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

where

*Proof.* Set

Then

The last inequality comes from the fact

Now

Hence

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,

(1)

Indeed, suppose the above is true. Then note that

since .

Now Marcinkiewicz-Zygmund theorem gives

converges almost surely.

Now we left to show that (1) holds. Let .

Then observe that

So we are done.