Let \(\displaystyle \{Y_n\}_{n\geq 1}\) be a sequence of independent, identically distributed random variables.

\(\displaystyle P=(Y_i=1)=P(Y_i=-1)=\frac12\)

Set \(\displaystyle S_0=0\) and \(\displaystyle S_n=Y_1+...+Y_n \) if \(\displaystyle n\geq 1\)

I want to check if the following sequences are martingales.

\(\displaystyle M_n^{(1)}=\frac {e^{\theta S_n}}{(\cosh{\theta})^n}\)

\(\displaystyle M_n^{(2)}=\displaystyle\sum_{k=1}^n sign{(S_{(k-1)})} Y_k, n\geq 1,M_0^{(2)}=0\)

\(\displaystyle M_n^{(3)}=S_n^2-n\)

I have no idea to answer these questions. I think to answer these questions, one must Moment generating functions and Cumulants in detail.