The Anatomy of Accident as a Deviation from Random Walk

This chapter depicts the anatomy of accident as deviation. The classical origin of the problem of random walk is attributed to Pearson’s letter to Nature’s editor. The problem of random walk as it is originally stated by Pearson (Nature 72:294–294, 1905a) applied to Donsker’s (Mem Am Math Soc 6:1–10, 1951) invariance principle as a functional extension of central limit theorem. The Kalman filter approach together with its extended dynamic version captures the practical mapping of random walk. Separate experimental designs for deterministic, stochastic and random events help us understand the clear distinction between definitions of these terms. This experimental designation is especially noteworthy since the concepts of stochastic and random variables are generally misused as synonymous. From the historical perspective, Bortkiewicz’s (Das Gesetz der kleinen Zahlen. B.G. Teubner, 1898) the so-called ‘law of small numbers’ and the data of Prussian cavaliers died by warhorse kick provide evidence for Poisson random distribution which can be used as a model for wide range of random ‘economic’ events as opposed to random ‘outright’ events. The definition of random walk as a short-run trend that is an unpredictable drift requires further clarification. The discussion about the real meaning of the law of small numbers (oft-times misinterpreted) sheds light into the difference between rare random events and stochastic processes.