martingale proof of Kolmogorov’s strong law for square integrable variables

We apply the martingale convergence theorem to prove the following result.


Let X1,X2, be independent random variablesMathworldPlanetmath such that nVar[Xn]/n2<. Then, setting


we have Sn0 as n, with probability one.

To prove this, we start by constructing a martingaleMathworldPlanetmath,


If n is the σ-algebra ( generated by X1,Xn then


Here, the independence of Xn+1 and n has been used to imply that 𝔼[Xn+1n]=𝔼[Xn+1]. So, M is a martingale with respect to the filtrationPlanetmathPlanetmath (n)n.

Also, by the independence of the Xn, the variance of Mn is


So, the inequalityMathworldPlanetmath 𝔼[|Mn|]𝔼[Mn2]=Var[Mn] shows that M is an L1-bounded martingale, and the martingale convergence theorem says that the limit M=limnMn exists and is finite, with probability one.

The strong law now follows from Kronecker’s lemma, which states that for sequencesMathworldPlanetmath of real numbers x1,x2, and 0<b1,b2, such that bn strictly increases to infinityMathworldPlanetmathPlanetmath and nxn/bn converges to a finite limit, then bn-1k=1nxk tends to 0 as n. In our case, we take xn=Xn-𝔼[Xn] and bn=n to deduce that n-1k=1n(Xk-𝔼[Xk]) 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
Canonical name MartingaleProofOfKolmogorovsStrongLawForSquareIntegrableVariables
Date of creation 2013-03-22 18:33:51
Last modified on 2013-03-22 18:33:51
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Proof
Classification msc 60F15
Classification msc 60G42
Related topic MartingaleConvergenceTheorem
Related topic KolmogorovsStrongLawOfLargeNumbers
Related topic StrongLawOfLargeNumbers
Related topic ProofOfKolmogorovsStrongLawForIIDRandomVariables