  To every vertex algebra $V$ we associate a canonical decreasing sequence of
subspaces and prove that the associated graded vector space $gr(V)$ is
naturally a vertex Poisson algebra, in particular a commutative vertex algebra.
We establish a relation between this sequence and the sequence $C_{n}$
introduced by Zhu. By using the (classical) algebra $gr(V)$, we prove that for
any vertex algebra $V$, $C_{2}$-cofiniteness implies $C_{n}$-cofiniteness for
all $n\ge 2$. We further use $gr(V)$ to study generating subspaces of certain
types for lower truncated $Z$-graded vertex algebras.
