Stein's Wishart identity yields a degree-lowering recursion that exactly evaluates joint moments of disjoint principal minors for nonnegative integer powers and any finite number of blocks.
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On Wilks' problem: Exact recursive formulas via Stein's method for the joint moments of disjoint principal minors of Wishart random matrices
Stein's Wishart identity yields a degree-lowering recursion that exactly evaluates joint moments of disjoint principal minors for nonnegative integer powers and any finite number of blocks.