martingale proof of Kolmogorov’s strong law for square integrable variables
We apply the martingale convergence theorem to prove the following result.
To prove this, we start by constructing a martingale,
If is the -algebra (http://planetmath.org/SigmaAlgebra) generated by then
Here, the independence of and has been used to imply that . So, is a martingale with respect to the filtration .
Also, by the independence of the , the variance of is
The strong law now follows from Kronecker’s lemma, which states that for sequences of real numbers and such that strictly increases to infinity and converges to a finite limit, then tends to as . In our case, we take and to deduce that converges to zero with probability one.
- 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||martingale proof of Kolmogorov’s strong law for square integrable variables|
|Date of creation||2013-03-22 18:33:51|
|Last modified on||2013-03-22 18:33:51|
|Last modified by||gel (22282)|