Ordinal pattern probabilities for symmetric random walks

2019/07/16 by Denoncourt, Hugh
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1907.07172

Abstract

An ordinal pattern for a finite sequence of real numbers is a permutation that records the relative positions in the sequence. For random walks with steps drawn uniformly from [-1,1], we show an ordinal pattern occurs with probability (|[1,w]|)/(2ⁿ n!), where [1,w] is a weak order interval in the affine Weyl group \widetildeA_n. For random walks with steps drawn from a symmetric Laplace distribution, the probability is \frac12ⁿ ∏_j=1ⁿ lev(π)_j, where lev(π)_j measures how often j occurs between consecutive values in π. Permutations whose consecutive values are at most two positions apart in π are shown to occur with the same probability for any choice of symmetric continuous step distribution. For random walks with steps from a mean zero normal distribution, ordinal pattern probabilities are determined by a matrix whose ij-th entry measures how often i and j are between consecutive values.

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