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

Subhotosh Khan

Super Moderator
Staff member
Joined
Jun 18, 2007
Messages
18,457
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
Joined
Aug 14, 2018
Messages
30
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
Joined
Jun 18, 2007
Messages
18,457
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?
 
Top