# How to check these sequences generated by i.i.d random variables are martingales?

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

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

Set $$S_0=0$$ and $$S_n=Y_1+...+Y_n$$ if $$n\geq 1$$

I want to check if the following sequences are martingales. $$M_n^{(1)}=\frac{e^{\theta S_n}}{(\cosh{\theta})^n}$$

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

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

I have no idea how to answer these questions. I think to answer these questions, knowledge of moment generating functions and its natural logarithm Cumulant is necessary.

If any member knows the correct answers to these questions may reply with correct answers.

• The very first thing to consider when given such simple random variables is to apply the definition. – whuber Mar 19 at 13:53
• what does superscripts (1),(2),(3) of $M_n$ indicate? – Dhamnekar Winod Mar 25 at 6:24

$$M_n^{(1)}$$ is a martingale. It satisfies the conditional expectation:
$$E\left(M_{n+1}^{(1)} | S_n \right)= M_n^{(1)}$$