Cyclic Permutations of Sequences and Uniform Partitions

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Abstract

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 $s_i=r_1+\ldots +r_i$ for every $i\in\{1,2,\ldots,n\}$. 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 $s_1,\ldots,s_n$ and $m(\vec{r})$ to be the smallest index $i$ with $s_i=\max\limits_{0\leq k\leq n}s_k$. 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 $\vec{r}_i=(r_i,\ldots,r_n,r_1,\ldots,r_{i-1})$ be the $i$-th cyclic permutation of $\vec{r}$. For $s>0$, we give the necessary and sufficient conditions for $\{ m(\vec{r}_i)\mid 1\leq i\leq n\}=\{1,2,\ldots,n\}$ and $\{ p(\vec{r}_i)\mid 1\leq i\leq n\}=\{1,2,\ldots,n\}$; for $s\leq 0$, we give the necessary and sufficient conditions for $\{ m(\vec{r}_i)\mid 1\leq i\leq n\}=\{0,1,\ldots,n-1\}$ and $\{ p(\vec{r}_i)\mid 1\leq i\leq n\}=\{0,1,\ldots,n-1\}$. We also give an analogous result for the class of all permutations of $\vec{r}$.

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