  We study different notions of pointwise redundancy in variable-length lossy
source coding. We present a construction of one-shot variable-length lossy
source coding schemes using the Poisson functional representation, and give
bounds on its pointwise redundancy for various definitions of pointwise
redundancy. This allows us to describe the distribution of the encoding length
in a precise manner. We also generalize the result to the one-shot lossy
Gray-Wyner system.
