Theory of Probability &Its Applications, Volume 65, Issue 4, Page 665-672, January 2021.
We consider the sequential $r$-stage chi-square test. For $r=2$, we study the asymptotic properties of the error probabilities as a function of the sizes of the rectangular critical domain, which via the Bonferroni inequality makes it possible to derive asymptotic properties of the error probability for an arbitrary $r$. For this purpose, we obtain some properties of the Infeld function, whose derivation is of independent interest. Based on the results obtained, the asymptotic behavior of the tails of two-dimensional distributions of a Bessel process is found.

中文翻译：

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贝塞尔过程的两阶段卡方检验和二维分布

Theory of Probability & Its Applications，第 65 卷，第 4 期，第 665-672 页，2021 年 1 月。
我们考虑顺序 $r$-stage 卡方检验。对于 $r=2$，我们研究了作为矩形临界域大小函数的误差概率的渐近特性，通过 Bonferroni 不等式可以推导出任意 $r$ 误差概率的渐近特性. 为此，我们获得了 Infeld 函数的一些性质，其推导是独立的。基于获得的结果，发现了贝塞尔过程的二维分布尾部的渐近行为。