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Extra info for Recent developments in nonparametric inference and probability: festschrift for Michael Woodroofe
The particle starts at 0 and thereafter at each step it moves to the adjacent vertex, going clockwise with a known probability p, or counterclockwise with probability 1 − p. The directions of successive movements are independent. What is the expected number of moves needed to visit all vertices? This and other related questions are answered using recursive relations. 1. Introduction Consider a particle subjected to a random walk over the vertices of an (m + 1)-gon which are labeled 0, 1, 2, . .
They follow the same approach as in Feller’s classical treatise . f. in terms of trigonometric functions with complex arguments (as done by Feller) and also alternatively in terms of hyperbolic functions with real arguments. Thereafter, they obtain marginal distributions and moments of each of these component random variables. In this paper we resolve Questions 1–3 by dissecting each question into parts that resemble the setup of the celebrated Gambler’s Ruin Problem. We identify a clockwise movement of the particle with the gambler winning a dollar, and a counterclockwise movement with her losing a dollar.
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