Abstract: In this paper the well-known Skorokhod embedding theorem is used to consider the increments of partial sums of independent （not necessarily identically distributed） random variables when the r-th （0＜r≤1）moment generating function exists or the 2nd moment exists but （2+δ）th does not for any δ＞0. The results obtained are ideal. It is particularly worth to mentioning that the method in this paper is also available for weakly dependent random variables.