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

#### Subhotosh Khan

##### Super Moderator
Staff member
Which sequences - post those here using ASCII or LaTex, along with your attempts/thoughts.

Most of us are not willing to step into an unknown web-site .

#### Dhamnekar Winod

##### New member
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.

#### Subhotosh Khan

##### Super Moderator
Staff member
In your first line you wrote:

P = (Yi = 1) = ..... Is that correct?

In your own words, what is a Martingale sequence?

What property you would need show to claim a sequence to be Martingale?