  Conditional local independence is an asymmetric independence relation among
continuous time stochastic processes. It describes whether the evolution of one
process is directly influenced by another process given the histories of
additional processes, and it is important for the description and learning of
causal relations among processes. We develop a model-free framework for testing
the hypothesis that a counting process is conditionally locally independent of
another process. To this end, we introduce a new functional parameter called
the Local Covariance Measure (LCM), which quantifies deviations from the
hypothesis. Following the principles of double machine learning, we propose an
estimator of the LCM and a test of the hypothesis using nonparametric
estimators and sample splitting or cross-fitting. We call this test the
(cross-fitted) Local Covariance Test ((X)-LCT), and we show that its level and
power can be controlled uniformly, provided that the nonparametric estimators
are consistent with modest rates. We illustrate the theory by an example based
on a marginalized Cox model with time-dependent covariates, and we show in
simulations that when double machine learning is used in combination with
cross-fitting, then the test works well without restrictive parametric
assumptions.
