  Asymptotics of maximum likelihood estimation for $\alpha$-stable law are
analytically investigated with a continuous parameterization. The consistency
and asymptotic normality are shown on the interior of the whole parameter
space. Although these asymptotics have been provided with Zolotarev's $(B)$
parameterization, there are several gaps between. Especially in the latter, the
density, so that scores and their derivatives are discontinuous at $\alpha=1$
for $\beta\neq 0$ and usual asymptotics are impossible. This is considerable
inconvenience for applications. By showing that these quantities are smooth in
the continuous form, we fill gaps between and provide a convenient theory. We
numerically approximate the Fisher information matrix around the Cauchy law
$(\alpha,\beta)=(1,0)$. The results exhibit continuity at $\alpha=1,\,\beta\neq
0$ and this secures the accuracy of our calculations.
