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The sample covariance — γ* sweeps the bridge with T

The estimation model taken literally: Σ̂ is the sample covariance of T draws from N(0,Σ), the setting in which Antonov, Lipton, and López de Prado prove HRP more robust than Markowitz. The browser runs a seeded Monte Carlo (4,000 replications, common draws across the γ grid; the paper's table uses 200,000).

Two honest wrinkles, both findings of the paper rather than bugs. First, on a sample covariance the implemented recursion at γ = 1 is not the sample minimum-variance portfolio (symmetrization and the naive splits break the exact recovery once exchangeability is gone), so the right end of the chart is full coupling, and the paper carries sample Markowitz as a separate benchmark. Second, the mean of the raw recursion's loss diverges at small T: rare draws drive the fitness sum through zero, the pole is non-integrable, and a sample mean never settles. The plot therefore shows the median curve, which is stable at any sample count, with the tail frequency reported separately.

What to look for: at T = 15 the median minimum sits near γ ≈ 0.6 and roughly 2% of draws at full coupling blow past ten times the population minimum variance; by T = 250 the minimum has slid to γ ≈ 0.96 and the tail is gone. The readout collects γ*(T) as you click through the sample sizes.

The widget runs the recursion of the paper's equation (4) with the naive fitness (5), ported directly from the reference implementations, live in your browser. Nothing is pre-baked.