  The asymptotic efficiency, ARE_{p,2}, of the tests for multivariate means
theta in \R^d based on the p-means relative to the standard 2-mean,
(approximate) likelihood ratio test (LRT), is considered for large dimensions
d. It turns out that these p-mean tests for p>2 may greatly outperform the LRT
while never being significantly worse than the LRT. For instance, ARE_{p,2} for
p=3 varies from about 0.96 to \infty, depending on the direction of the
alternative mean vector theta_1, for the null hypothesis H_0: theta=\0. These
results are based on a complete characterization, under certain general and
natural conditions, of the varying pairs (n,theta_1) for which the values of
the power of the p-mean test for theta=\0 and theta=theta_1 tend, respectively,
to prescribed values alpha and beta. The proofs use such classic results as the
Berry-Esseen bound in the central limit theorem and the conditions of
convergence to a given infinitely divisible distribution, as well as a recent
result by the author on the Schur^2-concavity properties of Gaussian measures.
