Random Walk

Random walks (RW’s) appeared in the mathematical and statistical literature in 1905 when KarlPearson, in a letter to the journal Nature, introduced the name for the first time. They are a simple kind of stochasticprocesses and describe the random movements of an object in a set of possible positions. RW’s are Markov processesas the conditional distribution of a future state given the present and the past depends only on the present state. Asa consequence the classification of Markov chains as irreducible, recurrent and periodic can be applied to characterizetheir limiting behavior. An important role in this regard is played by asymptotic results in probability theory concerningsums of i.i.d. random variables, such as the laws of large numbers and the central limit theorem. At present a largenumber of papers in environmental sciences make explicit or implicit use of RW based models. Their applicationhas mainly to do with studies of animal movements and microscopic motility and of particle diffusion in fluids. Theimplicit use of RW models arises when computational algorithms or complex, possibly hierarchical, statistical modelsare employed.