Testing that marginal sequences of data are not independent via self-normalization

Abstract

A statistic is proposed to test whether data arising from a sequence of (not necessarily independent) 2-vectors {(X i , Y i ), i≥1} have the property that its marginal sequences {X i , i≥1} and {Y i , i≥1} are not independent of each other. When the 2-vectors are i.i.d. with marginals in the domain of attraction of the normal law, the statistic is essentially √n·r n , where r n is Pearson’s correlation coefficient when n pairs of random variables are used. Asymptotic normality is shown under the same assumptions, thereby providing a rate of convergence of Pearson’s correlation coefficient r n to 0. The statistic is also shown to be asymptotically normal in several cases where the 2-vector sequences are dependent, including exchangeable and stationary uniform mixing. Under the condition of exchangeability, an example shows that r n →0, whereas the true correlation coefficient is not zero. The results suggest that Pearson’s r n deserves further study in testing whether marginal sequences are not independent when either second moments fail to exist or when the sequences are dependent.