Average of t-distributed random variables I have 10 t-distributed random variables that I'm averaging over. They are unlikely to be independent but for simplicity let's just assume that they are. Each random variable is parameterised by mean $\mu$, degrees of freedom $\nu$ and scale $\sigma^2$,
$$ x_i \sim St(\mu_i,\nu_i,\sigma_i^2). $$
Is the distribution of $\bar{x}$ t-distributed? If so, what are it's $\mu,\nu,\sigma^2$?
Based on this answer, I'm guessing that it's not t-distributed but it's not clear what it should be.


I have 10 t-distributed random variables that I'm averaging over. They
are unlikely to  be independent but for simplicity let's just assume
that they are. ...  I'm guessing that it's [their mean] not
t-distributed but it's not clear what it should be.

If all $t_i$ variables have tail parameter $v_i>2$ then all variables have finite variance. So considering that all are assumed independent each other it follow that the distribution of their mean tend to be Normal. The exact distribution have not closed form in general.
If all $t_i$ variables have tail parameter $v_i=1$ then all variables is Cauchy. So considering that all are assumed independent each other it follow that the distribution of their mean remain Cauchy.
If all $t_i$ variables share tail parameter $v_i=v$ then it is possible that them can be characterized by a Multivariate t-distribution. If it is so them cannot have independent marginals, even if covariance matrix can be diagonal. In this case even their mean have t-student distrbution.
