In this paper we give a precise mathematical formulation of the relation
between Bose condensation and long cycles and prove its validity for the
perturbed mean field model of a Bose gas. We decompose the total density
$\rho=\rho_{{\rm short}}+\rho_{{\rm long}}$ into the number density of
particles belonging to cycles of finite length ($\rho_{{\rm short}}$) and to
infinitely long cycles ($\rho_{{\rm long}}$) in the thermodynamic limit. For
this model we prove that when there is Bose condensation, $\rho_{{\rm long}}$
is different from zero and identical to the condensate density. This is
achieved through an application of the theory of large deviations. We discuss
the possible equivalence of $\rho_{{\rm long}}\neq 0$ with off-diagonal long
range order and winding paths that occur in the path integral representation of
the Bose gas.