Tail asymptotic behavior of the supremum of a class of chi-square processes

We analyze in this paper the supremum of a class of chi-square processes over non-compact intervals, which can be seen as a multivariate counterpart of the generalized weighted Kolmogorov–Smirnov statistic. The boundedness and the exact tail asymptotic behavior of the supremum are derived. As examples, the chi-square process generated from the Brownian bridge and the fractional Brownian motion are discussed.