as example (convergence in distributiondefine the random variables ! xk yn : = iid where ber( / we now show that yn - ( firstnote that exp(ityn ( exp(it/ ) exp(itxk / - = = secondfrom the collapsing product( exp(it/ )nk= ( exp(it/ ) exp(it)we have / exp(ityn ( exp(it) exp(it/ it follows that limne exp(ityn (exp(it )/(it)which we recognize as the characteristic function of the ( distribution yet another mode of convergence is the following definition convergence in -norm the sequence of random variables converges in -norm to random variable if lim |xn xp nlp convergence in -norm we denote the convergence in -norm as xn - the case for corresponds to convergence in mean squared error the following example illustrates that convergence in -norm is qualitatively different from convergence in distribution example (comparison of modes of convergencedefine xn : xwhere has uniform distribution on the interval ( , clearlyxn - ( howevere|xn - | / and so the sequence does not converge in -norm in additionp[|xn xe- and so xn does not converge in probability as well thusin general xn - implies neither xn -xnor xn - we mentionhoweverthat if xn - for some constant cthen xn - as well to see thisnote that xn - stands for lim [xn nin other wordswe can writep[|xn ce [xn ep[xn - which shows that xn - by definition