Almost yes; the assumption that  " y  is a real vector" is absolutely not applicable, since the indicated set  DOES NOT DEPEND ON   y !

Let me add, that the finite convex combinations of the form [yj yk ; j,k=1,2,...,n], where  n   is the number of  X-s,  may be not sufficient. However,  by the possibility of approximating every pd by the discrete ones,  the statement with "suitable closure of the convex combinations..." would be much more reliable.

Remark: I do not have a proof of the conjecture, yet:)

Regards

 Thank you Joachim, I edited the question according to your remark.

The phrase "the possibility of approximating every pd by the discrete ones.." is not clear.

Dear Maxim,

The phrase "the possibility of approximating every pd by the discrete ones.." is added to suggest the way of proving the following conjecture:

" E(x*x^T) belongs to the set closure of conv{yy^T | y >= 0, Ay=b, y is in R^n} "

Obviously, the formulation requires further details like assumption on boundedness of the second moments of the X-s. Now few words about the proof:  let us note, that if  the sequence of  X-s  assumes only finite number of values , say  from the finite set of   M  vectors 

x1, x2, x3, . . . , xM, with probabilities  p1, p2, p3, . . ., pM, respectively, 

then the random matrix  X*XT  assumes values  

x1*x1T, x2*x2T , x3*x3T , . . . ,  xM*xMTwith the same probabilities,

and the expectation  equals the corresponding convex combination 

E{ X* XT } =p1* x1*x1T+ p2* x2*x2T + p3*x3*x3T +...+pM*xM*xMT .  

But, if the pd of the vector  X   is not discrete, then one needs for the proof some approximation of the continuous pd  by  a sequence of finite discrete ones. And then apply some limit procedure to obtain the  matrix of the expectations. Namely this limit procedure requires the indicated change of your conjecture by adding the words closure of  in front of the mathematical symbol  \conv

Remark: I do not claim that the correction of the conjecture is necessary, but currently I cannot see any sketch of a proof of your original claim (suggested by your question).

Regards, Joachim

Thank you Joachim, your approach is clear. 

The second moments and expectations of X are finite. It seems to me that the set conv{ yy^T , y >= 0, Ay=b} is closed.

Just to explain why I ask these not very important  in general questions I can add that there are no specialists in the area to talk with in reality thus I ask these questions in the forum.

Dear Maxim. Yes, you are right! Thus only some details of the proof should be completed. I think this all I can suggest. Good luck in closing the considerations.

Regards, Joachim

Dear Joachim,

I'm completely disagree with the processes of getting information here in my country and I wonder to what extend the situation is the same in other countries, in particular in EU. Normally the process must bring satisfaction to all those who share information especially to the people as you who help with advises. Finally I hope that my questions will not divert you from your  duties and enrich your expertise.

With respect,

Maksim

Dear Maksim,

>"Normally the process must bring satisfaction to all those who share information"  - YES

>". .      .to what extend the situation is the same in other countries, in particular in EU" - It happens that all types of relations appear: everybody love somebody, or not:)

Thanks for kind words. I just like to share my knowledge to those who care, whenever I have some advices possible to explain something. The most difficult side is that we should take into account that the knowledge can also be a dangerous mean when applied by bad guys against other people.

Best regards, Joachim