Where does walk collide?

In finance the random walk names a hypothesis about prices: that successive changes in an asset's price are independent and unpredictable, so past movements carry no usable information about future ones. Louis Bachelier's 1900 Théorie de la spéculation first modelled speculative prices as a random process and concluded that the expected gain from speculation is zero; Maurice Kendall's 1953 The Analysis of Economic Time-Series found the same statistical wandering in real price data, and Eugene Fama's 1965 The Behavior of Stock-Market Prices gave the hypothesis its standard modern statement. The referent here is a testable economic claim about markets, not a mathematical object studied for its own properties or a description of how an animal moves across a landscape.[1][2][3]

Unit

One price series judged against the claim that its increments are independent and unforecastable

Recovery means

Not applicable

Measured by

Tested with autocorrelation, runs, and variance-ratio tests on returns to check for serial dependence

In probability the random walk is a mathematical object: a path built by adding successive random steps, so the position after n steps is the running total of n independent random displacements. Karl Pearson posed and named it in his 1905 note The Problem of the Random Walk in Nature, asking where a walker taking steps of fixed length in random directions is likely to end up, and Lord Rayleigh promptly connected it to his earlier work on the sum of many random vibrations. The referent is a formal process studied for properties such as how far it drifts and whether it returns to its start, distinct from a claim about market prices or an observed pattern of animal search.[4]

Unit

One trajectory, the running sum of n independent random steps

Recovery means

Recurrence: the simple walk returns to its origin with probability one in one and two dimensions, less than one in higher dimensions

Measured by

Characterised by its mean squared displacement, which grows in proportion to the number of steps, and by first-return and hitting times

In ecology a Lévy walk, also called a Lévy flight, is a model of how an animal moves while searching: most steps are short, but the step lengths follow a heavy-tailed distribution, so occasional very long moves punctuate many small ones. Viswanathan and colleagues' 1996 Lévy flight search patterns of wandering albatrosses in Nature reported this pattern in foraging seabirds and argued it can be an efficient way to find sparse, patchy food. The referent is an empirical description of movement behaviour in a real organism, not a formal process defined on paper or a hypothesis about the unpredictability of asset prices.[5]

Unit

One animal's movement path, described by the distribution of its step lengths

Recovery means

Not applicable

Measured by

Fitted by the power-law exponent of the step-length distribution, estimated from tracking or telemetry data