I performed a fit of some SP500 returns with two heavy tailed distributions, using MATLAB. These are like two guess about what distribution has generated the data. This is the output
In both cases standard error for estimated parameters are reported. This sound me as suggestion to use z-test for hypothesis testing about parameters. It seems me that in t-Student case, at least for tail index greater than $2$ (finite variance), this can work. However stable distribution deal with infinite variance (exception if tail index equal to $2$). This is not a problem? ML estimator maintains asymptotic normality? If not, exist one way for parameters inference (for example $\beta$) with above output only?