  For a set of dependent random variables, without stationary or the strong
mixing assumptions, we derive the asymptotic independence between their sums
and maxima. Then we apply this result to high-dimensional testing problems,
where we combine the sum-type and max-type tests and propose a novel test
procedure for the one-sample mean test, the two-sample mean test and the
regression coefficient test in high-dimensional setting. Based on the
asymptotic independence between sums and maxima, the asymptotic distributions
of test statistics are established. Simulation studies show that our proposed
tests have good performance regardless of data being sparse or not. Examples on
real data are also presented to demonstrate the advantages of our proposed
methods.
