Random walk

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Random walk is a mathematical concept that describes a path consisting of a succession of random steps on some mathematical space such as the integers or a Euclidean space. The term is often used in the context of stochastic or random processes in mathematics and various scientific disciplines, including physics, economics, and biology. The concept of a random walk is a fundamental idea in the theory of Markov chains and is used to model various phenomena in the natural and social sciences.

Definition

A random walk can be formally defined in several ways, depending on the context and the specific properties of the walk being modeled. In its simplest form, a one-dimensional random walk on the integers can be described as follows: starting at 0, at each step move +1 or -1 with equal probability. This process can be generalized to higher dimensions and different step distributions.

Types of Random Walks

There are several types of random walks, categorized based on their step rules, dimensionality, and boundary conditions. Some common types include:

• Simple Random Walk: The most basic form, usually on a lattice, where each step is taken in a random direction with equal probability.
• Biased Random Walk: A variation where there is a greater probability of moving in one direction over others.
• Continuous Random Walk: Instead of discrete steps, the path is continuous, modeled using stochastic differential equations.
• Lévy Flight: A random walk where the step lengths have a heavy-tailed probability distribution, leading to occasional long jumps.

Applications

Random walks have applications in various fields:

• In physics, they model phenomena such as Brownian motion and the diffusion of particles.
• In finance, the random walk hypothesis is a theory that stock market prices evolve according to a random walk and thus cannot be predicted.
• In computer science, random walks are used in algorithms for network analysis and optimization.
• In ecology, they model animal movement patterns and the spread of diseases.

Mathematical Properties

Random walks exhibit several interesting mathematical properties, such as the law of large numbers and the central limit theorem in the context of their long-term behavior. For example, in a simple symmetric random walk in one dimension, the expected distance from the starting point grows as the square root of the number of steps.

Random Walks in Physics

In physics, random walks are a model for various diffusion processes. The classic example is Brownian motion, where a particle undergoes a random walk due to collisions with molecules in a fluid. This phenomenon is described by the Einstein relation, which links the diffusion coefficient to the temperature and viscosity of the fluid.

Random Walks in Economics

The random walk hypothesis in economics suggests that asset prices evolve randomly, implying that it is impossible to consistently outperform the market through stock selection or market timing. This hypothesis is a foundational concept in the efficient-market hypothesis.

See Also

• Markov Chain
• Brownian Motion
• Efficient-Market Hypothesis
• Stochastic Process

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