that by definition the average converges to in -norm as that isx - in factto obtain convergence in probability the variance need not be finite -it is sufficient to assume that ex theorem weak law of large numbers law of large numbers if xn are iid with finite expectation uthen for all lim | ue np in other wordsx - theh theorem has ia natural generalization for random vectors namelyif ex then kxn uk where is the euclidean norm we give proof in the scalar case prooflet zk :xk for all kso that ez we thus need to show that - we use the properties of the characteristic function of denoted as psz due to the iid assumptionwe have psz (te eitz eitzi / = = eizi / psz ( / [psz ( / )] ( = an application of taylor' theorem in the neighborhood of yields psz ( /npsz ( ( /nsince psz ( we havepsz ( [psz ( / )] [ ( / )] - the characteristic function of random variable that always equals zero is therefored theorem implies that - howeveraccording to example convergence in distribution to constant implies convergence in probability hencez - there is also stronger version of this theoremas follows theorem strong law of large numbers strong law of large numbers if xn are iid with expectation and ex lim sup | ue na in other wordsx - >