the generalization to random vectors replaces ( with lim [xn ap[ afor all rn such that [ ( where denotes the boundary of the set useful tool for demonstrating convergence in distribution is the characteristic function ps of random vector xdefined as the expectationps ( : eit rn ( the moment generating function in ( is special case of the characteristic function evaluated at -is note that while the moment generating function of random variable may not existits characteristic function always exists the characteristic function of random vector is closely related to the fourier transform of its pdf characteristic function example (characteristic function of multivariate gaussian random vectorthe density of the multivariate standard normal distribution is given in ( and thus the characteristic function of ( in is it - / ps (te ( peit kzk dz rn -ktk / - / = ( pe kz-it dz -ktk / rn rn hencethe characteristic function of the random vector bz in ( with multivariate normal distribution (usis given by ps (te eit eit ( +bzeit ei( tz eit ps (bt eit -kb tk / eit - st/ the importance of the characteristic function is mainly derived from the following resultfor which proof can be foundfor examplein [ theorem characteristic function suppose that ps ( )ps ( )are the characteristic functions of the sequence of random vectors and ps (tis the characteristic function of thenthe following three statements are equivalent limnps xn (tps (tfor all rn xn - limneh(xn eh(xfor all bounded continuous functions rd