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

[Win_odd Dhamnekar]

How to answer this question$\displaystyle \Rightarrow$Are these sequences Martingales?

 

[Deleted member 4993]

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 .

 

[Win_odd Dhamnekar]

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.

 

[Deleted member 4993]

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?

 

[Win_odd Dhamnekar]

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?

$P(Y_i=1)=P(Y_i=-1)=\frac12$ is correct.

 

[HallsofIvy]

But you still haven't stated the definition of "martingale"!