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Verify that a stochastic process derived from a branching process is a martingale

0

$begingroup$

I was working on a problem while studying martingales and couldn't verify one of the necessary properties to determine if this is in fact a martingale. Given $X_n$ denotes the size of the nth generation of a branching process, I need to verify that the following process is a martingale:

$$ Z_n = X_n/m^n , space space Z_n, n geq 1 $$

To show that $mathbbE[|Z_n|] < infty$:

$$
mathbbE[|Z_n|] = mathbbEbigg[bigglvertfracXm^nbiggrvertbigg]
$$
$$
= fracmathbbE[m^n
$$
Since $X$ is a branching process with mean offspring per individual $m$:
$$
= fracm^nm^n
$$
$$
= 1
$$

I have trouble proving the second condition, that $mathbbE[Z_n+1|Z_1,dotsm,Z_n] = Z_n$

Currently I have:
$$
mathbbE[Z_n+1|Z_1,dotsm,Z_n] = mathbbEbigg[fracX_n+1m^n+1 bigg| Z_1,dotsm,Z_n bigg]
$$
$$
= fracmathbbE[X_n+1] m^n+1
$$
$$
= fracm mathbbE[X_n] m^n+1
$$
$$
= fracmathbbE[X_n] m^n
$$

Which is where I ended up for the proof of the first condition. Something must not be right - I suspect the step where the conditional expectation disappeared could be the problem.

I'd appreciate any and all help understanding where I'm going wrong.

stochastic-processes random-variables self-learning martingales

share|cite|improve this question

edited Mar 28 at 0:29

CLL

asked Mar 26 at 17:40

CLLCLL

1376

$endgroup$

add a comment | 

0

$begingroup$

I was working on a problem while studying martingales and couldn't verify one of the necessary properties to determine if this is in fact a martingale. Given $X_n$ denotes the size of the nth generation of a branching process, I need to verify that the following process is a martingale:

$$ Z_n = X_n/m^n , space space Z_n, n geq 1 $$

To show that $mathbbE[|Z_n|] < infty$:

$$
mathbbE[|Z_n|] = mathbbEbigg[bigglvertfracXm^nbiggrvertbigg]
$$
$$
= fracmathbbE[m^n
$$
Since $X$ is a branching process with mean offspring per individual $m$:
$$
= fracm^nm^n
$$
$$
= 1
$$

I have trouble proving the second condition, that $mathbbE[Z_n+1|Z_1,dotsm,Z_n] = Z_n$

Currently I have:
$$
mathbbE[Z_n+1|Z_1,dotsm,Z_n] = mathbbEbigg[fracX_n+1m^n+1 bigg| Z_1,dotsm,Z_n bigg]
$$
$$
= fracmathbbE[X_n+1] m^n+1
$$
$$
= fracm mathbbE[X_n] m^n+1
$$
$$
= fracmathbbE[X_n] m^n
$$

Which is where I ended up for the proof of the first condition. Something must not be right - I suspect the step where the conditional expectation disappeared could be the problem.

I'd appreciate any and all help understanding where I'm going wrong.

stochastic-processes random-variables self-learning martingales

share|cite|improve this question

edited Mar 28 at 0:29

CLL

asked Mar 26 at 17:40

CLLCLL

1376

$endgroup$

add a comment | 

$begingroup$

I was working on a problem while studying martingales and couldn't verify one of the necessary properties to determine if this is in fact a martingale. Given $X_n$ denotes the size of the nth generation of a branching process, I need to verify that the following process is a martingale:

$$ Z_n = X_n/m^n , space space Z_n, n geq 1 $$

To show that $mathbbE[|Z_n|] < infty$:

$$
mathbbE[|Z_n|] = mathbbEbigg[bigglvertfracXm^nbiggrvertbigg]
$$
$$
= fracmathbbE[m^n
$$
Since $X$ is a branching process with mean offspring per individual $m$:
$$
= fracm^nm^n
$$
$$
= 1
$$

I have trouble proving the second condition, that $mathbbE[Z_n+1|Z_1,dotsm,Z_n] = Z_n$

Currently I have:
$$
mathbbE[Z_n+1|Z_1,dotsm,Z_n] = mathbbEbigg[fracX_n+1m^n+1 bigg| Z_1,dotsm,Z_n bigg]
$$
$$
= fracmathbbE[X_n+1] m^n+1
$$
$$
= fracm mathbbE[X_n] m^n+1
$$
$$
= fracmathbbE[X_n] m^n
$$

Which is where I ended up for the proof of the first condition. Something must not be right - I suspect the step where the conditional expectation disappeared could be the problem.

I'd appreciate any and all help understanding where I'm going wrong.

stochastic-processes random-variables self-learning martingales

share|cite|improve this question

edited Mar 28 at 0:29

CLL

asked Mar 26 at 17:40

CLLCLL

1376

$endgroup$

share|cite|improve this question

edited Mar 28 at 0:29

CLL

asked Mar 26 at 17:40

CLLCLL

1376

share|cite|improve this question

edited Mar 28 at 0:29

CLL

asked Mar 26 at 17:40

CLLCLL

1376

asked Mar 26 at 17:40

CLLCLL

1376

asked Mar 26 at 17:40

CLLCLL

1376