prooffirstnote that any random variable can be written as the difference of two nonnegative random variablesx xxwhere :max{ and :min{ thuswithout loss of generalitywe assume that the random variables in the theorem above are nonnegative secondfrom the sequence { we can pick up the subsequence { ={ thenfrom chebyshev' inequality ( and the iid conditionwe have var = = cpl thereforeby definition - and from theorem we conclude that - thirdfor any arbitrary nwe can find ksay ncso that ( ) for such and nonnegative it holds that ( ) ( + ( ) since and ( + ) converge almost surely to as (and hence ngoes to infinitywe conclude that - note that the condition ex in theorem can be weakened to |xand the iid condition on the variables xn can be relaxed to mere pairwise independence the corresponding proofhoweveris significantly more difficult the central limit theorem describes the approximate distribution of and it applies to both continuous and discrete random variables looselyit states that central limit theorem the average of large number of iid random variables approximately has normal distribution specificallythe random variable has distribution that is approximately normalwith expectation and variance / theorem central limit theorem if xn are iid with finite expectation and finite variance then for all rx lim ph( )nsn where ph is the cdf of the standard normal distribution prooflet zk :(xk )/ for all kso that and we thus need to show that - ( we again use the properties of the characteristic function let psz be the characteristic function of an iid copy of zthen due to the iid assumption similar calculation to the one in ( yieldsps (te eit [psz (tn)]