Pairwise Independent Random Walks can be Slightly Unbounded
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A family of problems that have been studied in the context of various streaming algorithms are generalizations of the fact that the expected maximum distance of a 4-wise independent random walk on a line over n steps is O(√(n)). For small values of k, there exist k-wise independent random walks that can be stored in much less space than storing n random bits, so these properties are often useful for lowering space bounds. In this paper, we show that for all of these examples, 4-wise independence is required by demonstrating a pairwise independent random walk with steps uniform in ± 1 and expected maximum distance O(√(n) n) from the origin. We also show that this bound is tight for the first and second moment, i.e. the expected maximum square distance of a 2-wise independent random walk is always O(n ^2 n). Also, for any even k > 4, we show that the kth moment of the maximum distance of any k-wise independent random walk is O(n^k/2). The previous two results generalize to random walks tracking insertion-only streams, and provide higher moment bounds than currently known. We also prove a generalization of Kolmogorov's maximal inequality by showing an equivalent statement that requires only 4-wise independent random variables with bounded second moments, which also generalizes a result of [5].
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