For two independent Gaussian groups with variances 1 and σ²>1, the probability that the low-variance group supplies the overall maximum has a non-degenerate limit precisely when n1 ∼ C n2^σ² (log n2)^−(σ²−1)/2.
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Asymptotic Probabilities of Attaining the Maximum in Heterogeneous Gaussian Samples
For two independent Gaussian groups with variances 1 and σ²>1, the probability that the low-variance group supplies the overall maximum has a non-degenerate limit precisely when n1 ∼ C n2^σ² (log n2)^−(σ²−1)/2.