On the frequencies of patterns of rises and falls
Jean-Marc Luck, IPhT Saclay
We investigate the probability of observing a given pattern of rises and falls in a random stationary data series. The data can be modelled as a sequence of independent and identically distributed random numbers or, equivalently, as a uniform random permutation. The probability of observing a long pattern decays exponentially with its length in general. The associated decay rate is interpreted as the embedding entropy of the pattern. This rate is evaluated exactly for all periodic patterns, generalizing thus the pioneering work by André on alternating permutations. In the most general case, it is expressed in terms of a determinant of generalized hyperbolic or trigonometric functions. The probabilities of uniformly chosen random patterns are observed to obey multifractal statistics. The typical value of the rate, corresponding to the endpoint of the multifractal spectrum, plays the role of a Lyapunov exponent. A wide range of examples of patterns, either deterministic or random, is also investigated (Physica A 407 (2014) 252-275).