Suppose f is bounded, Xn converges to X weakly, and also that P(X ∈ Df) = 0, where is not continuous at x}. Show that f (Xn) converges weakly to f (X). Suppose a sequence {Xn} is uniformly...

Suppose f is bounded, X_n
converges to X weakly, and also that P(X ∈ D_f) = 0, where

is not continuous at x}. Show that f (X_n) converges weakly to f (X).

Suppose a sequence {Xn} is uniformly integrable and X_n
converges to X weakly. Show E X_n
→
EX.

Give an example of a sequence of random variables X_n
converging weakly to X and where each X_n
is integrable, but X is not integrable.