The strong law of large numbers (SLLN) is usually stated in the following way:
Theorem: For such that the ‘s are independent and identically distributed (i.i.d.) with finite mean , as ,
What if the ‘s are independent but not identically distributed? Can we say anything in that setting? We can if we add a condition on the sum of the variances of the ‘s. This is sometimes known as Kolmogorov’s strong law of large numbers or the Kolmogorov criterion.
Theorem: Assume that are independent with means and variances such that . Then
- WolframMathWorld. Strong Law of Large Numbers.