发表机构
UFABC - Centro de Matemática, Computação e Cognição; UFRJ - Departamento de Métodos Estatísticos do Instituto de Matemática（联邦圣保罗大学数学、计算与认知中心; 里约热内卢联邦大学数学研究所统计方法系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出广义遗忘象随机游走模型，结合记忆、遗忘与随机跳跃，证明中心极限定理与泛函极限定理，识别扩散、临界及超扩散状态，并给出高斯波动及稳定极限定理。
AI 中文摘要
我们引入了一种广义的遗忘象随机游走，其中在每个时刻，底层遗忘象随机游走的方向选择两种增量分布之一，而实际增量则从相应的独立序列中采样。所选择的分布不需要决定增量的符号。这种构造结合了记忆强化、遗忘和跳跃大小的随机性。假设存在有限的二阶矩，我们建立了中心极限定理和泛函极限定理，识别出由记忆和遗忘参数决定的扩散、临界和超扩散状态。我们进一步证明了围绕随机超扩散极限的高斯波动定理，其显式极限方差捕捉了底层遗忘游走和随机跳跃幅度的贡献。对于没有遗忘的特殊情况，当增量属于非高斯稳定律的吸引域时，在重尾跳跃主导记忆贡献的区域内，我们还建立了稳定极限定理及其泛函对应版本。
英文摘要
We introduce a generalized amnesic elephant random walk in which, at each time, the direction of an underlying amnesic elephant random walk selects one of two increment distributions, while the actual increment is sampled from the corresponding independent sequence. The selected distribution does not need to determine the sign of the increment. This construction combines memory reinforcement, amnesia, and randomness in jump sizes. Assuming finite second moments, we establish central limit theorems and functional limit theorems, identifying diffusive, critical, and superdiffusive regimes determined by the memory and amnesia parameters. We further prove a Gaussian fluctuation theorem around the random superdiffusive limit, with an explicit limiting variance that captures the contributions of both the underlying amnesic walk and the random jump magnitudes. For the special case without amnesia, we also establish stable limit theorems and their functional counterparts when the increments belong to the domain of attraction of a non-Gaussian stable law, in the regime where heavy-tailed jumps dominate the memory contribution.
Comments33 pages
