(Solved): Detailed and step-by-step proof of theorem 19.1. Theorem 19.1 (Lvy's Continuity Theorem). Let (n ...

Detailed and step-by-step proof of theorem 19.1.

Theorem 19.1 (Lévy's Continuity Theorem). Let (n)n21 be a se- quence of probability measures on Rd, and let (n)n21 denote their Fourier transforms, or characteristic functions. a) If ?n converges weakly to a probability measure ?, then in (u) ? (u) for all u € Rd; b) If in (u) converges to a function f(u) for all u € Rª, and if in addition f is continuous at 0, then there exists a probability ? on Rd such that f(u)(u), and un converges weakly to ?.