I understand that in the simple branching process, each object gives birth to its children according to the same distribution. However, I now need to handle a special branching process in which objects generate decedents according to different distributions. For example, objects generate objects according to a Gaussian distribution N(μ,σ)， where μ is decided by the generation of the object. Or, μ might be a function of the life time of the object. Are there existing studies on this kind of branching process, which is an extention of the simple Galton-Watson or Bellman-Harris process? Thank you!
Hey albertshx and welcome to the forums. What is your mathematical background? How much do you know about stochastic processes? You could for example do a random walk where each branch point has a distribution based on the current value in the way that we do it using a Wiener process but instead change your distribution to something more along the lines of what you had in mind. One suggestion you could do is to use some kind of transform on a particular random variable that does what you want it to do. So instead of using a normal 'linear' combination of random variables with the same distribution, you might want to introduce some kind of transformed behaviour that introduces complex mechanisms that mirror what you had in mind.
Hey albertshx and welcome to the forums. What is your mathematical background? How much do you know about stochastic processes? You could for example do a random walk where each branch point has a distribution based on the current value in the way that we do it using a Wiener process but instead change your distribution to something more along the lines of what you had in mind. One suggestion you could do is to use some kind of transform on a particular random variable that does what you want it to do. So instead of using a normal 'linear' combination of random variables with the same distribution, you might want to introduce some kind of transformed behaviour that introduces complex mechanisms that mirror what you had in mind.
Dear Chiro, I know Markovian processes well. But I really can't catch your idea of random walking with Wiener process. To be more specific, I want to describe a branching process whose distibution of number of progeny is determined by time or the generation the parent object is in. Simple random numbers may not fit my need. So-called branching process with random environment is very similar to the process I'm studying. But after all they are different.
The first thing is to consider the local changes first. You want to consider how the local changes are in different generations. This will give you an idea of the distribution at each 'generation' using your terminology. That will give you a way to express the overall behaviour in some way analytically that represents 'all generations' by using current information like 'current generation' which is some parameter or property of the process (it would help if you tell us what this is). Once this is done, then add your function of time and modify the stochastic process to get your overall process. For your information, a Weiner process is basically the continuous analog of the random walk which relates a delta from any point to a Normal distribution with mean 0 and variance h where h is the delta. It would be helpful if you gave us detailed characteristics of what you want to happen at each generation and how time affects the behaviour. Without this its really hard to give any specific qualitative (let alone specific mathematical) advice regarding your problem.
The general results for a branching process do not depend on the distribution. I've never thought of your question but it's clear that the mean number off offspring in the nth generation is just the product of the mean number in each previous generation as long as each generation's distribution is independent of the previous. You can verify this easily. The variance is not so clear to me at a first glance.
On the other hand, it's probably more realistic that each generation's distribution depends on the current population. For example, a birth process that depends not only on the current population but also on limited resources. I think a numerical study would be my first endeavor because the general results depend on independence. What is your problem?
Thank you for your patience. To be more specific, I want to start the branching process below: the distribution of the progency number of an object in generation d is P(d,0) = 1- [itex]\sum_{i>0} P(d,i)[/itex] P(d,r) = βP(d-1,r) , for r>0. r has a limited range of values. P(0,r) is known. β < 1is a constant for all generations. It's even better if the one below concerning time can be studied: P(r) = β(t)Pr , for r>0. P(0) = 1- [itex]\sum_{i>0} P(i)[/itex] r has a limited range of values. Pr's are concrete values known. And one can take time = 0 as the moment that the initial seed object is born. β(t) < 1here becomes a function of time, may be exponential. Also the life time (from the time it's born till the moment it splits) distribution of objects is known and it's i.i.d among all objects. As β < 1 the objects are sure to extinct. I wish to study the some properties of the processes. Most complicated one is the distribution of number of objects ever born up to time t in case 2.