Strong Law of Large Numbers¶

This law is similar to the weak law, but deals with the convergence of the mean. Let \(X_{i}\) be \(n\) independent identically distributed random variables with mean \(\mu\). Then the mean \(M_{n}\),

\[ \begin{aligned} \lim_{n \to \infty} P(M_{n} = \mu) = 1 \end{aligned} \]

That is, \(M_{n}\) converges to \(\mu\) with probability \(1\) or almost surely. Convergence with probability \(1\) implies convergence in probability, but the converse is not always true.

There is a subtle differnce between WLLN and SLLN. WLLN states that the probability of deviation of \(M_{n}\) from the true mean is always finite, although the probability of deviation converges to \(0\) in the limit. On the other hand, SLLN states with absolute certainty that in infinite experiments, the sample mean will converge to the true mean with probability \(1\).