proof of martingale convergence theorem

Let (Xn)n be a supermartingalePlanetmathPlanetmath such that 𝔼|Xn|M, and let a<b. We define a random variableMathworldPlanetmath counting how many times the process crosses the stripe between a and b:


Obviously Un+1Un therefore U=limnUn exists almost surely. Next we will construct a new process that mirrors the movement of Xn but only if the original process is underway of going from below a to over b, and is constant otherwise. To do this let C1:=χ[X0<a], Ck:=χ[Ck-1=0Xk-1<a]+χ[Ck-1=1Xk-1b] for k2, and define Y0:=0, Yn:=k=1nCk(Xk-Xk-1). Then Yn is also a supermartingale, and the inequalityMathworldPlanetmath Yn(b-a)Un-|Xn-a| holds, which gives 0𝔼(Yn)(b-a)𝔼(Un)-𝔼|Xn-a|. After rearrangement we get


Therefore by the monotone convergence theoremMathworldPlanetmath


which means (U=)=0. Since a and b were arbitrary X=limnXn exists almost surely. Now the Fatou lemmaMathworldPlanetmath gives

𝔼|X|=𝔼(limn|Xn|)lim infn𝔼|Xn|M<.

Thus Xn is in fact convergentMathworldPlanetmathPlanetmath almost surely, and XL1.

Title proof of martingale convergence theorem
Canonical name ProofOfMartingaleConvergenceTheorem
Date of creation 2013-03-22 18:34:33
Last modified on 2013-03-22 18:34:33
Owner scineram (4030)
Last modified by scineram (4030)
Numerical id 4
Author scineram (4030)
Entry type Proof
Classification msc 60F15
Classification msc 60G44
Classification msc 60G46
Classification msc 60G42