Problem concerning martingale convergence theorem

[koobstrukcja]

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.

 

[mmwave]

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