What is the distribution of the difference of two-t-distributions ... and why ?
Assuming $X_1$,$X_2$ are independent random-variables with mean $\mu_1,\mu_2$ and variance $\sigma^2_1,\sigma^2_2$ respectively. My basic statistics book tells me that the distribution of the $X_1-X_2$ has the following properties: 


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*$E(X_1-X_2)=\mu_1-\mu_2$

*$Var(X_1-X_2)=\sigma^2_1 +\sigma^2_2$


Now let's say $X_1$, $X_2$  are t-distributions with $n_1-1$, $n_2-2$ degrees of freedom. What is the distribution of $X_1-X_2$ ? 
This question has been edited: The original question was "What are the degrees of freedom of the difference of two t-distributions ?". mpiktas has already pointed out that this makes no sense since $X_1-X_2$ is not t-distributed, no matter how approximately normal $X_1,X_2$ (i.e. high df) may be.
 A: Agree the answers above, the difference of two independent t-distributed random variables are not t distributed. But I want to add some ways of calculating this.


*

*The easiest way of calculating this is using a Monte Carlo method. In R, for example, you random sample 100,000 numbers from the first t distribution, then you random sample another 100,000 numbers from the second t distribution. You let the first set of 100,000 numbers minus the second set of 100,000 numbers. The obtained 100,000 new numbers are the random samples from the distribution of the difference between the two distribution. You can calculate the mean and variance by simply using mean() and var().


*This is called Behrens–Fisher distribution. You can refer to the Wiki page: https://en.wikipedia.org/wiki/Behrens%E2%80%93Fisher_distribution. The CI given by this distribution is called "fiducial interval", this is not a CI. 

*Numerical integration might work. This is continued as the bullet point 2. You might refer to the Section 2.5.2 in Bayesian Inference in Statistical Analysis by Box, George E. P., Tiao, George C. It has the detailed steps of integration, and how this is approximated to be a Behrens–Fisher distribution.
A: The sum of two independent t-distributed random variables is not t-distributed. Hence you cannot talk about degrees of freedom of this distribution, since the resulting distribution does not have any degrees of freedom in a sense that t-distribution has.
