Convergence of Characteristic Functions

Recall that the characteristic function of a random vector X in Kk is defined as

p(t) = E [exp(itTX)] = E [cos(tTX)] + i ■ E [sin(tTX)]

for t e Kk, where i = */—1. The last equality obtains because exp(i ■ x) = cos(x) + i ■ sin(x).

Also recall that distributions are the same if and only if their characteristic functions are the same. This property can be extended to sequences of random variables and vectors:

Theorem 6.22: Let Xn and X be random vectors in Kk with characteristic functions pn (t) and (p(t), respectively. Then Xn ^d X if and only if (p(t) = limn^TOpn(t) for all t e Kk.

Proof: See Appendix 6.C for the case k = 1.

Note that the “only if” part of Theorem 6.22 follows from Theorem 6.18: Xn ^d X implies that, for any t e Kk,

lim E [cos(tTXn)] = E [cos(tTX)];

n^TO

lim E [sin(tTXn)] = E [sin(tTX)];

n^TO

hence,

lim pn (t) = lim E [cos(tTXn)] + i ■ lim E [sin(tTXn)]

n^TO n^TO n^TO

= E[cos(tTX)] + i ■ E[sin(tTX)] = p(t).

Theorem 6.22 plays a key role in the derivation of the central limit theorem in the next section.