Characteristic functions of probability matter

1. Given a random variable X , the symmetrized random variable, designated X8 , is defined by the equality X8=XY where Y is a random variable independent of X. and with its same distribution,

Prove that if X has a characteristic function f(t) , then X8 has a characteristic function f(t)2.

2. Let us consider a characteristic function f(t) Prove the inequality 1f(2t)24(1f(t)2) valid for all real t

3. Consider characteristic functions f1(t),...,fn(t) , and positive constants bi,..,bn , which verify b1++bn=1 . Show that b1f1(t)++bnfn(t) is a characteristic function.

More information at http://www.cmat.edu.uy/~mordecki/courses/procesos-2010/Petrov-Mordecki-cap6-7.pdf

https://web.ma.utexas.edu/users/gordanz/notes/characteristic.pdf

https://kurser.math.su.se/pluginfile.php/5149/mod_resource/content/1/lecture-10b.pdf

http://math.mit.edu/~sheffield/175/Lecture15.pdf