Cyclic permutations of sequences and uniform partitions
Summary: Let \(\vec r=(r_i)_{i=1}^n\) be a sequence of real numbers of length \(n\) with sum \(s\). Let \(s_0=0\) and \(s_i=r_1+\cdots+r_i\), for every \(i\in\{1,2,\dots,n\}\). Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums \(s_i\). Define \(p(\vec r)\) to be the number of positive sum \(s_i\) among \(s_1,\dots,s_n\) and \(m(\vec r)\) to be the smallest index \(i\) with \(s_i=\max_{0\leq k\leq n} s_k\). 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 \(\vec r_i= (r_i,\dots,r_n,r_1,\dots,r_{i-1})\) be the \(i\)-th cyclic permutation of \(\vec r\). For \(s>0\), we give the necessary and sufficient conditions for \(\{m(\vec r_i)\mid 1\leq i\leq n\}= \{1,2,\dots,n\}\) and \(\{p(\vec r_i)\mid 1\leq i\leq n\}= \{1,2,\dots, n\}\); for \(s\leq 0\), we give the necessary and sufficient conditions for \(\{m(\vec r_i)\mid 1\leq i\leq n\}= \{0,1,\dots,n-1\}\) and \(\{p(\vec r_i)\mid 1\leq i\leq n\}= \{0,1,\dots,n-1\}\). We also give an analogous result for the class of all permutations of \(\vec r\).
scientific article; zbMATH DE number 5799445