Ratliff-Rush Filtration, Regularity and depth of Higher Associated   graded modules: Part I
  In this paper we introduce a new technique to study associated graded modules.   Let $(A,\m)$ be a   Noetherian local ring with $\depth A \geq 2$. Our techniques gives a necessary and sufficient condition for $\depth G_{\m^n}(A) \geq 2$ for all $n \gg 0$.   Other applications are also included; most notable is an upper bound regarding the Ratliff-Rush filtration. 
Tony J. Puthenpurakal