Some Examples of the Weak and Strong Laws of Large Numbers for Averages of Mutually Independent Random Variables

Several theorems are stated which are useful in establishing whether a given sequence of averages of independent but not identically distributed random variables does or does not satisfy the weak and/or strong laws of large numbers.The theorems are illustrated with two families of examples, one discrete and one continuous.In each family, by varying one parameter we obtain cases in which the strong law holds, neither the strong nor the weak law holds, and only the weak law holds.