First, we present well-known Khintchine-Kolmogorov Theorem.
Theorem (Khintchine-Kolmogorov). Let be independent random variables with finite expectation. If ,
then converges a.e.
Now we present the Marcinkiewicz-Zygmund theorem which is our object of this article.
Theorem (Marcinkiewicz-Zygmund Theorem). Suppose that are i.i.d with for some . Then
The last inequality comes from the fact
converges a.e. Note that
Hence by Borel-Cantelli’s lemma, a.e. for all but finitely many . Hence
converges a.e. This completes the proof.
Now we give some application of this theorem.
Problem. Given i.i.d sequence of random variables , assume and
Show converges almost surely.
Proof. Note that . Indeed,
Indeed, suppose the above is true. Then note that
Now Marcinkiewicz-Zygmund theorem gives
converges almost surely.
Now we left to show that (1) holds. Let .
Then observe that
So we are done.