Extend chordal-sparsity correlation-matrix generation beyond chordal patterns

Determine whether and under what conditions the chordal sparsity method of Kurowicka (2014) for generating random correlation matrices can be extended to patterns of unspecified correlations that do not exhibit chordal sparsity, and, if extension is feasible, construct the methodology and characterize its validity and constraints.

Background

In the review of sampling algorithms for random correlation matrices, the monograph discusses the chordal sparsity method of Kurowicka (2014) as a generalization of the vine-based approach by Lewandowski et al. (2009). While effective for chordal sparsity patterns, the text explicitly notes uncertainty regarding extending the method to other correlation patterns.

This uncertainty highlights a concrete gap in the current methodology for generating valid correlation matrices under broader structural assumptions, which is important for financial applications where dependence structures may not follow chordal sparsity. Establishing such an extension would enhance the versatility of sampling algorithms used in stress testing and scenario analysis.