  We propose a novel method for testing serial independence of object-valued
time series in metric spaces, which is more general than Euclidean or Hilbert
spaces. The proposed method is fully nonparametric, free of tuning parameters,
and can capture all nonlinear pairwise dependence. The key concept used in this
paper is the distance covariance in metric spaces, which is extended to auto
distance covariance for object-valued time series. Furthermore, we propose a
generalized spectral density function to account for pairwise dependence at all
lags and construct a Cramer-von Mises type test statistic. New theoretical
arguments are developed to establish the asymptotic behavior of the test
statistic. A wild bootstrap is also introduced to obtain the critical values of
the non-pivotal limiting null distribution. Extensive numerical simulations and
two real data applications are conducted to illustrate the effectiveness and
versatility of our proposed method.
