On the depth of the tangent cone and the growth of the Hilbert function

For a d-dimensional Cohen-Macaulay local ring (R; m) we study the depth of the associated graded ring of R with respect to an m-primary ideal I in ter ms of the Vallabrega-Valla conditions and the length of It+1/JI(t), where J is a J minimal reduction of I and t greater than or equal to 1. As a corol lary we generalize Sally's conjecture on the depth of the associated graded ring with respect to a maximal ideal to m-primary ideals. We also study th e growth of the Hilbert function.