Let $(X_{n})$ be a sequence of independent integrable random variables with zero mean and $Y_{n}=\sum_{k=1}^{n}\prod_{j=1}^{k}X_{j}$, I would like to ask:
$1.$ What does it mean zero mean? It is just another word for expecting value, that is $\mathbb{E}X_{n}=0?$
$2.$ If I want to show that $Y_{n}$ is integrable, is this idea correct? $\mathbb{E}[\sum_{k=1}^{n}\prod_{j=1}^{k}X_{j}]\le\sum_{k=1}^{n}\mathbb{E}|\prod_{j=1}^{k}X_{j}|=\sum_{k=1}^{n}\mathbb{E}\prod_{j=1}^{k}|X_{j}|=\sum_{k=1}^{n}\prod_{j=1}^{k}\mathbb{E}|X_{j}|<\infty$ In the first term I use monotonicity of expected value, the third term (if $(X_{n})$ are independent random variable then also $(|X_{n}|)$ are independent random variables) and in the fourth term I use the fact that all $X_{n}$ are integrable random variables.
$3.$ If I want to show (later) that $(Y_{n})$ is a martingale, I need to show that it is $\mathcal{F}_{n}$-adapted . What is the form of filtration which I will use? I would say canonical filtration $\mathcal{F}_{n}=\sigma(Y_{1},\cdots,Y_{n})$. What is then relation between $\sigma(Y_{1},\cdots,Y_{n})$ and $\sigma(X_{1},\cdots,X_{n})$ Thanks for any advice.