Problem concerning martingale convergence theorem

In summary: L^1$ to a random variable $M_{\infty}$ such that $\mathbb{E}[M_{\infty}|\mathcal{F}_n]=M_n$ for all $n$. In our case, we can see that $M_n=S_n-S$ satisfies all these properties, and therefore converges almost surely to a random variable $M_{\infty}$.Finally, we can use the fact that $M_{\infty}=0$ almost surely to prove that $S_n$ converges almost certainly. To see this, note that $S_n=M_n+S=\mathbb{E}[S_n]+\epsilon
  • #1
koobstrukcja
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My goal is providing a proof based on martingale convergence theorems for the following fact:

Series $S_n:=\sum\limits_{k=1}^n X_k$ of independenet random variables converges in distribution. Prove that $S_n$ converges almost certainly.

I suppose these are not sufficent assuptions about $X_n$ and a distriubiution limit $S$ to provide a convergernt martingale related to this sequence. They should at least be integrable...

I've thought about following martingale: $S_n-\mathbb{E}S_n-S+\mathbb{E}S$ adapted to filtration $\mathcal{F}_n=\sigma (S,X_1,X_2,\ldots ,X_n)$, where $S$ denotes a random variable, having distribution $S$, independant from $\sigma (X_1,X_2,\ldots ,X_n\ldots)$, but I can't provide proof of its boundeness in $L^1(\Omega)$.

I know how to prove it without any martigale theorems, in full generality- the first step is proving that the series converge in probability, which might be usefull for a further martingale proof.

Thanks in andvance for any suggestions and ideas.
 
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  • #2


Dear poster,

Thank you for your question. I understand your goal is to provide a proof based on martingale convergence theorems for the fact that if the series $S_n=\sum\limits_{k=1}^n X_k$ of independent random variables converges in distribution, then it also converges almost certainly. I would like to offer some suggestions and ideas that may help you in your proof.

Firstly, I agree with your statement that the assumptions about $X_n$ and the distribution limit $S$ are not sufficient to provide a convergent martingale related to this sequence. As you mentioned, they should at least be integrable. In fact, we can make a stronger assumption that the random variables $X_n$ are uniformly integrable. This means that for any $\epsilon>0$, there exists a constant $M>0$ such that $\mathbb{E}[|X_n|\mathbb{I}_{|X_n|>M}]<\epsilon$ for all $n$. This assumption will ensure the boundedness of the martingale you proposed, as I will explain below.

Now, let's consider the martingale $M_n=S_n-\mathbb{E}[S_n|\mathcal{F}_{n-1}]$. By definition, this martingale satisfies the following properties:

1) $M_n$ is adapted to the filtration $\mathcal{F}_n=\sigma(S,X_1,\ldots,X_n)$
2) $\mathbb{E}[|M_n|]<\infty$ for all $n$
3) $\mathbb{E}[M_n|\mathcal{F}_{n-1}]=M_{n-1}$ for all $n$

Now, since $S_n$ converges in distribution, we can write $S_n=S+\epsilon_n$ where $\epsilon_n$ converges to $0$ in distribution. This means that for any $\delta>0$, we have $\mathbb{P}(|\epsilon_n|>\delta)\rightarrow 0$ as $n\rightarrow\infty$. In other words, the sequence $\{\epsilon_n\}$ is a tight sequence. This fact, along with the assumption of uniform integrability, allows us to apply the martingale convergence theorem.

Specifically, the martingale convergence theorem states that if a sequence of uniformly integrable martingales satisfies
 

Related to Problem concerning martingale convergence theorem

What is the martingale convergence theorem?

The martingale convergence theorem is a fundamental result in probability theory that states that under certain conditions, a sequence of random variables that satisfy the martingale property will converge to a limit with probability 1.

What is a martingale?

A martingale is a stochastic process where the expected value of the next value in the sequence is equal to the current value. In other words, a martingale is a fair game where the average outcome is always the same.

What are the conditions for the martingale convergence theorem to hold?

The martingale convergence theorem holds if the sequence of random variables is a martingale, the variables are bounded, and the variables are independent of each other.

How is the martingale convergence theorem used in finance?

In finance, the martingale convergence theorem is used to prove the existence of a risk-neutral probability measure, which is a key concept in option pricing and portfolio optimization.

What are some real-world applications of the martingale convergence theorem?

The martingale convergence theorem has various applications in fields such as economics, statistics, and engineering. It is used to analyze stock prices, model random walks, and study the behavior of queues, among other things.

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