Random Walk (English: Random Walk, abbreviated as RW), is a mathematical statistical model, which is composed of a series of trajectories, each of which is random. [1] [2] It can be used to represent irregular changes, like a random process record formed by a person walking after drinking. It was first proposed by Karl Pearson in 1905. [3]
Random walks can be carried out in various spaces: commonly studied include graphs, integer or real number lines, vector spaces, surfaces, high-dimensional Riemannian manifolds, and groups, finitely generated groups or Lie groups. In the simplest case, time is discrete, and the path of a random walk is a sequence of random variables indexed by natural numbers (Xt) = (X1, X2, …). However, it is also possible to define a random walk that takes steps at random times. In this case,
all times t ∈ [0,+∞) of X t must be defined .
Usually, we can assume that random walks appear in the form of Markov chains or Markov processes, but more complex random walks do not necessarily appear in this form. Under certain restricted conditions, there will be some more special modes, such as the model Brownian motion of diffusion, drunkard's walk or Levi flight.
Random walk has many applications in various fields, such as in engineering and many scientific fields, including ecology, psychology, computer science, physics, chemistry, biology, and economics. In mathematics, we can use the random walk of the individual-based model to estimate the value of π. It can be used to simulate the path of a molecule as it travels in liquid or gas, the search path of foraging animals, fluctuating stock prices and the financial situation of gamblers. In these fields, random walk can be used to explain many observed phenomena, so it is a basic statistical model for recording random activities. [1]
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