1. Branching process with martingale

Let $\{ X_k^n : n,k \geq 1 \}$ be i.i.d. positive interger-value random variables with $EX_k^n = \mu < \infty$ and $Var(X_k^n) = \sigma ^2 > 0$. Define $Y_0 =1$ and recursively define $Y_{n+1}=X^{n+1}_1+ . . . +X_{Y_n}^{n+1} \ \ \ n \geq 0$

a) Show that $M_n = \frac {Y_n}{ \mu ^n }$ is a martingale with respect to the filtration $\sigma (Y_0, Y_1, . . . , Y_n)$

b) Find $E(Y_{n+1}^2 | Y_0,...,Y_n)$ and deduce that M_n has uniformly bounded variance if and only if $\mu > 1$

c) For $\mu > 1$, find $Var(M_ \infty )$

My proof so far.

a) This one is easy, $E(M_{n+1}|Y_1,...,Y_n) = \frac {1}{ \mu ^{n+1} } E(Y_{n+1} | Y_1,...,Y_n)$

$= \frac {1}{ \mu ^{n+1} }E(X_1^{n+1}+...+X_{Y_n}^{n+1})$

$= \frac {1}{ \mu ^{n+1} }(E(X_1^{n+1})+...+E(X_{Y_n}^{n+1}))$

$= \frac {1}{ \mu ^{n+1} } \frac {Y_n}{ \mu ^n } = M_n$. So that proves that M_n is a martingale.

b) I'm having problem trying to break down this thing...

$E(Y_{n+1}^2 | Y_0,...,Y_n) = E[(X_1^{n+1}+...+X_{Y_n}^{n+1})^2|Y_0,...,Y_n]$

But how would I proceed from here? Thanks.

2. Re: Branching process with martingale

Hello,

You know the conditional variance : Var[Y|X]=E[Y^2|X]-E[Y|X]^2
This will give you E[Y^2|X], and the conditional variance is the sum of the variances, because the variables are independent. Sorry I don't use the latex, nor did I use the correct names for the variables, but it takes too much time to write them down