Kolmogorov’s strong law of large numbers

The strong law of large numbers (SLLN) is usually stated in the following way:

Theorem: For X_1, X_2, \dots such that the X_i‘s are independent and identically distributed (i.i.d.) with finite mean \mu, as n \rightarrow \infty,

\begin{aligned} \frac{X_1 + X_2 + \dots + X_n}{n} \stackrel{a.s.}{\rightarrow} \mu. \end{aligned}

What if the X_i‘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 X_i‘s. This is sometimes known as Kolmogorov’s strong law of large numbers or the Kolmogorov criterion.

Theorem: Assume that X_1, X_2, \dots are independent with means \mu_1, \mu_2, \dots and variances \sigma_1^2, \sigma_2^2, \dots such that \displaystyle\sum_{k=1}^\infty \dfrac{\sigma_k^2}{k^2} < \infty. Then

\begin{aligned} \frac{X_1 + X_2 + \dots + X_n - (\mu_1 + \mu_2 + \dots + \mu_n)}{n} \stackrel{a.s.}{\rightarrow} 0. \end{aligned}


  1. WolframMathWorld. Strong Law of Large Numbers.

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