Cyclic Permutations of Sequences and Uniform Partitions

Let ~r = (ri) n i=1 be a sequence of real numbers of length n with sum s. Let s0 = 0 and si = r1 + . . . + ri 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 si. Define p(~r) to be the number of positive sum si among s1, . . . , sn and m(~r) to be the smallest index i with si = max 06k6n sk. 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 ~ri = (ri, . . . , rn, r1, . . . , ri−1) be the i-th cyclic permutation of ~r. For s > 0, we give the necessary and sufficient conditions for {m(~ri) | 1 6 i 6 n} = {1, 2, . . . , n} and {p(~ri) | 1 6 i 6 n} = {1, 2, . . . , n}; for s 6 0, we give the necessary and sufficient conditions for {m(~ri) | 1 6 i 6 n} = {0, 1, . . . , n − 1} and {p(~ri) | 1 6 i 6 n} = {0, 1, . . . , n − 1}. We also give an analogous result for the class of all permutations of ~r.