  Strong approximations of uniform transport processes to the standard Brownian
motion rely on the Skorokhod embedding of random walk with centered double
exponential increments. In this note we make such an embedding explicit by
means of a Poissonian scheme, which both simplifies classic constructions of
strong approximations of uniform transport processes (Griego et al. (1971)) and
improves their rate of strong convergence (Gorostiza et al. (1980)). We
finalise by providing an extension regarding the embedding of a random walk
with asymmetric double exponential increments.
