Question

In the branching process with immigration (Problem 11) assume that \(\varphi^{\prime}(1)=m<1\). Prove that the associated Markov chain has a stationary probability distribution with probability generating function \(\pi(s)=\sum_{r=0}^{\infty} \pi_{r} s^{\prime}\) that satisfies the functional equation, $$ \pi(\varphi(s)) h(s)=\pi(s) $$

Short answer

The procedure involves the understanding of generating functions, computation of derivatives, checking against properties of generating functions, and solving functional equations to determine the stationary probability distribution of the Markov chain in a branching process with immigration. The series expansion \(\pi(s)=\sum_{r=0}^{\infty} \pi_{r} s^{\prime}\) satisfying the functional equation, confirms the existence of a stationary probability distribution.

Step-by-step solution

Understanding the Given Parameters and Construct the Fundamental Equations

The process has immigration if \(\varphi^{\prime}(1)=m<1\). A generating function \(\pi(s)\) is given as the solution of a functional equation \( \pi(\varphi(s)) h(s)=\pi(s)\). From the equation, we can multiply both sides by \(1/\pi(s)\) (provided that \(\pi(s) \neq 0\)) to obtain a functional equation for \(h(s)\) as \( h(s)=\varphi(s)/s\).

Compute for the derivative of h(s)

From \( h(s)=\varphi(s)/s\), you can compute \( h'(s) \) by applying the quotient rule for differentiation: \( h'(s) = [-\varphi(s) + s\varphi'(s)]/{s^2} \). Since \( \varphi'(1) \) is given, you can compute \( h'(1) \).

Check the result using the properties of generating functions

Given by the properties of generating functions, \( h'(1) \) should be equal to 1 if h(s) is the generating function of a probability distribution. Check to confirm if the computed \( h'(1) \) equals to 1.

Solve for Functional Equations Determining Pi(s)

Assuming the validity after the verification in step 3, you can then proceed to solve \( \pi(\varphi(s)) h(s)=\pi(s) \) to find a function \(\pi(s)\) that satisfies this equation.