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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?

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    $\begingroup$ 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? $\endgroup$
    – Xi'an
    Sep 14, 2015 at 8:39
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    $\begingroup$ 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? $\endgroup$
    – undefined
    Sep 14, 2015 at 8:50
  • $\begingroup$ Please amend your question to include your clarification in comments $\endgroup$
    – Glen_b
    Sep 14, 2015 at 9:44
  • $\begingroup$ 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 $|t|$. $\endgroup$
    – whuber
    Sep 14, 2015 at 13:43
  • $\begingroup$ @whuber I assumed the motivating intent on the mention of t was multivariate-t otherwise there's not much point raising elliptical at all. OP should definitely clarify $\endgroup$
    – Glen_b
    Sep 15, 2015 at 4:23

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The family of elliptical distributions is not closed under convolution. e.g. see Sec 5.3.4 (p90) of Prestele, C. (2007), "Credit Portfolio Modelling with Elliptically Contoured Distributions", Doctoral thesis, Institute for Finance Mathematics, University of Ulm. However, some subfamilies of elliptical distributions are closed under convolution; for example the class of elliptical stable distributions (ESD) are closed under convolutions.

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  • $\begingroup$ I previously converted this to a comment but since the Q. remains unanswered, I've decided to reopen it. $\endgroup$
    – Glen_b
    Nov 23, 2015 at 0:05

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