Fdtxjr

Am I breaking OOP practice with this architecture?

OP Amp not amplifying audio signal

What Exploit Are These User Agents Trying to Use?

Is it possible to map the firing of neurons in the human brain so as to stimulate artificial memories in someone else?

Why was Sir Cadogan fired?

Did 'Cinema Songs' exist during Hiranyakshipu's time?

Is "/bin/[.exe" a legitimate file? [Cygwin, Windows 10]

Why are UK visa biometrics appointments suspended at USCIS Application Support Centers?

Send out email when Apex Queueable fails and test it

What is the fastest integer factorization to break RSA?

Machine learning testing data

Can I hook these wires up to find the connection to a dead outlet?

Does int main() need a declaration on C++?

Can a virus destroy the BIOS of a modern computer?

Processor speed limited at 0.4 Ghz

How to travel to Japan while expressing milk?

Sums of two squares in arithmetic progressions

Why is the sentence "Das ist eine Nase" correct?

Getting extremely large arrows with tikzcd

files created then deleted at every second in tmp directory

How can I prove that a state of equilibrium is unstable?

Is it possible to create a QR code using text?

Knowledge-based authentication using Domain-driven Design in C#

Could the museum Saturn V's be refitted for one more flight?

Understanding an Identity in Arthreya's “Branching Processes”

Find the probability generating function of a GW processDifference between $lim P[…]$ and $P[ lim ]$Branching Process in simple random walkShow that in a Galton-Walton Branching Process, $phi_n'(s)to0$ for every $sin(0,1)$ if $p_0>0$Relation between stationary distribution of a D/M/1 queue and extinction probability in a branching process?Branching Process; understanding theorem proofDeeper intuition about probability generating function through extinction probability?Proofs (involving Branching Processes)Extinction problem for a generalisation of Galton-Watson branching processesIs $C^X_n_n = 1^infty$ is a martingale?

0

$begingroup$

I'm currently reading from Chapter 1: The Galton-Watson Process in Arthreya's "Branching Processes," and I'm not sure that I'm understanding this identity correctly. We're given our generating function for the linear fractional Galton-Watson Process as $f$. We denote $mathbbEZ_1 = f'(1) = m$.

We have observed that for any two points $u,v$ that
$$
fracf(s)-f(u)f(s)-f(v)=fracs-us-vfrac1-pv1-pu.
$$

We've shown that $f(s)=s$ has roots $s_0$ and $1$. The text claims the following:

Taking $u=s_0, v=1$, if $mneq 1$ we get
$$
frac1-p1-ps_0=lim_sto 1left(fracf(s)-s_0s-s_0right)left(fracf(s)-1s-1right)^-1 = frac1m.
$$

I would like to confirm that my understanding of this is correct. From the first identity, we can rearrange to get the terms that are in the second. Then, we can observe that if $mneq1$ then $s_0 neq 1$, then the left factor has a defined limit as $sto1$, and the right factor is the definition of the derivative at 1, only inverted. So we can break our limit into
$$
lim_sto 1left(fracf(s)-s_0s-s_0right)cdotlim_sto 1left(fracf(s)-1s-1right)^-1.
$$
Then since $f(s)$ is continuous and $f(1)=1$, we have
$$
lim_sto 1left(fracf(s)-s_0s-s_0right)cdotlim_sto 1left(fracf(s)-1s-1right)^-1 =
left(frac1-s_01-s_0right)cdotlim_sto 1left(fracf(s)-1s-1right)^-1 = lim_sto 1left(fracf(s)-1s-1right)^-1 = frac1m.
$$

Is this a correct assessment of the argument?

probability-theory generating-functions

share|cite|improve this question

asked Mar 20 at 18:10

perpetuallyconfusedperpetuallyconfused

126

$endgroup$

add a comment | 

0

$begingroup$

I'm currently reading from Chapter 1: The Galton-Watson Process in Arthreya's "Branching Processes," and I'm not sure that I'm understanding this identity correctly. We're given our generating function for the linear fractional Galton-Watson Process as $f$. We denote $mathbbEZ_1 = f'(1) = m$.

We have observed that for any two points $u,v$ that
$$
fracf(s)-f(u)f(s)-f(v)=fracs-us-vfrac1-pv1-pu.
$$

We've shown that $f(s)=s$ has roots $s_0$ and $1$. The text claims the following:

Taking $u=s_0, v=1$, if $mneq 1$ we get
$$
frac1-p1-ps_0=lim_sto 1left(fracf(s)-s_0s-s_0right)left(fracf(s)-1s-1right)^-1 = frac1m.
$$

I would like to confirm that my understanding of this is correct. From the first identity, we can rearrange to get the terms that are in the second. Then, we can observe that if $mneq1$ then $s_0 neq 1$, then the left factor has a defined limit as $sto1$, and the right factor is the definition of the derivative at 1, only inverted. So we can break our limit into
$$
lim_sto 1left(fracf(s)-s_0s-s_0right)cdotlim_sto 1left(fracf(s)-1s-1right)^-1.
$$
Then since $f(s)$ is continuous and $f(1)=1$, we have
$$
lim_sto 1left(fracf(s)-s_0s-s_0right)cdotlim_sto 1left(fracf(s)-1s-1right)^-1 =
left(frac1-s_01-s_0right)cdotlim_sto 1left(fracf(s)-1s-1right)^-1 = lim_sto 1left(fracf(s)-1s-1right)^-1 = frac1m.
$$

Is this a correct assessment of the argument?

probability-theory generating-functions

share|cite|improve this question

asked Mar 20 at 18:10

perpetuallyconfusedperpetuallyconfused

126

$endgroup$

add a comment | 

$begingroup$

I'm currently reading from Chapter 1: The Galton-Watson Process in Arthreya's "Branching Processes," and I'm not sure that I'm understanding this identity correctly. We're given our generating function for the linear fractional Galton-Watson Process as $f$. We denote $mathbbEZ_1 = f'(1) = m$.

We have observed that for any two points $u,v$ that
$$
fracf(s)-f(u)f(s)-f(v)=fracs-us-vfrac1-pv1-pu.
$$

We've shown that $f(s)=s$ has roots $s_0$ and $1$. The text claims the following:

Taking $u=s_0, v=1$, if $mneq 1$ we get
$$
frac1-p1-ps_0=lim_sto 1left(fracf(s)-s_0s-s_0right)left(fracf(s)-1s-1right)^-1 = frac1m.
$$

I would like to confirm that my understanding of this is correct. From the first identity, we can rearrange to get the terms that are in the second. Then, we can observe that if $mneq1$ then $s_0 neq 1$, then the left factor has a defined limit as $sto1$, and the right factor is the definition of the derivative at 1, only inverted. So we can break our limit into
$$
lim_sto 1left(fracf(s)-s_0s-s_0right)cdotlim_sto 1left(fracf(s)-1s-1right)^-1.
$$
Then since $f(s)$ is continuous and $f(1)=1$, we have
$$
lim_sto 1left(fracf(s)-s_0s-s_0right)cdotlim_sto 1left(fracf(s)-1s-1right)^-1 =
left(frac1-s_01-s_0right)cdotlim_sto 1left(fracf(s)-1s-1right)^-1 = lim_sto 1left(fracf(s)-1s-1right)^-1 = frac1m.
$$

Is this a correct assessment of the argument?

probability-theory generating-functions

share|cite|improve this question

asked Mar 20 at 18:10

perpetuallyconfusedperpetuallyconfused

126

$endgroup$

share|cite|improve this question

asked Mar 20 at 18:10

perpetuallyconfusedperpetuallyconfused

126

share|cite|improve this question

asked Mar 20 at 18:10

perpetuallyconfusedperpetuallyconfused

126

asked Mar 20 at 18:10

perpetuallyconfusedperpetuallyconfused

126

asked Mar 20 at 18:10

perpetuallyconfusedperpetuallyconfused

126