We develop a new multi-scale framework flexible enough to solve a number
of problems involving embedding random sequences into random sequences. As
an example we show that there exists an increasing M-Lipschitz embedding
from one i.i.d. Bernoulli sequences into an independent copy with positive
probability provided that M is large enough. In a closely related problem
we show that two independent Poisson processes on R are roughly isometric
(or quasi-isometric).