In this paper, we provide an extension of confidence sequences for settings where the variance of the data-generating distribution does not exist or is infinite. Confidence sequences furnish confidence intervals that are valid at arbitrary data-dependent stopping times, naturally having a wide range of applications. We first establish a lower bound for the width of the Catoni-style confidence sequences for the finite variance case to highlight the looseness of the existing results. Next, we derive tight Catoni-style confidence sequences for data distributions having a relaxed bounded~$p^{th}-$moment, where~$p \in (1,2]$, and strengthen the results for the finite variance case of~$p =2$. The derived results are shown to better than confidence sequences obtained using Dubins-Savage inequality.
翻译：在本文中, 我们为不存在或无限数据生成分布差异的设置提供了信任序列的延伸。 信任序列提供了在任意数据依赖性停止时有效的信任间隔, 自然具有广泛的应用范围。 我们首先为有限差异案例的Catoni 式信任序列的宽度设定了一个较低的约束线, 以突出现有结果的松散性。 其次, 我们为数据发布设定了严格的 Catoni 式信任序列, 其约束性为: $- moment, 即~ p p $\ in ( 1, 2美元), 并强化了 ~ p = 2 美元的有限差异案例的结果。 所得结果比使用 Dubins- Savage 不平等性获得的信任序列要好。
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