11  Why More Data Doesn’t Always Fix It

A short, pointed chapter built entirely around one simulation: convergence rate as a function of tail weight.

def convergence_rate_experiment(sample_fn, sizes):
    return {n: variance([mean([sample_fn() for _ in range(n)]) for _ in range(300)])
            for n in sizes}

For Gaussian data, the variance of the estimate shrinks like 1/n, exactly as theory predicts. For alpha=1.5 Pareto data, it shrinks far more slowly and erratically. The reader leaves with a rule of thumb rather than a false comfort: “more data helps — but how much it helps depends entirely on the tail you’re fighting.”