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The most common formulation of a branching process is that of the Galton–Watson process.Let Z n denote the state in period n (often interpreted as the size of generation n), and let X n,i be a random variable denoting the number of direct successors of member i in period n, where X n,i are independent and identically distributed random variables over all n ∈{ 0, 1, 2, ...} and i ∈ {1 ...
A branching process (BP) (see e.g. Jagers (1975)) is a mathematical model to describe the development of a population. Here population is meant in a general sense, including a human population, animal populations, bacteria and others which reproduce in a biological sense, cascade process, or particles which split in a physical sense, and others.
The Galton–Watson process is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names. [ 1 ] [ 2 ] The process models family names as patrilineal (passed from father to son), while offspring are randomly either male or female, and names become extinct if the family name line dies ...
In probability theory, a branching random walk is a stochastic process that generalizes both the concept of a random walk and of a branching process.At every generation (a point of discrete time), a branching random walk's value is a set of elements that are located in some linear space, such as the real line.
Y n are independent identically distributed random variables whose common distribution is the offspring distribution of the branching process. In the case where this common distribution is Poisson with mean μ , the random variable S n has Poisson distribution with mean μn , leading to the mass function of the Borel distribution given above.
A stochastic process with this semigroup is called a Brownian snake. We may again find a duality between this process and a branching process. Here the branching process will be a super-Brownian motion + with branching mechanism () =, started on a Dirac in 0.
The integral () is the average number of daughters of each arrival and is called the branching ratio. Thus viewing some arrivals as descendants of earlier arrivals, we have a Galton–Watson branching process. The number of such descendants is finite with probability 1 if branching ratio is 1 or less.
The renewal process is a generalization of the Poisson process. In essence, the Poisson process is a continuous-time Markov process on the positive integers (usually starting at zero) which has independent exponentially distributed holding times at each integer i {\displaystyle i} before advancing to the next integer, i + 1 {\displaystyle i+1} .