  In this work we examine the impact of our motion with respect to the CMB rest
frame on statistics of CMB maps by examining the one-, two-, three- and four-
point statistics of simulated maps of the CMB and Sunyaev-Zeldovich (SZ)
effects. We validate boosting codes by comparing their outcomes for temperature
and polarization power spectra up to $\ell \simeq 6000$. We derive and validate
a new analytical formula for the computation of the boosted power spectrum of a
signal with a generic frequency dependence. As an example we show how this
increases the boosting correction to the power spectrum of CMB intensity
measurements by $\sim 30\%$ at 150 GHz. We examine the effect of boosting on
thermal and kinetic SZ power spectra from semianalytical and hydrodynamical
simulations; the boosting correction is generally small for both simulations,
except when considering frequencies near the tSZ null. For the non-Gaussian
statistics, in general we find that boosting has no impact with two exceptions.
We find that, whilst the statistics of the CMB convergence field are
unaffected, quadratic estimators that are used to measure this field can become
biased at the $O(1)\%$ level by boosting effects. We present a simple
modification to the standard estimators that removes this bias. Second,
bispectrum estimators can receive a systematic bias from the Doppler induced
quadrupole when there is anisotropy in the sky -- in practice this anisotropy
comes from masking and inhomogenous noise. This effect is unobservable and
already removed by existing analysis methods.
