Question

Let \(\varphi(s)\) be the generating function of the number of progeny of a single individual in a branching process that starts with one individual at time zero, and let \(\varphi_{n}(s)\) denote its \(n\)th iterate. Suppose in addition to the ordinary branching process there also exists some immigration into the population during a single generation described by the probability generating function \(h(s)\). Consider the branching process with immigration whose transition probability matrix is defined by $$ \sum_{j=0}^{\infty} P_{i j} s^{\prime}=[\varphi(s)]^{i} \cdot h(s) $$

Short answer

The short answer for this problem is to find the transition probability matrix \(P_{ij}\) by using the given progeny generating function \(\varphi(s)\) and the immigration generating function \(h(s)\) in the formula: \[P_{i j} =\frac{1}{s^{\prime}}\left([\varphi(s)]^{i} \cdot h(s)\right)\] For specific values of i and j, substitute the given functions \(\varphi(s)\) and \(h(s)\), compute the coefficient of \(s^{\prime j}\) in the resulting series, and assemble the transition probability matrix by calculating \(P_{ij}\) for the desired range of i and j values.

Step-by-step solution

Identify the given functions

We are given the progeny generating function \(\varphi(s)\) and the immigration generating function \(h(s)\) for the branching process with immigration. #Step 2: Find the nth iterate of the progeny generating function#

Find the nth iterate of the progeny generating function

We have to find \(\varphi_{n}(s)\), which is the nth iterate of the progeny generating function \(\varphi(s)\). To find this, repeat the function n times: \[\varphi_{n}(s) = \varphi(\varphi(\cdots\varphi(s)\cdots))\] where the function is applied n times. #Step 3: Write the transition probability matrix formula#

Write the transition probability matrix formula

We have to express the transition probability matrix with the given PGFs. The formula to achieve this is given by the equation: \[\sum_{j=0}^{\infty} P_{i j} s^{\prime}=[\varphi(s)]^{i} \cdot h(s)\] #Step 4: Calculate the transition probability matrix using the given functions#

Calculate the transition probability matrix using the given functions

To find the transition probability matrix \(P_{ij}\) using the formula mentioned above, rewrite the equation: \[P_{i j} =\frac{1}{s^{\prime}}\left([\varphi(s)]^{i} \cdot h(s)\right)\] To find \(P_{ij}\) for specific values of i and j, substitute the given functions \(\varphi(s)\) and \(h(s)\), and the appropriate value of i, then compute the coefficient of \(s^{\prime j}\) in the resulting series. The transition probability matrix can then be assembled by computing \(P_{ij}\) for any desired range of i and j values.