Two-stage chi-square test and two-dimensional distributions of a Bessel process

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.