Third order central moment of a positive linear combination of log-normal random variables

What is the sign ($\pm$) of the third order central moment of a positive linear combination of log-normal random variables?

It seems to be a common notion that the skewness of random variables with longer tails to the right is positive. Is it correct? If so, how do you prove it?

• (1) Please don't cross-post. Pick a site. If you change your mind about the best location, ask for the original post to be moved. (2) your first question is quite different from the two later (closely related) questions. They should probably be separate posts. – Glen_b Jun 4 '14 at 6:05
• Sure, Glen. I will remove the first post. I was advised to ask the same question here. – Shravan Mohan Jun 4 '14 at 6:17