Many Limit Theorems, Convergence Theorems and Characterization Theorems in Probability and Statistics, in particular those related to normal distribution , are based on the assumption of independence of two or more random variables.
However, the full power of independence is not used in the proofs of these Theorems, since it is the distribution of summation of the random variables which is needed and not the joint distribution of the variables. A concept is reintroduced, which is quite weak in comparison to independence, and can replace the concept of independence in most of the above mentioned theorems. Another relatively new concept will also be mentioned and some related results are discussed.
