  Recently, Chatterjee (2021) introduced a new rank-based correlation
coefficient which can be used to measure the strength of dependence between two
random variables. This coefficient has already attracted much attention as it
converges to the Dette-Siburg-Stoimenov measure (see Dette et al. (2013)),
which equals $0$ if and only if the variables are independent and $1$ if and
only if one variable is a function of the other. Further, Chatterjee's
coefficient is computable in (near) linear time, which makes it appropriate for
large-scale applications. In this paper, we expand the theoretical
understanding of Chatterjee's coefficient in two directions: (a) First we
consider the problem of testing for independence using Chatterjee's
correlation. We obtain its asymptotic distribution under any changing sequence
of alternatives converging to the null hypothesis (of independence). We further
obtain a general result that gives exact detection thresholds and limiting
power for Chatterjee's test of independence under natural nonparametric
alternatives converging to the null. As applications of this general result, we
prove a $n^{-1/4}$ detection boundary for this test and compute explicitly the
limiting local power on the detection boundary for popularly studied
alternatives in the literature. (b) We then construct a test for non-trivial
levels of dependence using Chatterjee's coefficient. In contrast to testing for
independence, we prove that, in this case, Chatterjee's coefficient indeed
yields a minimax optimal procedure with a $n^{-1/2}$ detection boundary. Our
proof techniques rely on Stein's method of exchangeable pairs, a non-asymptotic
projection result, and information theoretic lower bounds.
