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Branching random walk
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. Each element of a given generation can have several descendants in the next generation. The location of any descendant is the sum of its parent's location and a random variable.
==Example==
An example of branching random walk can be constructed where the branching process generates exactly two descendants for each element, a ''binary ''branching random walk. Given the initial condition that ''X''ϵ = 0, we suppose that ''X''1 and ''X''2 are the two children of ''X''ϵ. Further, we suppose that they are independent random variables. Consequently, in generation 2, the random variables ''X''1,1 and ''X''1,2 are each the sum of ''X''1 and a (0, 1) random variable. In the next generation, the random variables ''X''1,2,1 and ''X''1,2,2 are each the sum of ''X''1,2 and a (0, 1) random variable. The same construction produces the values at successive times.
Each lineage in the infinite "genealogical tree" produced by this process, such as the sequence ''X''ϵ, ''X''1, ''X''1,2, ''X''1,2,2, ..., forms a conventional random walk.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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