Limit involving extinction probability of branching process

[JanetJanet]

Let x(a) be the extinction probability of a branching process whose offspring is Poisson distributed with parameter a. I need to find the limit as a approaches infinity x(a)e^a. I tried computing x(a) directly using generating functions, and I found that it's the solution to e^(a(s-1))=s, but I couldn't solve for x(a). Thus, I couldn't compute the limit. What other ways could I consider going about this problem?

 

[mfb]

I moved the thread to the homework section.

I need to find the limit as a approaches infinity x(a)e^a

What does the formula at the end of the quote do there?

What is s and how did you get that equation?

 

[JanetJanet]

The LHS is what the generating function adds up to. s is the argument of the generating function. x(a) is just the solution to that formula at the end with s between 0 and 1.

 

[Ray Vickson]

Let x(a) be the extinction probability of a branching process whose offspring is Poisson distributed with parameter a. I need to find the limit as a approaches infinity x(a)e^a. I tried computing x(a) directly using generating functions, and I found that it's the solution to e^(a(s-1))=s, but I couldn't solve for x(a). Thus, I couldn't compute the limit. What other ways could I consider going about this problem?

There are standard derivative tests to determine whether or not extinction takes place, and then by solving an equation (numerically, if necessary) to find the extinction probability if it is in $(0,1)$. If $f(s) = e^{a(s-1)}$, just look at $f'(1)$. If $f'(1) \leq 1$, extinction is certain; if $f'(1) > 1$ extinction is not certain. In the latter case, if the extinction probability is $p$, the population will grow without bound with probability $1-p$. See, eg.,
http://wwwf.imperial.ac.uk/~ejm/M3S4/NOTES2.PDF (p. 29) or
http://en.wikipedia.org/wiki/Branching_process
The latter link does not make the derivative condition explicit, but a glance at the displayed graphs should make its applicability clear enough.

 

[JanetJanet]

The problem is that I have to solve the limit as a approaches infinity of x(a)e^a. Not just x(a). That means I have to know how fast x(a) goes to 0 as a goes to infinity.

 

[Ray Vickson]

The problem is that I have to solve the limit as a approaches infinity of x(a)e^a. Not just x(a). That means I have to know how fast x(a) goes to 0 as a goes to infinity.

You can solve the equation $e^{a(s-1)} = s$ in terms of the so-called Lambert W-function; see, eg., http://en.wikipedia.org/wiki/Lambert_W_function . The series expansion of the Lambert function for small argument, and an asymptotic expansion for large argument are known and documented, so that information should be enough for your purposes.