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arXiv · 1503.08790

Analysis of Bidirectional Ballot Sequences and Random Walks Ending in their Maximum

Benjamin Hackl·Clemens Heuberger·Helmut Prodinger·Stephan Wagner
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Abstract

Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the expected height are analyzed asymptotically. Additionally, we use a bijection between admissible random walks and special binary sequences to prove a recent conjecture by Zhao on ballot sequences.

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Benjamin Hackl, Clemens Heuberger, Helmut Prodinger, Stephan Wagner. 2015-03-30. Analysis of Bidirectional Ballot Sequences and Random Walks Ending in their Maximum. https://doi.org/10.1007/s00026-016-0330-0

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