MHW
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MHW asked in Science & MathematicsMathematics · 1 decade ago

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.

1 Answer

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  • 1 decade ago
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    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.

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