Weak Law of Large Numbers
For i.i.d. with mean , the sample averages satisfy
that is, for every .
Theorem 5.2 (Weak Law of Large Numbers)
Let be i.i.d. random variables with mean . Let . Then
i.e.
This is called the weak law of large numbers (WLLN).
Proof under finite variance
Assume additionally . For ,
where the inequality is Chebyshev’s. Since the summands are independent, , so
which is exactly in probability.
Related
Stated in
- Theorem 5.2 (Weak Law of Large Numbers)§5.1 Convergence Results