### Abstract

Let r = (r_{i})_{i=1}^{n} be a sequence of real numbers of length n with sum s. Let s_{0} = 0 and s_{i} = r_{1} +. . .+ r_{i} for every i ∈ {1, 2,. . ., n}. Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums s_{i}. Define p(r) to be the number of positive sum si among s_{1},. . ., s_{n} and m(r) to be the smallest index i with s_{i} = max s_{k}. An important problem in fluctuation theory is that of showing that in a random path the number of steps on the positive half-line has the same distribution as the index where the maximum is attained for the first time. In this paper, let r_{i} = (r_{i},. . ., r_{n}, r_{1},. . ., r_{i-1}) be the i-th cyclic permutation of r. For s > 0, we give the necessary and sufficient conditions for {m(r_{i}) | 1 ≤ i ≤ n} = {1, 2,. . ., n} and {p(r_{i}) | 1 ≤ i ≤ n} = {1, 2,. . ., n}; for s ≤ 0, we give the necessary and sufficient conditions for {m(r_{i}) | 1 ≤ i ≤ n} = {0, 1,..., n - 1} and {p(r_{i}) | 1 ≤ i ≤ n} = {0, 1,. . ., n-1}. We also give an analogous result for the class of all permutations of r.

Original language | English |
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Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Electronic Journal of Combinatorics |

Volume | 17 |

Issue number | 1 |

Publication status | Published - 2010 Oct 1 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Electronic Journal of Combinatorics*,

*17*(1), 1-12.