# proof of martingale convergence theorem

Let ${({X}_{n})}_{n\in \mathbb{N}}$ be a supermartingale^{} such that $\mathbb{E}|{X}_{n}|\le M$, and let $$. We define a random variable^{} counting how many times the process crosses the stripe between $a$ and $b$:

$$ |

Obviously ${U}_{n+1}\ge {U}_{n}$ therefore ${U}_{\mathrm{\infty}}={lim}_{n\to \mathrm{\infty}}{U}_{n}$ exists almost surely. Next we will construct a new process that mirrors the movement of ${X}_{n}$ but only if the original process is underway of going from below $a$ to over $b$, and is constant otherwise. To do this let $$, $$ for $k\ge 2$, and define ${Y}_{0}:=0$, ${Y}_{n}:=\sum _{k=1}^{n}{C}_{k}({X}_{k}-{X}_{k-1})$. Then ${Y}_{n}$ is also a supermartingale, and the inequality^{} ${Y}_{n}\ge (b-a){U}_{n}-|{X}_{n}-a|$ holds, which gives $0\ge \mathbb{E}({Y}_{n})\ge (b-a)\mathbb{E}({U}_{n})-\mathbb{E}|{X}_{n}-a|$. After rearrangement we get

$$\mathbb{E}({U}_{n})\le \frac{\mathbb{E}|{X}_{n}-a|}{b-a}\le \frac{\mathbb{E}|{X}_{n}|+|a|}{b-a}\le \frac{M+|a|}{b-a}.$$ |

Therefore by the monotone convergence theorem^{}

$$ |

which means $\mathbb{P}({U}_{\mathrm{\infty}}=\mathrm{\infty})=0$. Since $a$ and $b$ were arbitrary $X={lim}_{n\to \mathrm{\infty}}{X}_{n}$ exists almost surely. Now the Fatou lemma^{} gives

$$ |

Thus ${X}_{n}$ is in fact convergent^{} almost surely, and $X\in {L}^{1}.\mathit{\u220e}$

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 |