  Let $(A, \mathfrak{m})$ be a Gorenstein local ring, and $\mathcal{F} =\{F_n
\}_{n\in \mathbb{Z}}$ a Hilbert filtration. In this paper, we give a criterion
for Gorensteinness of the associated graded ring of $\mathcal{F}$ in terms of
the Hilbert coefficients of $\mathcal{F}$ in some cases. As a consequence we
recover and extend a result proved by Okuma, Watanabe and Yoshida. Further, we
present ring-theoretic properties of the normal tangent cone of the maximal
ideal of $A=S/(f)$ where $S=K[\![x_0,x_1,\ldots, x_m]\!]$ is a formal power
series ring over an algebraically closed field $K$, and
$f=x_0^a-g(x_1,\ldots,x_m)$, where $g$ is a polynomial with $g \in
(x_1,\ldots,x_m)^b \setminus (x_1,\ldots,x_m)^{b+1}$, and $a, \, b, \, m$ are
integers. We show that the normal tangent cone $\overline{G}(\mathfrak{m})$ is
Cohen-Macaulay if $A$ is normal and $a \le b$. Moreover, we give a criterion of
the Gorensteinness of $\overline{G}(\mathfrak{m})$.
