# 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.

• Right now it looks like you are summing them (divided by 10 for the mean), in which case the CLT could kick in and hence... Jan 27, 2022 at 12:09
• No, only in case of a $t_1$-distribution, aka Cauchy: stats.stackexchange.com/questions/366178/… or stats.stackexchange.com/questions/238246/… Jan 27, 2022 at 13:08
• Thanks both. Just eyeballing the data, $\nu$ ranges from 5 to 9 Jan 28, 2022 at 10:33

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.
• Thanks for your answer. The random variables are actually the output of a prediction model where there is a constraint on $\nu > 1$. However, just from eyeballing the data, $\nu$ ranges from 5 to 9. I guess that means it's safe to assume the distribution will be Normal? Jan 28, 2022 at 10:34