Average of t-distributed random variables

Question

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... — user2974951, Jan 27 '22 at 12:09
No, only in case of a $t_1$-distribution, aka Cauchy: https://stats.stackexchange.com/questions/366178/are-there-any-distributions-other-than-cauchy-for-which-the-arithmetic-mean-of-a/366198#366198 or https://stats.stackexchange.com/questions/238246/what-is-the-distribution-of-sample-means-of-a-cauchy-distribution — Christoph Hanck, Jan 27 '22 at 13:08
Thanks both. Just eyeballing the data, $\nu$ ranges from 5 to 9 — stevew, Jan 28 '22 at 10:33

score 1 · Accepted Answer · answered Jan 27 '22 at 14:06

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.

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? — stevew, Jan 28 '22 at 10:34
This is not an easy question. If the first case I mentioned is yours, remain the problem of approximation accuracy that with 10 variables can be not so good. Moreover with financial data some dependence usually appear. I warn you to not undervalue the third case. — markowitz, Jan 28 '22 at 13:18

Average of t-distributed random variables

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