilinear checkerboard extension of discrete random copulas and derive E[X̂_k(u,v)]=uv and Var[X̂_k(u,v)]=(1/(k−1))(u(1−u)−t(1−t)/k)(v(1−v)−s(1−s)/k).

Figure 1. Mass distribution of discrete copula $C_{\pi}$ from Example 3.3 .

Experimental results

Research questions

• RQ1 How can randomization be incorporated into copula models to reflect uncertainty or time-varying dependence?
• RQ2 What are the distributional properties (distribution, mean, variance) of random discrete copulas defined on an equidistant mesh?
• RQ3 How do convex mixtures of base discrete copulas (permutation-based) behave in terms of moments and dependence structure?
• RQ4 How can discrete random copulas be extended to the entire unit square via bilinear (checkerboard) extension while preserving key moment properties?
• RQ5 What is the form of the cumulative distribution function for aggregated random discrete copulas at a fixed mesh point?

Key findings

• For a random discrete copula induced by a permutation, the value at a mesh point has mean uv and variance uv(1−u)(1−v)/(k−1).
• For a random discrete copula formed by uniform Dirichlet weights over permutation-based copulas, the mean remains uv and the variance scales by 1/(k!+1) relative to the permutation-based case: uv(1−u)(1−v)/((k!+1)(k−1)).
• The distribution at any rectangle [x,x+u]×[y,y+v] matches the same as at [0,u]×[0,v], showing translation-invariant behavior on the mesh.
• A bilinear extension to the full unit square preserves E[̂X_k(u,v)]=uv and yields an explicit variance formula depending on the fractional parts t and s of u and v within their grid cells.
• The framework enables construction of random copulas via convex combinations of base copulas, enabling Bayesian-like or imprecise modeling of dependence without specifying a single copula.

Figure 2. The densities of the random variables $Y_{4}(u,v)$ for all $(u,v)\in\Delta_{4}$ with $u,v\notin\{0,1\}$ .