If are pairwise independent and then as

Proof:

Let and . Since are pairwise independent, the are uncorrelated and thus

Since

Now using the Chebyshev’s inequality

Hence

For a.s. convergence we have to take subsequences. Let

Then

So

and the Borel-Cantelli lemma implies

which means that

Now pick an s.th. and observe that if then

or

But since

we have that and hence the terms at the right and left ends both .

**References**:

A.N. Shiryaev (1996), *Probability, *Springer

W. Feller (1971), *An Introduction to Probability Theory and Its Applications,* Vol.2. John Wiley & Sons (NY)

R. Durrett (2010). *Probability: Theory and Examples, *Cambridge University Press

O. Kallenberg (2006)**, ***Foundations of modern probability, *Springer

Advertisements