  We propose a novel statistical test to assess the mutual independence of
multidimensional random vectors. Our approach is based on the $L_1$-distance
between the joint density function and the product of the marginal densities
associated with the presumed independent vectors. Under the null hypothesis, we
employ Poissonization techniques to establish the asymptotic normal
approximation of the corresponding test statistic, without imposing any
regularity assumptions on the underlying Lebesgue density function, denoted as
$f(\cdot)$. Remarkably, we observe that the limiting distribution of the
$L_1$-based statistics remains unaffected by the specific form of $f(\cdot)$.
This unexpected outcome contributes to the robustness and versatility of our
method. Moreover, our tests exhibit nontrivial local power against a subset of
local alternatives, which converge to the null hypothesis at a rate of {${\tiny
n^{\tiny -1/2}h_n^{\tiny -{d/4}}}$}, $d\geq 2$, where $n$ represents the sample
size and $h_n$ denotes the bandwidth. Finally, the theory is supported by a
comprehensive simulation study to investigate the finite-sample performance of
our proposed test. The results demonstrate that our testing procedure generally
outperforms existing approaches across various examined scenarios.
