Question

Let \(X_{n}\) be a discrete branching process with associated probability generating function \(\varphi(s)\) and let \(\varphi_{n}(s)=\sum_{k=0}^{\infty} \operatorname{Pr}\left\\{X_{n}=k\right\\} s^{k}\). Assume that \(\varphi^{\prime}(1)>1\) Let \(\tilde{X}_{n}\) denote the number of all the particles in the nth generation which have an infinite line of descent. Show that the probability generating function for \(\tilde{X}_{n}\) is $$ \sum_{k=0}^{\infty} \operatorname{Pr}\left\\{\bar{X}_{n}=k \mid \bar{X}_{0}=X_{0}=1\right\\} s^{k}=\frac{\varphi_{n}(s(1-q)+q)-q}{1-q} $$ where \(q\) is the probability of extinction. Hint: Note that for \(k \geq 1\) $$ \operatorname{Pr}\left\\{\tilde{X}_{n}=k \mid \dot{X}_{0}=1, X_{0}=1\right\\}=\frac{\sum_{i=k}^{\infty} \operatorname{Pr}\left\\{\tilde{X}_{n}=k, X_{n}=l \mid X_{0}=1\right\\}}{\operatorname{Pr}\left\\{\bar{X}_{0}=1 \mid X_{0}=1\right\\}} $$

Short answer

The probability generating function for \(\tilde{X}_{n}\) is \(\frac{\varphi_{n}(s(1-q)+q)-q}{1-q}\), where \(\varphi_{n}(s)\) is the probability generating function for \(X_{n}\), and \(q\) is the probability of extinction. It is derived based on the principles of branching processes, where \(\tilde{X}_{n}\) represents the number of particles in \(X_{n}\) that do not go extinct and thus have an infinite line of descent.

Step-by-step solution

Understand the probability generating function

By definition, the probability generating function (pgf) \(\varphi(s)\) of a random variable X is defined as \(\varphi(s)=E[s^{X}]\), which is the expected value of \(s^{X}\). Here, we're given that \(\varphi_{n}(s) = \sum_{k=0}^{\infty} \operatorname{Pr}\{X_{n}=k\} s^{k}\). So our main goal is to find the pgf for \(\tilde{X}_{n}\) but this is expressed in terms of \(X_{n}\), or the total number of particles in the nth generation.

Breakdown of the expression for \(\tilde{X}_{n}\)

Let's now look at the expression for \(\tilde{X}_{n}\). The probability \(\operatorname{Pr}\{\tilde{X}_{n}=k \mid \dot{X}_{0}=1, X_{0}=1\}\) appears to be conditioned on \(\dot{X}_{0}\) and \(X_{0}\), both of which uniquely determine \(\tilde{X}_{0}\). By simplifying and rearranging the right-hand side of the equation in the hint, you get \(\frac{\sum_{l=k}^{\infty} \operatorname{Pr}\{\tilde{X}_{n}=k, X_{n}=l \mid X_{0}=1\}}{\operatorname{Pr}\{\bar{X}_{0}=1 \mid X_{0}=1\}}\). This essentially implies that the total number of particles at the nth generation with infinite descent is a sum of conditional probabilities.

Use the definition of the extinction probability

The extinction probability \(q\) is the smallest root of the equation \(\varphi(s)=s\). As \(\varphi^{\prime}(1) > 1\), it ensures that \(q < 1\), or in other words, extinction is not certain.

Define the probability generating function for \(\tilde{X}_{n}\)

Given that \(\tilde{X}_{n}\) essentially counts the number of descendants in \(X_{n}\) that do not go extinct, we know that, because of what was defined before, the pgf for \(\tilde{X}_{n}\) is defined as \(\sum_{k=0}^{\infty} \operatorname{Pr}\{\bar{X}_{n}=k \mid \bar{X}_{0}=X_{0}=1\} s^{k}\). This pgf should reflect the fact that all the particles counted in \(\tilde{X}_{n}\) survive extinction and keep proliferating.

Derive the expression for the pgf of \(\tilde{X}_{n}\)

The branching process theory tells us that if \(X_{0}=1\), then \(X_{n}\) has the same distribution as the sum of \(X_{1}\) independent copies of \(X_{0}\). Thus, the survival of the branching process is equivalent to the survival of at least one of these copies. When \(s(1-q)+q\) is plugged into \(\varphi_{n}(s)\), it gives the pgf of the sum of \(X_{1}\) independent processes, each of which survives with probability \(1-q\). The expression for the pgf of \(\tilde{X}_{n}\) then becomes \(\frac{\varphi_{n}(s(1-q)+q)-q}{1-q}\).

Key concepts

Probability Generating Function

Understanding the concept of a probability generating function (pgf) is crucial for working with discrete stochastic processes like branching processes. In simple terms, the pgf is a concise way to encode all the probabilities of different outcomes into a single function. For any non-negative integer-valued random variable, say X, the pgf \( \varphi(s) \) is defined as follows:

\[ \varphi(s) = E[s^{X}] = \sum_{k=0}^{ewline\infty} \operatorname{Pr}\{X=k\} s^{k} \]

Here, \( E[s^{X}] \) represents the expected value of \( s \) raised to the power of X, for some parameter \( s \) that lies between 0 and 1. This mathematical object is powerful because it allows for the compact representation and easier manipulation of the entire distribution of X.

Calculating the pgf can serve a variety of purposes:

• It helps in determining moments of the random variable. For instance, the first derivative of the pgf evaluated at \( s=1 \) gives the expected value of X, while higher-order derivatives provide information about variance and other moments.
• It simplifies the calculation of the distribution of the sum of independent random variables. If two discrete random variables X and Y are independent, the pgf of their sum is simply the product of their individual pgfs, i.e., \( \varphi_{X+Y}(s) = \varphi_{X}(s) \cdot \varphi_{Y}(s) \).

Now, let's tie back to the given exercise. We are examining the pgf for the number of individuals in a branch that have an infinite line of descent, a concept crucial to the study of population growth and genetics. It is a twist on the straightforward pgf, as we must account for the survival of each individual across generations, leading us to the nuanced expression presented in the problem statement.

Extinction Probability

The extinction probability is a pivotal concept in the theory of branching processes. It refers to the likelihood that the process will eventually die out, meaning that there will be no individuals left after some point in time. For a branching process defined by the probability generating function \( \varphi(s) \) the extinction probability, denoted as \( q \) is the smallest non-negative root of the equation:

\[ \varphi(s) = s \]

This equation intuitively states that for the branching process to die out, the expected number of offspring must, on average, equal the current generation. When \( \varphi'(1) > 1 \), this indicates that the expected number of offspring of each individual exceeds one. Hence, we expect the population to grow on average, but there's still a chance of eventual extinction.

The probability of extinction plays a central role in the formulation of the pgf for \( \tilde{X}_{n} \) given in the exercise. It represents a modified pgf which only counts individuals that have an infinite line of descent—that is, their lineage will never go extinct. The calculations needed to determine the distribution of \( \tilde{X}_{n} \) take into account both the original distribution of offspring and the possibility of extinction.

Importantly, understanding extinction probability helps us in conservation biology for predicting the fate of small populations, and in epidemiology for anticipating the potential eradication of a virus or the persistence of an infection within a host population.

Discrete Stochastic Processes

A discrete stochastic process is a collection of random variables that represent the evolution of some process over time, where time can be measured in discrete steps. The 'stochastic' part essentially means that the process is random; various outcomes can occur, each with its own probability.

Branching processes are a prime example of discrete stochastic processes. They describe scenarios like population growth, nuclear chain reactions, and the spread of diseases. Each individual in a branching process can give birth to a random number of offspring, and this randomness is what defines the stochastic nature of the process.

When dealing with such processes, the goal is often to predict the future state of the process or to understand its long-term behavior. Probability generating functions are instrumental for this purpose, as they allow us to deal with the process's evolution compactly and efficiently. Other important concepts include the expected number of individuals in the future, the variance in that number, and as previously discussed, the extinction probability.

These processes are also a fundamental concept in fields like genetic algorithms, queuing theory, and financial modeling, where they help describe the behavior of complex systems over time. More broadly, understanding discrete stochastic processes like branching processes enables us to forecast, control, and optimize various real-world phenomena that are inherently uncertain.