satunnaiskävelyiden

Satunnaiskävelyiden, known in English as random walks, are mathematical objects that describe a path consisting of a succession of random steps. At each step, the direction of movement is chosen randomly. The simplest and most common form is the one-dimensional random walk, where a particle moves either left or right with equal probability at each time step. Imagine a person standing at a point on a line, flipping a coin to decide whether to take one step forward or one step backward.

The concept of random walks has applications across various fields. In physics, they are used to model phenomena like Brownian motion, the erratic movement of particles suspended in a fluid. In finance, they form the basis of the efficient-market hypothesis, suggesting that stock prices follow a random walk and cannot be predicted. Computer science utilizes random walks for algorithms like PageRank, which determines the importance of web pages.

The behavior of a random walk is characterized by its statistical properties. For instance, in a one-dimensional symmetric random walk (where steps left and right are equally likely), the expected position of the walker after n steps is zero. However, the expected squared displacement grows linearly with the number of steps, indicating that the walker tends to drift away from the origin over time. Higher-dimensional random walks exhibit similar diffusive behavior. The mathematical study of random walks involves analyzing probabilities of returning to the origin, the distribution of distances from the starting point, and convergence to continuous probability distributions.