I have found something interesting in the ‘asymptotic independence’ order statistics literature: apparently it’s been proven since 1960 that the extremes of two correlated distributions are asymptotically independent (obviously when r != 1 or −1). So as you increase n, the probability of double-maxima decreases to the lower bound of 1/n.
The intuition here seems to be that n increases faster than increased deviation for any r, which functions as a constant-factor boost; so if you make n arbitrarily large, you can arbitrarily erode away the constant-factor boost of any r, and thus decrease the max-probability.
I suspected as much from my Monte Carlo simulations (Figure 2), but nice to have it proven for the maxima and minima. (I didn’t understand the more general papers, so I’m not sure what other order statistics are asymptotically independent: it seems like it should be all of them? But some papers need to deal with multiple classes of order statistics, so I dunno—are there order statistics, like maybe the median, where the probability of being the same order in both samples doesn’t converge on 1/n?)
I have found something interesting in the ‘asymptotic independence’ order statistics literature: apparently it’s been proven since 1960 that the extremes of two correlated distributions are asymptotically independent (obviously when r != 1 or −1). So as you increase n, the probability of double-maxima decreases to the lower bound of 1/n.
The intuition here seems to be that n increases faster than increased deviation for any r, which functions as a constant-factor boost; so if you make n arbitrarily large, you can arbitrarily erode away the constant-factor boost of any r, and thus decrease the max-probability.
I suspected as much from my Monte Carlo simulations (Figure 2), but nice to have it proven for the maxima and minima. (I didn’t understand the more general papers, so I’m not sure what other order statistics are asymptotically independent: it seems like it should be all of them? But some papers need to deal with multiple classes of order statistics, so I dunno—are there order statistics, like maybe the median, where the probability of being the same order in both samples doesn’t converge on 1/n?)