  We characterize the growth of the Sibson and Arimoto mutual informations and
$\alpha$-maximal leakage, of any order that is at least unity, between a random
variable and a growing set of noisy, conditionally independent and
identically-distributed observations of the random variable. Each of these
measures increases exponentially fast to a limit that is order- and
measure-dependent, with an exponent that is order- and measure-independent.
