Martingales, finite expectation I have some uncertainties about one of the requirements for martingale, i.e. showing that $\mathbb{E}|X_n|<\infty,n=0,1,\dots$ when $(X_n,n\geq 0)$ is some stochastic process. In particularly, in some solutions I find that lets say $\mathbb{E}|X_n|<n$ is accepted, for example here (2nd slide, 1.2 example). So my question is: what is the way of thinking if n goes to infinity? Why are we accepting n as a boundary or maybe I misunderstood something? 

For a definition of a martingale you only want each member $X_n$ of the sequence $\{X_n\}_{n\geq 0}$ to be integrable and you don't care about the growth of $\mathsf E|X_n|$, i.e. you only need to show that $\mathsf E|X_n|$ is finite - no matter how big is it or how fast does it grow with the growth of $n$.
