I notice that every Martingale Process refers somehow to an iid. I was wondering why do we need that and what role does the iid actually play in the process.
Sometimes when the iid is not obvious to be spotted but it is there, is there a way to identify it?
For example, in the problem below, I am not sure how to relate the iid ($X_n$) to the sequence ($L_n$) even though they mentioned ($X_k$) in the definition of ($L_n$).
Thank you very much for your help
Let $(X_n,n\geqslant1)$ be i.i.d. random variables, each uniformly distributed on the set $\{-1,0,1\}$. The sequence of random variables $(L_n,n\geqslant0)$ is defined as follows: $L_0=1$ and for $n\geqslant1$: $$L_n = \begin{cases} 0,& \text{ if $X_k=0$ for some $1\leqslant k\leqslant n$}\\ \left(\frac32\right)^n,& \text{ otherwise}. \end{cases}$$
Verify that $(L_n,n\geqslant0)$ is a martingale.
In the given example, it is clear that the $L_n$ are integrable, since $\mathbb P(L_n\geqslant 0)=1$ and $$ \mathbb E[L_n] \leqslant \left(\frac32\right)^n<\infty. $$ To show the martingale condition, consider that if $X_{n+1}=0$ then $L_{n+1}=0$, and otherwise \begin{align} \mathbb E[L_{n+1}\mid L_n] &= 0\cdot\mathbb P(L_{n+1}=0) + \left(\frac32\right)^{n+1}\cdot\mathbb P(X_{n+1}\ne 0)\\ &= 0 + \left(\frac 32\right)^{n+1}\cdot \frac 23\\ &= \left(\frac 32\right)^n\\ &= L_n. \end{align}