  Let R be a commutative noetherian ring, I an ideal of R, and M a finitely
generated R-module. The asymptotic behavior of the quotient modules M/I^n M of
M is an actively studied subject in commutative algebra. The main result of
this paper asserts that the depth of the localization of M/I^n M at any prime
ideal of R is stable for large integers n that do not depend on the prime
ideal, if the module M or M/I^n M is Cohen-Macaulay for some n>0, or the ring R
is one of the following: a homomorphic image of a Cohen-Macaulay ring, a
semi-local ring, an excellent ring, a quasi-excellent and catenary ring, and an
acceptable ring.
