In a branching process, the generating function $H_n(t)$ for the total number of individuals up to and including generation $n$ is defined as:

$H_n(t) = \mathbb{E} \left( t^{Z_1 + Z_2 + \cdots + Z_n} \right)$

where $Z_1, Z_2, \dots, Z_n$ represent the number of individuals in each generation. The goal is to find a recursive equation for $H_n(t)$ in terms of $t$, $H_{n-1}(t)$, and the generating function of the offspring distribution, $G_X(t)$, where $G_X(t)$ is the probability generating function for the number of offspring of each individual.

Recursive Derivation:

1. Base Case for $n = 0$: The total number of individuals in generation 0 is simply the initial population, say 1 individual. Therefore, the generating function for $H_0(t)$ is:

   $H_0(t) = t$ This reflects that there is only one individual initially.

2. Recursive Step for $n \geq 1$: To build $H_n(t)$, note that the total number of individuals in generation $n$ depends on the number of offspring produced by each individual in generation $n-1$. Let $Z_{n-1}$ be the number of individuals in generation $n-1$. Then the total number of individuals in generation $n$ is the sum of the offspring of all individuals in generation $n-1$.

   The generating function for the total number of offspring of each individual is given by $G_X(t)$. Since each individual in generation $n-1$ produces a random number of offspring according to the same distribution, the generating function for the total number of individuals in generation $n$ is:

   $H_n(t) = H_{n-1}(G_X(t))$

Final Recursive Equation:

$H_n(t) = H_{n-1}(G_X(t))$

This recursion shows that the generating function for the total number of individuals up to and including generation $n$ is obtained by substituting the offspring generating function $G_X(t)$ into the generating function of the previous generation $H_{n-1}(t)$.

Summary of Recursive Process:

• Initial Condition: $H_0(t) = t$
• Recursive Formula: $H_n(t) = H_{n-1}(G_X(t))$

This recursion allows you to calculate the generating function for any generation $n$ using the offspring distribution's generating function $G_X(t)$.

Would you like further clarification on any of the steps or more details?

Related Questions:

1. What is the significance of the generating function $G_X(t)$ in branching processes?
2. How does the recursive equation change if the initial population is not 1 but some random variable?
3. What is the interpretation of the fixed point of the generating function $H_n(t)$?
4. How does this recursive process relate to the extinction probability in a branching process?
5. What is the expected total population size after $n$ generations?

Tip:

In branching processes, generating functions are extremely useful because they allow you to handle sums of random variables (like total population) in a manageable way by leveraging their properties such as composition and differentiation.