On the limit law of the superdiffusive elephant random walk

Hélène Guérin, Lucile Laulin, Kilian Raschel, Thomas Simon
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Abstract

When the memory parameter of the elephant random walk is above a critical threshold, the process becomes superdiffusive and, once suitably normalised, converges to a non-Gaussian random variable. In a recent paper by the three first authors, it was shown that this limit variable has a density and that the associated moments satisfy a nonlinear recurrence relation. In this work, we exploit this recurrence to derive an asymptotic expansion of the moments and the asymptotic behaviour of the density at infinity. In particular, we show that an asymmetry in the distribution of the first step of the random walk leads to an asymmetry of the tails of the limit variable. These results follow from a new, explicit expression of the Stieltjes transformation of the moments in terms of special functions such as hypergeometric series and incomplete beta integrals. We also obtain other results about the random variable, such as unimodality and, for certain values of the memory parameter, log-concavity.
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论超扩散大象随机漫步的极限规律
当大象随机游走的记忆参数超过临界阈值时,过程就会变得超扩散,一旦适当归一化,就会收敛为非高斯随机变量。在三位第一作者最近发表的一篇论文中,证明了这种极限变量具有密度,而且相关矩满足非线性递推关系。在本文中,我们利用这一递推关系推导出了矩的渐近展开和密度在无穷大时的渐近行为。特别是,我们证明了随机游走第一步分布的不对称性导致了极限变量尾部的不对称性。这些结果源于用特殊函数(如超几何级数和不完全贝特积分)对矩的斯蒂尔杰斯变换的新的明确表达。我们还得到了有关随机变量的其他结果,如单调性,以及在记忆参数的某些值下的对数凹性。
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