proofwe prove the theorem for xand scalar the proof for random vectors is analogxn ous firstwe show that :-= usingfor exampletheorem in yn other wordswe wish to show that the characteristic function of the joint distribution of xn and yn converges pointwise as psxn ,yn (te ei( xn + yn -eit eit psx, ( ) to show the limit aboveconsider |psxn ,yn (tpsx, ( ) |psxn , (tpsx, ( )|psxn ,yn (tpsxn , ( )|eit (eit xn eit )| ei( xn + (eit (yn - ) |eit | (eit xn eit ) |ei( xn + cx |eit (yn - |psxn ( psx ( ) |eit (yn - since xn -xtheorem implies that psxn ( -psx ( )and the first term |psxn ( psx ( )goes to zero for the second term we use the fact that rx rx |eix eith dth | eith dth | | to obtain the bounde|eit (yn - |eit (yn - | {|yn - |>ee|eit (yn - | {|yn - | {|yn - |>ee| (yn )| {|yn - | [|yn ce| | -| |en since is arbitrarywe can let to conclude that limn|psxn ,yn ( )-psx, ( ) in other wordsz -zand by the continuity of gwe have ( - (zor (xn - (xcexample (necessity of slutsky' conditionthe condition that yn converges in probability to constant cannot be relaxed for examplesuppose that (xyx yd xn - ( and yn - ( thenour intuition tempts us to incorrectly conclude that xn yn - ( this intuition is falsebecause we can have yn -xn for all so that xn yn while both and have the same marginal distribution (in this case standard normalc law of large numbers and central limit theorem two main results in probability are the law of large numbers and the central limit theorem both are limit theorems involving sums of independent random variables in particularconsider sequence of iid random variables with finite expectation and finite variance for each define :( +*xn )/ what can we say about the (randomsequence of averages by ( and ( we have and var / henceas increasesthe variance of the (randomaverage goes to this means