Convergence of sequences of sets of associated primes

It is a well-known result of M. Brodmann that if $\mathfrak{a}$ is an ideal of a commutative Noetherian ring $A$, then the set of associated primes $\operatorname{Ass} (A/\mathfrak{a}^n)$ of the $n$-th power of $\mathfrak{a}$ is constant for all large $n$. This paper is concerned with the following question: given a prime ideal $\mathfrak{p}$ of $A$ which is known to be in $\operatorname{Ass}(A/\mathfrak{a}^n)$ for all large integers $n$, can one identify a term of the sequence $(\operatorname{Ass} (A/\mathfrak{a}^n))_{n \in \mathbb{N} }$ beyond which $\mathfrak{p}$ will subsequently be an ever-present? This paper presents some results about convergence of sequences of sets of associated primes of graded components of finitely generated graded modules over a standard positively graded commutative Noetherian ring; those results are then applied to the above question.