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.

中文翻译：

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无限方差下的 Catoni 式置信序列

在本文中，我们为数据生成分布的方差不存在或无限的设置提供了置信序列的扩展。置信序列提供了在任意数据相关的停止时间有效的置信区间，自然具有广泛的应用。我们首先为有限方差情况下的 Catoni 式置信序列的宽度建立下界，以突出现有结果的松散性。接下来，我们为具有松弛有界~$p^{th}-$moment，其中~$p \in (1,2]$ 的数据分布推导紧 Catoni 式置信序列，并加强有限方差情况下的结果of~$p =2$. 得出的结果显示比使用 Dubins-Savage 不等式获得的置信度序列更好。