This chapter addresses various questions related to bridge paths, which are random walks on the integers that return to zero after a specified even number of steps. Such bridge paths arise, among other contexts, in the two-sample problem of nonparametric statistics. Topics include the number of zero-crossings, sojourn times, first-passage times, the maximum and minimum values, changes of sign, the absolute maximum, and the Kolmogorov-Smirnov test. Various limit theorems are also derived in this chapter. The Weibull distribution, the continuous uniform distribution, and the Kolmogorov distribution appear in this context. The chapter concludes with an outlook on the Brownian bridge, which is closely linked to the Brownian motion (Wiener process).

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Bridges: The Tied-down Random Walk

  • Norbert Henze

摘要

This chapter addresses various questions related to bridge paths, which are random walks on the integers that return to zero after a specified even number of steps. Such bridge paths arise, among other contexts, in the two-sample problem of nonparametric statistics. Topics include the number of zero-crossings, sojourn times, first-passage times, the maximum and minimum values, changes of sign, the absolute maximum, and the Kolmogorov-Smirnov test. Various limit theorems are also derived in this chapter. The Weibull distribution, the continuous uniform distribution, and the Kolmogorov distribution appear in this context. The chapter concludes with an outlook on the Brownian bridge, which is closely linked to the Brownian motion (Wiener process).