Problem 1. (15 points) Suppose that Z1,Z2,… are i.i.d. positive random variables, and X=ln(Z1). Let W=ln(X)=ln(ln(Z1)). Suppose that E(ln(Z1))<[infinity]. Use the central limit theorem and the continuous mapping theorem to show that X is asymptotically lognormal.
a) Explain the central limit theorem and its application in this context.
b) Apply the continuous mapping theorem to derive the distribution of W.
c) Discuss the implications of X being asymptotically lognormal.
d) Compare the properties of the lognormal distribution with those of the normal distribution.