proof of Kolmogorov’s strong law for IID random variables
Kolmogorov’s strong law for square integrable random variables states that if is a sequence of independent random variables with then converges to zero with probability one as (see martingale proof of Kolmogorov’s strong law for square integrable variables). We show that the following version of the strong law for IID random variables follows from this.
Let be independent and identically distributed random variables with . Then, as , with probability one.
Note that here, the random variables are not necessarily square integrable. Let us set , so that are IID random variables with . Then, set
Using the fact that has the same distribution as gives
Letting be the smallest integer greater than ,
So, putting this into equation (1),
Therefore, satisfies the required properties to apply the strong law for square integrable random variables,
as , with probability one. Also,
as with probability one.
We finally note that
so , and for large (with probability one). So, can be replaced by in (3), giving the result.
- 1 David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, 1991.
- 2 Olav Kallenberg, Foundations of modern probability, Second edition. Probability and its Applications. Springer-Verlag, 2002.
|Title||proof of Kolmogorov’s strong law for IID random variables|
|Date of creation||2013-03-22 18:33:57|
|Last modified on||2013-03-22 18:33:57|
|Last modified by||gel (22282)|