# Sum of two elliptical distributions will it be again elliptical distribution?

What will be the sum of two elliptical distributions will it also be elliptically distributed? from this What is the distribution of the difference of two-t-distributions It says that sum of t distribution with different degree of freedom is not always t distribution but does it still lie in elliptical distribution?

• In short, no it does not. Just apply the convolution formula to derive the density of the sum or difference of two elliptical variates and this should come out clearly. Elliptical distributions are not naturally associated with degrees of freedom, what do you mean by these? Sep 14, 2015 at 8:39
• Sorry for degree of freedom I meant when we use student t- distribution. But I wanted to know sum of two elliptical distribution also lie in family of elliptical distribution or not? Sep 14, 2015 at 8:50
• A Student t distribution is univariate. In this case an "elliptical" distribution simply is one that is symmetric. The sum (or, indeed, any linear combination) of any two Student t distributions will remain symmetric. One way to make this obvious is to recognize that the characteristic function of a symmetric distribution is a function of $|t|$ and the cf of a sum is the product of the cfs--and therefore is still a function of $|t|$.