Measure theoretic question?
Let X be a random variable and define g on the real numbers by
g(t) = E[ e^(itX) ] for t in R
(ie. g is the characteristic function of X).
(i) Suppose |g(t)| = 1 for some non zero t. Prove that there exists a real constant s such that
P(X = s + 2kπ/t for some integer k) = 1.
(ii) Suppose there exist non zero t and irrational u such that
|g(t)| = |g(ut)| = 1.
Use the result you proved in (i) to show that X is constant with probability 1.
- mathematicianLv 71 decade agoFavorite Answer
a) First notice that |e^(itX)|=1 for all t. If |g(t)|=1, then let z be such that zg(t)=1. Then |z|=1 and E(ze^(itx)]=1. But this requires that ze^(itX)=1 almost surely. Write z=e^(-is) to find that e^i(-s+tX)=1 a.s. so -s+tX is an integer a.s. Solve for X to get your result.
Well, if X=s+2pi*k/t=v+2pi*m/tu, there is just one pair of (k,m) where this can be satisfied (since u is irrational). But this shows that Xis constant a.s.